The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The...
Transcript of The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The...
![Page 1: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/1.jpg)
The Topologyof Chaos
RobertGilmore
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The Topology of Chaos
Robert Gilmore
Physics DepartmentDrexel University
Philadelphia, PA [email protected]
2011 Brazilian Conference on Dynamics, Control, and ApplicationsAguas de Lindoia, Sao Paulo, Brazil
August 29 — September 2, 2011
August 16, 2011
![Page 2: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/2.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
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Exp’tal-03
Exp’tal-04
Exp’tal-05
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Abstract
Data generated by a low-dimensional dynamical systemoperating in a chaotic regime can be analyzed using topologicalmethods. The process is (almost) straightforward. On a scalartime series, the following steps are taken:
1 Unstable periodic orbits are identified;
2 An embedding is constructed; ? ?
3 The topological organization of these periodic orbits isdetermined;
4 Some orbits are used to identify an underlying branchedmanifold;
5 The branched manifold is used as a tool to predict theremaining topological invariants.
This algorithm has its own built in rejection criterion.
![Page 3: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/3.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
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Exp’tal-03
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Program-01
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Summary-01
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Abstract - Key Point
One soft spot in this analysis program is the embedding step.Different embeddings can yield different topological results.This makes the following question exciting:
When you analyze embedded data: How much ofwhat you learn is about the embedding and howmuch is about the underlying dynamics?
This question has been answered by creating a representationtheory of low dimensional strange attractors. It is now possibleto totally disentangle the mechanism generating the underlyingdynamics from topological structure induced by the embeddingstep.
![Page 4: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/4.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
Exp’tal-08
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Topology ofOrbits-01
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Program-01
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Program-09a
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Program-11
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Reps-01
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Class-01
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Bases-01
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Tori-01
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Summary-01
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Table of Contents
Outline1 Overview
2 Experimental Challenge
3 Embedding Problems
4 Topological Analysis Program
5 Representation Theory of Strange Attractors
6 Classification of Strange Attractors
7 Basis Sets of Orbits
8 Bounding Tori
9 Summary
![Page 5: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/5.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
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Exp’tal-06
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Embed-01
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Topology ofOrbits-01
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Topology ofOrbits-04
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Program-01
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Summary-01
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Experimental Schematic
Laser Experimental Arrangement
![Page 6: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/6.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
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Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
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Embed-01
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Topology ofOrbits-01
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Program-01
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Program-11
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Reps-01
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Class-01
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Bases-01
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Tori-01
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Summary-01
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Real Data
Experimental Data: LSA
Lefranc - Cargese
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The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
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Exp’tal-01
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Program-01
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Summary-01
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Ask the Masters
Periodic Orbits are the Key
Joseph Fourier Henri PoincareLinear Systems Nonlinear Systems
![Page 8: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/8.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
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Exp’tal-01
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Program-01
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Reps-01
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Class-01
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Bases-01
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Tori-01
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Summary-01
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Periodic Orbit Surrogates
Searching for Periodic Orbits
Θ(i, i+ p) = |x(i)− x(i+ p)|
![Page 9: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/9.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
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Exp’tal-05
Exp’tal-06
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Topology ofOrbits-01
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Program-01
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Program-08
Program-09a
Program-09b
Program-10a
Program-10b
Program-11
Program-12
Reps-01
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Reps-04
Reps-05
Reps-06
Reps-07
Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
Tori-06
Tori-07
Tori-08
Tori-09
Tori-10
Tori-11
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Summary-01
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Real Data
“Periodic Orbits” in Real Data
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The Topologyof Chaos
RobertGilmore
Intro.-01
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Intro.-03
Exp’tal-01
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Exp’tal-06
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Embed-01
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Topology ofOrbits-01
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Topology ofOrbits-04
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Program-01
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Reps-01
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Class-01
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Bases-01
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Tori-01
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Tori-16c
Tori-17
Tori-18
Tori-19
Tori-20
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
Mechanism
Stretching & Squeezing in a Torus
![Page 11: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/11.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
Exp’tal-08
Embed-01
Embed-02
Embed-03
Embed-04a
Embed-04b
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05a
Topology ofOrbits-05b
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09a
Program-09b
Program-10a
Program-10b
Program-11
Program-12
Reps-01
Reps-02
Reps-03
Reps-04
Reps-05
Reps-06
Reps-07
Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
Tori-06
Tori-07
Tori-08
Tori-09
Tori-10
Tori-11
Tori-12
Tori-13
Tori-14
Tori-15
Tori-16a
Tori-16b
Tori-16c
Tori-17
Tori-18
Tori-19
Tori-20
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
Time Evolution
Rotating the Poincare Section
around the axis of the torus
![Page 12: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/12.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
Exp’tal-08
Embed-01
Embed-02
Embed-03
Embed-04a
Embed-04b
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05a
Topology ofOrbits-05b
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09a
Program-09b
Program-10a
Program-10b
Program-11
Program-12
Reps-01
Reps-02
Reps-03
Reps-04
Reps-05
Reps-06
Reps-07
Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
Tori-06
Tori-07
Tori-08
Tori-09
Tori-10
Tori-11
Tori-12
Tori-13
Tori-14
Tori-15
Tori-16a
Tori-16b
Tori-16c
Tori-17
Tori-18
Tori-19
Tori-20
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
Time Evolution
Rotating the Poincare Section
around the axis of the torus
Lefranc - Cargese
![Page 13: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/13.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
Exp’tal-08
Embed-01
Embed-02
Embed-03
Embed-04a
Embed-04b
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05a
Topology ofOrbits-05b
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09a
Program-09b
Program-10a
Program-10b
Program-11
Program-12
Reps-01
Reps-02
Reps-03
Reps-04
Reps-05
Reps-06
Reps-07
Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
Tori-06
Tori-07
Tori-08
Tori-09
Tori-10
Tori-11
Tori-12
Tori-13
Tori-14
Tori-15
Tori-16a
Tori-16b
Tori-16c
Tori-17
Tori-18
Tori-19
Tori-20
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
Embeddings
Creating Something from “Nothing”
scalar time series −→ vector time series
Embedding is an Art.
Perhaps more like Black Magic.
There are many ways to conjure an embedding.
![Page 14: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/14.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
Exp’tal-08
Embed-01
Embed-02
Embed-03
Embed-04a
Embed-04b
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05a
Topology ofOrbits-05b
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09a
Program-09b
Program-10a
Program-10b
Program-11
Program-12
Reps-01
Reps-02
Reps-03
Reps-04
Reps-05
Reps-06
Reps-07
Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
Tori-06
Tori-07
Tori-08
Tori-09
Tori-10
Tori-11
Tori-12
Tori-13
Tori-14
Tori-15
Tori-16a
Tori-16b
Tori-16c
Tori-17
Tori-18
Tori-19
Tori-20
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
Embeddings
Varieties of Embeddings
x(i)→ (y1(i), y2(i), y3(i), · · · )
Delay (x(i), x(i− τ1), x(i− τ2), x(i− τ3), · · · )Delay yj(i) = x(i− [j − 1] τ) τ,NDifferential y1 = x, y2 = dx/dt, y3 = d2x/dt2, · · ·Int.−Diff . y1 =
∫ x−∞ dx, y2 = x, y3 = dx/dt, · · ·
SVD EoMHilbert−Tsf .“Circular′′
KnottedOther
![Page 15: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/15.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
Exp’tal-08
Embed-01
Embed-02
Embed-03
Embed-04a
Embed-04b
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05a
Topology ofOrbits-05b
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09a
Program-09b
Program-10a
Program-10b
Program-11
Program-12
Reps-01
Reps-02
Reps-03
Reps-04
Reps-05
Reps-06
Reps-07
Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
Tori-06
Tori-07
Tori-08
Tori-09
Tori-10
Tori-11
Tori-12
Tori-13
Tori-14
Tori-15
Tori-16a
Tori-16b
Tori-16c
Tori-17
Tori-18
Tori-19
Tori-20
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
Embeddings
Circular and Knotted Embeddings
If there is a “hole in the middle” then parameterize the scalarobservable by an angle θ: x(t)→ x(θ)
Introduce Knot coordinates (“harmonic knots”)
K(θ) = (ξ(θ), η(θ), ζ(θ)) = K(θ + 2π)
Repere Mobile: t(θ),n(θ),b(θ)
x(t)→ x(θ)→ K(θ) + y1n(θ) + y2b(θ)
![Page 16: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/16.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
Exp’tal-08
Embed-01
Embed-02
Embed-03
Embed-04a
Embed-04b
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05a
Topology ofOrbits-05b
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09a
Program-09b
Program-10a
Program-10b
Program-11
Program-12
Reps-01
Reps-02
Reps-03
Reps-04
Reps-05
Reps-06
Reps-07
Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
Tori-06
Tori-07
Tori-08
Tori-09
Tori-10
Tori-11
Tori-12
Tori-13
Tori-14
Tori-15
Tori-16a
Tori-16b
Tori-16c
Tori-17
Tori-18
Tori-19
Tori-20
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
Some Knotted Embeddings
Simple “Unknot” Trefoil Knot
![Page 17: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/17.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
Exp’tal-08
Embed-01
Embed-02
Embed-03
Embed-04a
Embed-04b
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05a
Topology ofOrbits-05b
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09a
Program-09b
Program-10a
Program-10b
Program-11
Program-12
Reps-01
Reps-02
Reps-03
Reps-04
Reps-05
Reps-06
Reps-07
Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
Tori-06
Tori-07
Tori-08
Tori-09
Tori-10
Tori-11
Tori-12
Tori-13
Tori-14
Tori-15
Tori-16a
Tori-16b
Tori-16c
Tori-17
Tori-18
Tori-19
Tori-20
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
More Knotted Embeddings
Granny Knot Square Knot
![Page 18: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/18.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
Exp’tal-08
Embed-01
Embed-02
Embed-03
Embed-04a
Embed-04b
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05a
Topology ofOrbits-05b
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09a
Program-09b
Program-10a
Program-10b
Program-11
Program-12
Reps-01
Reps-02
Reps-03
Reps-04
Reps-05
Reps-06
Reps-07
Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
Tori-06
Tori-07
Tori-08
Tori-09
Tori-10
Tori-11
Tori-12
Tori-13
Tori-14
Tori-15
Tori-16a
Tori-16b
Tori-16c
Tori-17
Tori-18
Tori-19
Tori-20
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
Chaos
Chaos
Motion that is
•Deterministic: dxdt = f(x)
•Recurrent
•Non Periodic
• Sensitive to Initial Conditions
![Page 19: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/19.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
Exp’tal-08
Embed-01
Embed-02
Embed-03
Embed-04a
Embed-04b
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05a
Topology ofOrbits-05b
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09a
Program-09b
Program-10a
Program-10b
Program-11
Program-12
Reps-01
Reps-02
Reps-03
Reps-04
Reps-05
Reps-06
Reps-07
Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
Tori-06
Tori-07
Tori-08
Tori-09
Tori-10
Tori-11
Tori-12
Tori-13
Tori-14
Tori-15
Tori-16a
Tori-16b
Tori-16c
Tori-17
Tori-18
Tori-19
Tori-20
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
Strange Attractor
Strange Attractor
The Ω limit set of the flow. There areunstable periodic orbits “in” thestrange attractor. They are
• “Abundant”
•Outline the Strange Attractor
•Are the Skeleton of the StrangeAttractor
![Page 20: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/20.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
Exp’tal-08
Embed-01
Embed-02
Embed-03
Embed-04a
Embed-04b
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05a
Topology ofOrbits-05b
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09a
Program-09b
Program-10a
Program-10b
Program-11
Program-12
Reps-01
Reps-02
Reps-03
Reps-04
Reps-05
Reps-06
Reps-07
Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
Tori-06
Tori-07
Tori-08
Tori-09
Tori-10
Tori-11
Tori-12
Tori-13
Tori-14
Tori-15
Tori-16a
Tori-16b
Tori-16c
Tori-17
Tori-18
Tori-19
Tori-20
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
Skeletons
UPOs Outline Strange Attractors
BZ reaction
![Page 21: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/21.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
Exp’tal-08
Embed-01
Embed-02
Embed-03
Embed-04a
Embed-04b
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05a
Topology ofOrbits-05b
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09a
Program-09b
Program-10a
Program-10b
Program-11
Program-12
Reps-01
Reps-02
Reps-03
Reps-04
Reps-05
Reps-06
Reps-07
Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
Tori-06
Tori-07
Tori-08
Tori-09
Tori-10
Tori-11
Tori-12
Tori-13
Tori-14
Tori-15
Tori-16a
Tori-16b
Tori-16c
Tori-17
Tori-18
Tori-19
Tori-20
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
Skeletons
UPOs Outline Strange attractors
Laser w. Modulated Losses
![Page 22: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/22.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
Exp’tal-08
Embed-01
Embed-02
Embed-03
Embed-04a
Embed-04b
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05a
Topology ofOrbits-05b
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09a
Program-09b
Program-10a
Program-10b
Program-11
Program-12
Reps-01
Reps-02
Reps-03
Reps-04
Reps-05
Reps-06
Reps-07
Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
Tori-06
Tori-07
Tori-08
Tori-09
Tori-10
Tori-11
Tori-12
Tori-13
Tori-14
Tori-15
Tori-16a
Tori-16b
Tori-16c
Tori-17
Tori-18
Tori-19
Tori-20
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
Organization
How Are Orbits Organized
Ask the Master:
Carl Friedrich Gauss
![Page 23: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/23.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
Exp’tal-08
Embed-01
Embed-02
Embed-03
Embed-04a
Embed-04b
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05a
Topology ofOrbits-05b
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09a
Program-09b
Program-10a
Program-10b
Program-11
Program-12
Reps-01
Reps-02
Reps-03
Reps-04
Reps-05
Reps-06
Reps-07
Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
Tori-06
Tori-07
Tori-08
Tori-09
Tori-10
Tori-11
Tori-12
Tori-13
Tori-14
Tori-15
Tori-16a
Tori-16b
Tori-16c
Tori-17
Tori-18
Tori-19
Tori-20
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
Dynamics and Topology
Organization of UPOs in R3:
Gauss Linking Number
LN(A,B) =1
4π
∮ ∮(rA − rB)·drA×drB
|rA − rB|3
# Interpretations of LN ' # Mathematicians in World
![Page 24: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/24.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
Exp’tal-08
Embed-01
Embed-02
Embed-03
Embed-04a
Embed-04b
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05a
Topology ofOrbits-05b
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09a
Program-09b
Program-10a
Program-10b
Program-11
Program-12
Reps-01
Reps-02
Reps-03
Reps-04
Reps-05
Reps-06
Reps-07
Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
Tori-06
Tori-07
Tori-08
Tori-09
Tori-10
Tori-11
Tori-12
Tori-13
Tori-14
Tori-15
Tori-16a
Tori-16b
Tori-16c
Tori-17
Tori-18
Tori-19
Tori-20
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
Linking Numbers
Linking Number of Two UPOs
Lefranc - Cargese
![Page 25: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/25.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
Exp’tal-08
Embed-01
Embed-02
Embed-03
Embed-04a
Embed-04b
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05a
Topology ofOrbits-05b
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09a
Program-09b
Program-10a
Program-10b
Program-11
Program-12
Reps-01
Reps-02
Reps-03
Reps-04
Reps-05
Reps-06
Reps-07
Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
Tori-06
Tori-07
Tori-08
Tori-09
Tori-10
Tori-11
Tori-12
Tori-13
Tori-14
Tori-15
Tori-16a
Tori-16b
Tori-16c
Tori-17
Tori-18
Tori-19
Tori-20
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
Evolution in Phase Space
One Stretch-&-Squeeze Mechanism
“Rossler Mechanism”
![Page 26: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/26.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
Exp’tal-08
Embed-01
Embed-02
Embed-03
Embed-04a
Embed-04b
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05a
Topology ofOrbits-05b
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09a
Program-09b
Program-10a
Program-10b
Program-11
Program-12
Reps-01
Reps-02
Reps-03
Reps-04
Reps-05
Reps-06
Reps-07
Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
Tori-06
Tori-07
Tori-08
Tori-09
Tori-10
Tori-11
Tori-12
Tori-13
Tori-14
Tori-15
Tori-16a
Tori-16b
Tori-16c
Tori-17
Tori-18
Tori-19
Tori-20
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
Motion of Blobs in Phase Space
Another Stretch-&-Squeeze Mechanism
“Lorenz Mechanism”
![Page 27: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/27.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
Exp’tal-08
Embed-01
Embed-02
Embed-03
Embed-04a
Embed-04b
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05a
Topology ofOrbits-05b
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09a
Program-09b
Program-10a
Program-10b
Program-11
Program-12
Reps-01
Reps-02
Reps-03
Reps-04
Reps-05
Reps-06
Reps-07
Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
Tori-06
Tori-07
Tori-08
Tori-09
Tori-10
Tori-11
Tori-12
Tori-13
Tori-14
Tori-15
Tori-16a
Tori-16b
Tori-16c
Tori-17
Tori-18
Tori-19
Tori-20
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
Motion of Blobs in Phase Space
Stretching — Folding
![Page 28: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/28.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
Exp’tal-08
Embed-01
Embed-02
Embed-03
Embed-04a
Embed-04b
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05a
Topology ofOrbits-05b
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09a
Program-09b
Program-10a
Program-10b
Program-11
Program-12
Reps-01
Reps-02
Reps-03
Reps-04
Reps-05
Reps-06
Reps-07
Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
Tori-06
Tori-07
Tori-08
Tori-09
Tori-10
Tori-11
Tori-12
Tori-13
Tori-14
Tori-15
Tori-16a
Tori-16b
Tori-16c
Tori-17
Tori-18
Tori-19
Tori-20
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
Collapse Along the Stable Manifold
Birman - Williams Projection
Identify x and y if
limt→∞|x(t)− y(t)| → 0
![Page 29: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/29.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
Exp’tal-08
Embed-01
Embed-02
Embed-03
Embed-04a
Embed-04b
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05a
Topology ofOrbits-05b
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09a
Program-09b
Program-10a
Program-10b
Program-11
Program-12
Reps-01
Reps-02
Reps-03
Reps-04
Reps-05
Reps-06
Reps-07
Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
Tori-06
Tori-07
Tori-08
Tori-09
Tori-10
Tori-11
Tori-12
Tori-13
Tori-14
Tori-15
Tori-16a
Tori-16b
Tori-16c
Tori-17
Tori-18
Tori-19
Tori-20
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
Fundamental Theorem
Birman - Williams Theorem
If:
Then:
Certain Assumptions
Specific Conclusions
![Page 30: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/30.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
Exp’tal-08
Embed-01
Embed-02
Embed-03
Embed-04a
Embed-04b
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05a
Topology ofOrbits-05b
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09a
Program-09b
Program-10a
Program-10b
Program-11
Program-12
Reps-01
Reps-02
Reps-03
Reps-04
Reps-05
Reps-06
Reps-07
Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
Tori-06
Tori-07
Tori-08
Tori-09
Tori-10
Tori-11
Tori-12
Tori-13
Tori-14
Tori-15
Tori-16a
Tori-16b
Tori-16c
Tori-17
Tori-18
Tori-19
Tori-20
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
Fundamental Theorem
Birman - Williams Theorem
If:
Then:
Certain Assumptions
Specific Conclusions
![Page 31: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/31.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
Exp’tal-08
Embed-01
Embed-02
Embed-03
Embed-04a
Embed-04b
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05a
Topology ofOrbits-05b
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09a
Program-09b
Program-10a
Program-10b
Program-11
Program-12
Reps-01
Reps-02
Reps-03
Reps-04
Reps-05
Reps-06
Reps-07
Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
Tori-06
Tori-07
Tori-08
Tori-09
Tori-10
Tori-11
Tori-12
Tori-13
Tori-14
Tori-15
Tori-16a
Tori-16b
Tori-16c
Tori-17
Tori-18
Tori-19
Tori-20
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
Fundamental Theorem
Birman - Williams Theorem
If:
Then:
Certain Assumptions
Specific Conclusions
![Page 32: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/32.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
Exp’tal-08
Embed-01
Embed-02
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Embed-04a
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Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05a
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Topology ofOrbits-06
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Topology ofOrbits-08
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Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09a
Program-09b
Program-10a
Program-10b
Program-11
Program-12
Reps-01
Reps-02
Reps-03
Reps-04
Reps-05
Reps-06
Reps-07
Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
Tori-06
Tori-07
Tori-08
Tori-09
Tori-10
Tori-11
Tori-12
Tori-13
Tori-14
Tori-15
Tori-16a
Tori-16b
Tori-16c
Tori-17
Tori-18
Tori-19
Tori-20
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
Birman-Williams Theorem
Assumptions, B-W Theorem
A flow Φt(x)
• on Rn is dissipative, n = 3, so thatλ1 > 0, λ2 = 0, λ3 < 0.
•Generates a hyperbolic strangeattractor SA
IMPORTANT: The underlined assumptions can be relaxed.
![Page 33: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/33.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
Exp’tal-08
Embed-01
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Embed-04a
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Topology ofOrbits-01
Topology ofOrbits-02
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Topology ofOrbits-03b
Topology ofOrbits-04
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05a
Topology ofOrbits-05b
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09a
Program-09b
Program-10a
Program-10b
Program-11
Program-12
Reps-01
Reps-02
Reps-03
Reps-04
Reps-05
Reps-06
Reps-07
Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
Tori-06
Tori-07
Tori-08
Tori-09
Tori-10
Tori-11
Tori-12
Tori-13
Tori-14
Tori-15
Tori-16a
Tori-16b
Tori-16c
Tori-17
Tori-18
Tori-19
Tori-20
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
Birman-Williams Theorem
Conclusions, B-W Theorem
• The projection maps the strange attractor SA onto a2-dimensional branched manifold BM and the flow Φt(x)on SA to a semiflow Φ(x)t on BM.
• UPOs of Φt(x) on SA are in 1-1 correspondence withUPOs of Φ(x)t on BM. Moreover, every link of UPOs of(Φt(x),SA) is isotopic to the correspond link of UPOs of(Φ(x)t,BM).
Remark: “One of the few theorems useful to experimentalists.”
![Page 34: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/34.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
Exp’tal-08
Embed-01
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Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05a
Topology ofOrbits-05b
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09a
Program-09b
Program-10a
Program-10b
Program-11
Program-12
Reps-01
Reps-02
Reps-03
Reps-04
Reps-05
Reps-06
Reps-07
Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
Tori-06
Tori-07
Tori-08
Tori-09
Tori-10
Tori-11
Tori-12
Tori-13
Tori-14
Tori-15
Tori-16a
Tori-16b
Tori-16c
Tori-17
Tori-18
Tori-19
Tori-20
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
A Very Common Mechanism
Rossler:
Attractor Branched Manifold
![Page 35: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/35.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
Exp’tal-08
Embed-01
Embed-02
Embed-03
Embed-04a
Embed-04b
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05a
Topology ofOrbits-05b
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09a
Program-09b
Program-10a
Program-10b
Program-11
Program-12
Reps-01
Reps-02
Reps-03
Reps-04
Reps-05
Reps-06
Reps-07
Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
Tori-06
Tori-07
Tori-08
Tori-09
Tori-10
Tori-11
Tori-12
Tori-13
Tori-14
Tori-15
Tori-16a
Tori-16b
Tori-16c
Tori-17
Tori-18
Tori-19
Tori-20
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
A Mechanism with Symmetry
Lorenz:
Attractor Branched Manifold
![Page 36: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/36.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
Exp’tal-08
Embed-01
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Topology ofOrbits-01
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Topology ofOrbits-03b
Topology ofOrbits-04
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05a
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Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09a
Program-09b
Program-10a
Program-10b
Program-11
Program-12
Reps-01
Reps-02
Reps-03
Reps-04
Reps-05
Reps-06
Reps-07
Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
Tori-06
Tori-07
Tori-08
Tori-09
Tori-10
Tori-11
Tori-12
Tori-13
Tori-14
Tori-15
Tori-16a
Tori-16b
Tori-16c
Tori-17
Tori-18
Tori-19
Tori-20
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
Examples of Branched Manifolds
Inequivalent Branched Manifolds
![Page 37: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/37.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
Exp’tal-08
Embed-01
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Topology ofOrbits-01
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Topology ofOrbits-04
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Topology ofOrbits-05b
Topology ofOrbits-06
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Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09a
Program-09b
Program-10a
Program-10b
Program-11
Program-12
Reps-01
Reps-02
Reps-03
Reps-04
Reps-05
Reps-06
Reps-07
Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
Tori-06
Tori-07
Tori-08
Tori-09
Tori-10
Tori-11
Tori-12
Tori-13
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Tori-16a
Tori-16b
Tori-16c
Tori-17
Tori-18
Tori-19
Tori-20
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
Aufbau Princip for Branched Manifolds
Any branched manifold can be built up from stretchingand squeezing units
subject to the conditions:
•Outputs to Inputs
•No Free Ends
![Page 38: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/38.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
Exp’tal-08
Embed-01
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Topology ofOrbits-01
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Topology ofOrbits-04
Topology ofOrbits-04a
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Topology ofOrbits-06
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Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09a
Program-09b
Program-10a
Program-10b
Program-11
Program-12
Reps-01
Reps-02
Reps-03
Reps-04
Reps-05
Reps-06
Reps-07
Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
Tori-06
Tori-07
Tori-08
Tori-09
Tori-10
Tori-11
Tori-12
Tori-13
Tori-14
Tori-15
Tori-16a
Tori-16b
Tori-16c
Tori-17
Tori-18
Tori-19
Tori-20
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
Dynamics and Topology
Rossler System
![Page 39: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/39.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
Exp’tal-08
Embed-01
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Topology ofOrbits-01
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Topology ofOrbits-03a
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Topology ofOrbits-04
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Topology ofOrbits-06
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Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09a
Program-09b
Program-10a
Program-10b
Program-11
Program-12
Reps-01
Reps-02
Reps-03
Reps-04
Reps-05
Reps-06
Reps-07
Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
Tori-06
Tori-07
Tori-08
Tori-09
Tori-10
Tori-11
Tori-12
Tori-13
Tori-14
Tori-15
Tori-16a
Tori-16b
Tori-16c
Tori-17
Tori-18
Tori-19
Tori-20
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
Dynamics and Topology
Lorenz System
![Page 40: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/40.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
Exp’tal-08
Embed-01
Embed-02
Embed-03
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Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05a
Topology ofOrbits-05b
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09a
Program-09b
Program-10a
Program-10b
Program-11
Program-12
Reps-01
Reps-02
Reps-03
Reps-04
Reps-05
Reps-06
Reps-07
Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
Tori-06
Tori-07
Tori-08
Tori-09
Tori-10
Tori-11
Tori-12
Tori-13
Tori-14
Tori-15
Tori-16a
Tori-16b
Tori-16c
Tori-17
Tori-18
Tori-19
Tori-20
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
Dynamics and Topology
Poincare Smiles at Us in R3
•Determine organization of UPOs ⇒
•Determine branched manifold ⇒
•Determine equivalence class of SA
![Page 41: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/41.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
Exp’tal-08
Embed-01
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Embed-04a
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Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05a
Topology ofOrbits-05b
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09a
Program-09b
Program-10a
Program-10b
Program-11
Program-12
Reps-01
Reps-02
Reps-03
Reps-04
Reps-05
Reps-06
Reps-07
Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
Tori-06
Tori-07
Tori-08
Tori-09
Tori-10
Tori-11
Tori-12
Tori-13
Tori-14
Tori-15
Tori-16a
Tori-16b
Tori-16c
Tori-17
Tori-18
Tori-19
Tori-20
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
Topological Analysis Program
Topological Analysis Program
Locate Periodic Orbits
Create an Embedding
Determine Topological Invariants (LN)
Identify a Branched Manifold
Verify the Branched Manifold
—————————————————————————-
Model the Dynamics
Validate the Model
![Page 42: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/42.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
Exp’tal-08
Embed-01
Embed-02
Embed-03
Embed-04a
Embed-04b
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05a
Topology ofOrbits-05b
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09a
Program-09b
Program-10a
Program-10b
Program-11
Program-12
Reps-01
Reps-02
Reps-03
Reps-04
Reps-05
Reps-06
Reps-07
Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
Tori-06
Tori-07
Tori-08
Tori-09
Tori-10
Tori-11
Tori-12
Tori-13
Tori-14
Tori-15
Tori-16a
Tori-16b
Tori-16c
Tori-17
Tori-18
Tori-19
Tori-20
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
Locate UPOs
Method of Close Returns
![Page 43: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/43.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
Exp’tal-08
Embed-01
Embed-02
Embed-03
Embed-04a
Embed-04b
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05a
Topology ofOrbits-05b
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09a
Program-09b
Program-10a
Program-10b
Program-11
Program-12
Reps-01
Reps-02
Reps-03
Reps-04
Reps-05
Reps-06
Reps-07
Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
Tori-06
Tori-07
Tori-08
Tori-09
Tori-10
Tori-11
Tori-12
Tori-13
Tori-14
Tori-15
Tori-16a
Tori-16b
Tori-16c
Tori-17
Tori-18
Tori-19
Tori-20
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
Embeddings
Embeddings
Many Methods: Time Delay, Differential, Hilbert Transforms,SVD, Mixtures, ...
Tests for Embeddings: Geometric, Dynamic, Topological†
None Good
We Demand a 3 Dimensional Embedding
![Page 44: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/44.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
Exp’tal-08
Embed-01
Embed-02
Embed-03
Embed-04a
Embed-04b
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05a
Topology ofOrbits-05b
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09a
Program-09b
Program-10a
Program-10b
Program-11
Program-12
Reps-01
Reps-02
Reps-03
Reps-04
Reps-05
Reps-06
Reps-07
Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
Tori-06
Tori-07
Tori-08
Tori-09
Tori-10
Tori-11
Tori-12
Tori-13
Tori-14
Tori-15
Tori-16a
Tori-16b
Tori-16c
Tori-17
Tori-18
Tori-19
Tori-20
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
Determine Topological Invariants
Compute Table of Expt’l LN
![Page 45: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/45.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
Exp’tal-08
Embed-01
Embed-02
Embed-03
Embed-04a
Embed-04b
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05a
Topology ofOrbits-05b
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09a
Program-09b
Program-10a
Program-10b
Program-11
Program-12
Reps-01
Reps-02
Reps-03
Reps-04
Reps-05
Reps-06
Reps-07
Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
Tori-06
Tori-07
Tori-08
Tori-09
Tori-10
Tori-11
Tori-12
Tori-13
Tori-14
Tori-15
Tori-16a
Tori-16b
Tori-16c
Tori-17
Tori-18
Tori-19
Tori-20
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
Determine Topological Invariants
Compare w. LN From Various BM
![Page 46: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/46.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
Exp’tal-08
Embed-01
Embed-02
Embed-03
Embed-04a
Embed-04b
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05a
Topology ofOrbits-05b
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09a
Program-09b
Program-10a
Program-10b
Program-11
Program-12
Reps-01
Reps-02
Reps-03
Reps-04
Reps-05
Reps-06
Reps-07
Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
Tori-06
Tori-07
Tori-08
Tori-09
Tori-10
Tori-11
Tori-12
Tori-13
Tori-14
Tori-15
Tori-16a
Tori-16b
Tori-16c
Tori-17
Tori-18
Tori-19
Tori-20
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
Determine Topological Invariants
Guess Branched Manifold
Lefranc - Cargese
![Page 47: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/47.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
Exp’tal-08
Embed-01
Embed-02
Embed-03
Embed-04a
Embed-04b
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05a
Topology ofOrbits-05b
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09a
Program-09b
Program-10a
Program-10b
Program-11
Program-12
Reps-01
Reps-02
Reps-03
Reps-04
Reps-05
Reps-06
Reps-07
Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
Tori-06
Tori-07
Tori-08
Tori-09
Tori-10
Tori-11
Tori-12
Tori-13
Tori-14
Tori-15
Tori-16a
Tori-16b
Tori-16c
Tori-17
Tori-18
Tori-19
Tori-20
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
Determine Topological Invariants
Identification & ‘Confirmation’
• BM Identified by LN of small number of orbits
• Table of LN GROSSLY overdetermined
• Predict LN of additional orbits
• Rejection criterion (Reject or Fail to Reject)
![Page 48: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/48.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
Exp’tal-08
Embed-01
Embed-02
Embed-03
Embed-04a
Embed-04b
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05a
Topology ofOrbits-05b
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09a
Program-09b
Program-10a
Program-10b
Program-11
Program-12
Reps-01
Reps-02
Reps-03
Reps-04
Reps-05
Reps-06
Reps-07
Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
Tori-06
Tori-07
Tori-08
Tori-09
Tori-10
Tori-11
Tori-12
Tori-13
Tori-14
Tori-15
Tori-16a
Tori-16b
Tori-16c
Tori-17
Tori-18
Tori-19
Tori-20
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
Determine Topological Invariants
What Do We Learn?• BM Depends on Embedding• Some things depend on embedding, some don’t• Depends on Embedding: Global Torsion, Parity, ..• Independent of Embedding: Mechanism
![Page 49: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/49.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
Exp’tal-08
Embed-01
Embed-02
Embed-03
Embed-04a
Embed-04b
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05a
Topology ofOrbits-05b
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09a
Program-09b
Program-10a
Program-10b
Program-11
Program-12
Reps-01
Reps-02
Reps-03
Reps-04
Reps-05
Reps-06
Reps-07
Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
Tori-06
Tori-07
Tori-08
Tori-09
Tori-10
Tori-11
Tori-12
Tori-13
Tori-14
Tori-15
Tori-16a
Tori-16b
Tori-16c
Tori-17
Tori-18
Tori-19
Tori-20
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
Perestroikas of Strange Attractors
Evolution Under Parameter Change
Lefranc - Cargese
![Page 50: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/50.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
Exp’tal-08
Embed-01
Embed-02
Embed-03
Embed-04a
Embed-04b
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05a
Topology ofOrbits-05b
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09a
Program-09b
Program-10a
Program-10b
Program-11
Program-12
Reps-01
Reps-02
Reps-03
Reps-04
Reps-05
Reps-06
Reps-07
Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
Tori-06
Tori-07
Tori-08
Tori-09
Tori-10
Tori-11
Tori-12
Tori-13
Tori-14
Tori-15
Tori-16a
Tori-16b
Tori-16c
Tori-17
Tori-18
Tori-19
Tori-20
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
Perestroikas of Strange Attractors
Evolution Under Parameter Change
Lefranc - Cargese
![Page 51: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/51.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
Exp’tal-08
Embed-01
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Topology ofOrbits-01
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Topology ofOrbits-04
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Topology ofOrbits-06
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Topology ofOrbits-11
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Topology ofOrbits-13
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Topology ofOrbits-17
Program-01
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Program-04
Program-05
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Program-08
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Program-10a
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Program-11
Program-12
Reps-01
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Reps-06
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Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
Tori-06
Tori-07
Tori-08
Tori-09
Tori-10
Tori-11
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Tori-13
Tori-14
Tori-15
Tori-16a
Tori-16b
Tori-16c
Tori-17
Tori-18
Tori-19
Tori-20
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
Perestroikas of Strange Attractors
Evolution Under Parameter Change
Used and Martin — Zaragosa
![Page 52: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/52.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
Exp’tal-08
Embed-01
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Topology ofOrbits-01
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Topology ofOrbits-03b
Topology ofOrbits-04
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05a
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Topology ofOrbits-06
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Topology ofOrbits-11
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Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09a
Program-09b
Program-10a
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Program-11
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Reps-01
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Reps-07
Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
Tori-06
Tori-07
Tori-08
Tori-09
Tori-10
Tori-11
Tori-12
Tori-13
Tori-14
Tori-15
Tori-16a
Tori-16b
Tori-16c
Tori-17
Tori-18
Tori-19
Tori-20
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
Perestroikas of Strange Attractors
Evolution Under Parameter Change
Used and Martin — Zaragosa
![Page 53: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/53.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
Exp’tal-08
Embed-01
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Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05a
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Topology ofOrbits-06
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Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09a
Program-09b
Program-10a
Program-10b
Program-11
Program-12
Reps-01
Reps-02
Reps-03
Reps-04
Reps-05
Reps-06
Reps-07
Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
Tori-06
Tori-07
Tori-08
Tori-09
Tori-10
Tori-11
Tori-12
Tori-13
Tori-14
Tori-15
Tori-16a
Tori-16b
Tori-16c
Tori-17
Tori-18
Tori-19
Tori-20
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
An Unexpected Benefit
Analysis of Nonstationary Data
Lefranc - Cargese
![Page 54: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/54.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
Exp’tal-08
Embed-01
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Topology ofOrbits-01
Topology ofOrbits-02
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Topology ofOrbits-04
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Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09a
Program-09b
Program-10a
Program-10b
Program-11
Program-12
Reps-01
Reps-02
Reps-03
Reps-04
Reps-05
Reps-06
Reps-07
Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
Tori-06
Tori-07
Tori-08
Tori-09
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Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
Our Hope → Now a Result
Compare with
Original Objectives
Construct a simple, algorithmic procedure for:
Classifying strange attractors
Extracting classification information
from experimental signals.
![Page 55: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/55.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
Exp’tal-08
Embed-01
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Topology ofOrbits-01
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Topology ofOrbits-04
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Topology ofOrbits-06
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Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09a
Program-09b
Program-10a
Program-10b
Program-11
Program-12
Reps-01
Reps-02
Reps-03
Reps-04
Reps-05
Reps-06
Reps-07
Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
Tori-06
Tori-07
Tori-08
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Summary-01
Summary-02
Summary-03
Summary-04
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Summary-08
Summary-09
Summary-10
Representations
Representations
An embedding creates a diffeomorphism between an(‘invisible’) dynamics in someone’s laboratory and a (‘visible’)attractor in somebody’s computer.
Embeddings provide a representation of an attractor.
Equivalence is by Isotopy.
Irreducible is by Dimension
![Page 56: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/56.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
Exp’tal-08
Embed-01
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Topology ofOrbits-12
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Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09a
Program-09b
Program-10a
Program-10b
Program-11
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Reps-01
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Reps-05
Reps-06
Reps-07
Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
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Tori-20
Summary-01
Summary-02
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Summary-10
Representations
Representations
We know about representations from studies of groups andalgebras.
We use this knowledge as a guiding light.
![Page 57: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/57.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
Exp’tal-08
Embed-01
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Topology ofOrbits-01
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Topology ofOrbits-06
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Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09a
Program-09b
Program-10a
Program-10b
Program-11
Program-12
Reps-01
Reps-02
Reps-03
Reps-04
Reps-05
Reps-06
Reps-07
Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
Tori-06
Tori-07
Tori-08
Tori-09
Tori-10
Tori-11
Tori-12
Tori-13
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Tori-16a
Tori-16b
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Tori-20
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
Representation Labels
Inequivalent Irreducible Representations
Irreducible Representations of 3-dimensional Genus-oneattractors are distinguished by three topological labels:
ParityGlobal TorsionKnot Type
PNKT
ΓP,N,KT (SA)
Mechanism (stretch & fold, stretch & roll) is an invariant ofembedding. It is independent of the representation labels.
![Page 58: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/58.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
Exp’tal-08
Embed-01
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Topology ofOrbits-01
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Topology ofOrbits-03b
Topology ofOrbits-04
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05a
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Topology ofOrbits-06
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Topology ofOrbits-08
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Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09a
Program-09b
Program-10a
Program-10b
Program-11
Program-12
Reps-01
Reps-02
Reps-03
Reps-04
Reps-05
Reps-06
Reps-07
Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
Tori-06
Tori-07
Tori-08
Tori-09
Tori-10
Tori-11
Tori-12
Tori-13
Tori-14
Tori-15
Tori-16a
Tori-16b
Tori-16c
Tori-17
Tori-18
Tori-19
Tori-20
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
Representation Labels
Global Torsion & Parity
![Page 59: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/59.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
Exp’tal-08
Embed-01
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Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05a
Topology ofOrbits-05b
Topology ofOrbits-06
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Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09a
Program-09b
Program-10a
Program-10b
Program-11
Program-12
Reps-01
Reps-02
Reps-03
Reps-04
Reps-05
Reps-06
Reps-07
Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
Tori-06
Tori-07
Tori-08
Tori-09
Tori-10
Tori-11
Tori-12
Tori-13
Tori-14
Tori-15
Tori-16a
Tori-16b
Tori-16c
Tori-17
Tori-18
Tori-19
Tori-20
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
Inequivalence in R3
Inequivalence in R3
![Page 60: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/60.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
Exp’tal-08
Embed-01
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Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04
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Topology ofOrbits-05a
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Topology ofOrbits-06
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Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09a
Program-09b
Program-10a
Program-10b
Program-11
Program-12
Reps-01
Reps-02
Reps-03
Reps-04
Reps-05
Reps-06
Reps-07
Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
Tori-06
Tori-07
Tori-08
Tori-09
Tori-10
Tori-11
Tori-12
Tori-13
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Tori-16a
Tori-16b
Tori-16c
Tori-17
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Tori-19
Tori-20
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
Creating Isotopies
Equivalent Reducible Representations
Topological indices (P,N,KT) are obstructions to isotopy forembeddings of minimum dimension (irreduciblerepresentations).
Are these obstructions removed by injections into higherdimensions (reducible representations)?
Systematically?
![Page 61: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/61.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
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Exp’tal-06
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Program-01
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Program-10a
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Program-11
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Reps-01
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Reps-06
Reps-07
Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
Tori-06
Tori-07
Tori-08
Tori-09
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Tori-20
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
Equivalences
Crossing Exchange in R4
Parity reversal is also possible in R4 by isotopy.
![Page 62: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/62.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
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Program-01
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Program-10a
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Program-11
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Reps-01
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Reps-05
Reps-06
Reps-07
Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
Tori-06
Tori-07
Tori-08
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Summary-01
Summary-02
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Summary-05
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Summary-08
Summary-09
Summary-10
Isotopies
2 Twists = 1 Writhe = Identity
Z −→ Z2
Global Torsion −→ Binary Op
![Page 63: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/63.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
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Program-01
Program-02
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Program-08
Program-09a
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Program-10a
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Program-11
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Reps-01
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Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
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Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
Tori-06
Tori-07
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Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
Creating Isotopies
Equivalences by Injection
Obstructions to Isotopy
R3
Global TorsionParityKnot Type
→ R4
Global Torsion
→ R5
There is one Universal reducible representation in RN , N ≥ 5.In RN the only topological invariant is mechanism.
![Page 64: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/64.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
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Topology ofOrbits-17
Program-01
Program-02
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Program-04
Program-05
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Program-08
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Program-10a
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Program-11
Program-12
Reps-01
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Class-01
Class-02
Class-03
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Bases-01
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Bases-04
Tori-01
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Tori-04
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Tori-07
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Tori-16a
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Tori-19
Tori-20
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
Classifications
Can We Classify
Strange Attractors?
Chemists have their classification.
Nuclear Physicists have their classification.
Particle Physicists have their classification.
Astronomers have their classification.
![Page 65: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/65.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
Exp’tal-08
Embed-01
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Topology ofOrbits-01
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Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09a
Program-09b
Program-10a
Program-10b
Program-11
Program-12
Reps-01
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Reps-05
Reps-06
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Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
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Summary-01
Summary-02
Summary-03
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Summary-10
Mendeleev’s Table of the Chemical Elements
![Page 66: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/66.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
Exp’tal-08
Embed-01
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Topology ofOrbits-01
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Topology ofOrbits-13
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Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09a
Program-09b
Program-10a
Program-10b
Program-11
Program-12
Reps-01
Reps-02
Reps-03
Reps-04
Reps-05
Reps-06
Reps-07
Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
Tori-06
Tori-07
Tori-08
Tori-09
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Tori-13
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Tori-16a
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Tori-17
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Tori-19
Tori-20
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
Atomic Nuclei
Table of Atomic Nuclei
![Page 67: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/67.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
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Exp’tal-08
Embed-01
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Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09a
Program-09b
Program-10a
Program-10b
Program-11
Program-12
Reps-01
Reps-02
Reps-03
Reps-04
Reps-05
Reps-06
Reps-07
Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
Tori-06
Tori-07
Tori-08
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Tori-16a
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Tori-20
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
Stars
Hertzsprung-Russell Diagram
![Page 68: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/68.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
Exp’tal-08
Embed-01
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Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09a
Program-09b
Program-10a
Program-10b
Program-11
Program-12
Reps-01
Reps-02
Reps-03
Reps-04
Reps-05
Reps-06
Reps-07
Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
Tori-06
Tori-07
Tori-08
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Tori-16b
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Tori-19
Tori-20
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
Challenge
How Does it Work Out?
Chemical Elements 1 Integer: NP
Atomic Nuclei 2 Integers: NP , NN
Stars 1 Continuous variable M + exceptions
Strange Attractors ? ? ? ?
![Page 69: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/69.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
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Embed-01
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Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09a
Program-09b
Program-10a
Program-10b
Program-11
Program-12
Reps-01
Reps-02
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Reps-04
Reps-05
Reps-06
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Reps-08
Reps-09
Class-01
Class-02
Class-03
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Class-05
Bases-01
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Tori-01
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Summary-01
Summary-02
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Summary-04
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Summary-10
An Experimental Study
What We Did
We analyzed data from a Laser with Saturable Absorber (LSA).
3 Different absorbers were used.
For each absorber data were taken under 6 - 10 operatingconditions.
There was a total of 25 different data sets.
We wanted to “prove experimentally” that changing theabsorber/operating condition served merely to push to flowaround on the same branched manifold.
![Page 70: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/70.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
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Embed-01
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Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09a
Program-09b
Program-10a
Program-10b
Program-11
Program-12
Reps-01
Reps-02
Reps-03
Reps-04
Reps-05
Reps-06
Reps-07
Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
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Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
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Summary-01
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An Experimental Observation
What We Found
When certain orbits (UPOs) were present they were invariablyaccompanied by a specific set of other orbits.
This led us to propose that certain orbits ‘force’ other orbits.
Forcing is topological.
A discrete set of “basis orbits” serves to identify the completecollection of UPOs present in an attractor.
![Page 71: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/71.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
Exp’tal-08
Embed-01
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Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05a
Topology ofOrbits-05b
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
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Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09a
Program-09b
Program-10a
Program-10b
Program-11
Program-12
Reps-01
Reps-02
Reps-03
Reps-04
Reps-05
Reps-06
Reps-07
Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
Tori-06
Tori-07
Tori-08
Tori-09
Tori-10
Tori-11
Tori-12
Tori-13
Tori-14
Tori-15
Tori-16a
Tori-16b
Tori-16c
Tori-17
Tori-18
Tori-19
Tori-20
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
Basis Set of Orbits
Forcing Diagram - Horseshoe
![Page 72: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/72.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
Exp’tal-08
Embed-01
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Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05a
Topology ofOrbits-05b
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09a
Program-09b
Program-10a
Program-10b
Program-11
Program-12
Reps-01
Reps-02
Reps-03
Reps-04
Reps-05
Reps-06
Reps-07
Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
Tori-06
Tori-07
Tori-08
Tori-09
Tori-10
Tori-11
Tori-12
Tori-13
Tori-14
Tori-15
Tori-16a
Tori-16b
Tori-16c
Tori-17
Tori-18
Tori-19
Tori-20
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
An Ongoing Problem
Status of Problem
Horseshoe organization - active
More folding - barely begun
Circle forcing - even less known
Higher genus - new ideas required
Higher dimension - ???
![Page 73: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/73.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
Exp’tal-08
Embed-01
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Topology ofOrbits-01
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Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
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Program-05
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Program-08
Program-09a
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Program-10a
Program-10b
Program-11
Program-12
Reps-01
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Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
Tori-06
Tori-07
Tori-08
Tori-09
Tori-10
Tori-11
Tori-12
Tori-13
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Tori-16a
Tori-16b
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Tori-17
Tori-18
Tori-19
Tori-20
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
Perestroikas of Branched Manifolds
Constraints
Branched manifolds largely constrain the ‘perestroikas” thatforcing diagrams can undergo.
Is there some mechanism/structure that constrains the types ofperestroikas that branched manifolds can undergo?
![Page 74: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/74.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
Exp’tal-08
Embed-01
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Topology ofOrbits-01
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Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
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Topology ofOrbits-17
Program-01
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Program-10a
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Program-11
Program-12
Reps-01
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Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
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Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
Tori-06
Tori-07
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Tori-16a
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Tori-20
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
Perestroikas of Branched Manifolds
Constraints on Branched Manifolds
“Inflate” a strange attractor
Union of ε ball around each point
Boundary is surface of bounded 3D manifold
Torus that bounds strange attractor
![Page 75: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/75.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
Exp’tal-08
Embed-01
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Topology ofOrbits-01
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Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
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Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09a
Program-09b
Program-10a
Program-10b
Program-11
Program-12
Reps-01
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Reps-06
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Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
Tori-06
Tori-07
Tori-08
Tori-09
Tori-10
Tori-11
Tori-12
Tori-13
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Tori-15
Tori-16a
Tori-16b
Tori-16c
Tori-17
Tori-18
Tori-19
Tori-20
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
Torus and Genus
Torus, Longitudes, Meridians
Tori are identified by genus g and dressed with a surface flowinduced from that creating the strange attractor.
![Page 76: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/76.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
Exp’tal-08
Embed-01
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Topology ofOrbits-01
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Topology ofOrbits-04
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05a
Topology ofOrbits-05b
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09a
Program-09b
Program-10a
Program-10b
Program-11
Program-12
Reps-01
Reps-02
Reps-03
Reps-04
Reps-05
Reps-06
Reps-07
Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
Tori-06
Tori-07
Tori-08
Tori-09
Tori-10
Tori-11
Tori-12
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Tori-16a
Tori-16b
Tori-16c
Tori-17
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Tori-19
Tori-20
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
Flows on Surfaces
Surface Singularities
Flow field: three eigenvalues: +, 0, –
Vector field “perpendicular” to surface
Eigenvalues on surface at fixed point: +, –
All singularities are regular saddles∑s.p.(−1)index = χ(S) = 2− 2g
# fixed points on surface = index = 2g - 2
Singularities organize the surface flow dressing the torus
![Page 77: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/77.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
Exp’tal-08
Embed-01
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Topology ofOrbits-01
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Topology ofOrbits-04
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05a
Topology ofOrbits-05b
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09a
Program-09b
Program-10a
Program-10b
Program-11
Program-12
Reps-01
Reps-02
Reps-03
Reps-04
Reps-05
Reps-06
Reps-07
Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
Tori-06
Tori-07
Tori-08
Tori-09
Tori-10
Tori-11
Tori-12
Tori-13
Tori-14
Tori-15
Tori-16a
Tori-16b
Tori-16c
Tori-17
Tori-18
Tori-19
Tori-20
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
Flows in Vector Fields
Flow Near a Singularity
![Page 78: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/78.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
Exp’tal-08
Embed-01
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Topology ofOrbits-01
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Topology ofOrbits-12
Topology ofOrbits-13
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Topology ofOrbits-17
Program-01
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Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09a
Program-09b
Program-10a
Program-10b
Program-11
Program-12
Reps-01
Reps-02
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Reps-04
Reps-05
Reps-06
Reps-07
Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
Tori-06
Tori-07
Tori-08
Tori-09
Tori-10
Tori-11
Tori-12
Tori-13
Tori-14
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Tori-16a
Tori-16b
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Tori-17
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Tori-19
Tori-20
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
Some Bounding Tori
Torus Bounding Lorenz-like Flows
![Page 79: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/79.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
Exp’tal-08
Embed-01
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Topology ofOrbits-01
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Topology ofOrbits-13
Topology ofOrbits-14
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Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09a
Program-09b
Program-10a
Program-10b
Program-11
Program-12
Reps-01
Reps-02
Reps-03
Reps-04
Reps-05
Reps-06
Reps-07
Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
Tori-06
Tori-07
Tori-08
Tori-09
Tori-10
Tori-11
Tori-12
Tori-13
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Tori-15
Tori-16a
Tori-16b
Tori-16c
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Tori-19
Tori-20
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
Canonical Forms
Twisting the Lorenz Attractor
![Page 80: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/80.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
Exp’tal-08
Embed-01
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Topology ofOrbits-01
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Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04
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Topology ofOrbits-05a
Topology ofOrbits-05b
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09a
Program-09b
Program-10a
Program-10b
Program-11
Program-12
Reps-01
Reps-02
Reps-03
Reps-04
Reps-05
Reps-06
Reps-07
Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
Tori-06
Tori-07
Tori-08
Tori-09
Tori-10
Tori-11
Tori-12
Tori-13
Tori-14
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Tori-16a
Tori-16b
Tori-16c
Tori-17
Tori-18
Tori-19
Tori-20
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
Constraints Provided by Bounding Tori
Two possible branched manifolds
in the torus with g=4.
![Page 81: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/81.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
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Topology ofOrbits-01
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Topology ofOrbits-06
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Program-01
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Program-10a
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Program-11
Program-12
Reps-01
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Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
Tori-06
Tori-07
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Tori-20
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
Labeling Bounding Tori
Labeling Bounding Tori
Poincare section is disjoint union of g-1 disks.
Transition matrix sum of two g-1 × g-1 matrices.
Both are g-1 × g-1 permutation matrices.
They identify mappings of Poincare sections to P’sections.
Bounding tori labeled by (permutation) group theory.
![Page 82: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/82.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
Exp’tal-08
Embed-01
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Topology ofOrbits-01
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Topology ofOrbits-04
Topology ofOrbits-04a
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Topology ofOrbits-11
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Program-01
Program-02
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Program-08
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Program-10a
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Program-11
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Reps-01
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Reps-05
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Reps-07
Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
Tori-06
Tori-07
Tori-08
Tori-09
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Tori-16a
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Tori-19
Tori-20
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
Some Bounding Tori
Bounding Tori of Low Genus
![Page 83: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/83.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
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Exp’tal-08
Embed-01
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Program-01
Program-02
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Program-10a
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Reps-01
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Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
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Tori-03
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Tori-19
Tori-20
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
Exponential Growth
The Growth is Exponential
![Page 84: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/84.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
Exp’tal-08
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Topology ofOrbits-13
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Topology ofOrbits-17
Program-01
Program-02
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Program-08
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Program-10a
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Program-11
Program-12
Reps-01
Reps-02
Reps-03
Reps-04
Reps-05
Reps-06
Reps-07
Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
Tori-06
Tori-07
Tori-08
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Tori-16a
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Tori-19
Tori-20
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
Motivation
Some Genus-9 Bounding Tori
![Page 85: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/85.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
Exp’tal-08
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Topology ofOrbits-06
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Topology ofOrbits-11
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Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09a
Program-09b
Program-10a
Program-10b
Program-11
Program-12
Reps-01
Reps-02
Reps-03
Reps-04
Reps-05
Reps-06
Reps-07
Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
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Tori-19
Tori-20
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
Aufbau Princip for Bounding Tori
Aufbau Princip for Bounding Tori
These units (”pants, trinions”) surround the stretching andsqueezing units of branched manifolds.
![Page 86: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/86.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
Exp’tal-08
Embed-01
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Topology ofOrbits-11
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Topology ofOrbits-13
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Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09a
Program-09b
Program-10a
Program-10b
Program-11
Program-12
Reps-01
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Reps-09
Class-01
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Class-03
Class-04
Class-05
Bases-01
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Tori-01
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Tori-20
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
Aufbau Princip for Bounding Tori
Any bounding torus can be built upfrom equal numbers of stretching andsqueezing units
•Outputs to Inputs•No Free Ends• Colorless
![Page 87: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/87.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
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Embed-01
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Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09a
Program-09b
Program-10a
Program-10b
Program-11
Program-12
Reps-01
Reps-02
Reps-03
Reps-04
Reps-05
Reps-06
Reps-07
Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
Tori-06
Tori-07
Tori-08
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Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
Poincare Section
Construction of Poincare Section
![Page 88: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/88.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
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Exp’tal-08
Embed-01
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Topology ofOrbits-01
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Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09a
Program-09b
Program-10a
Program-10b
Program-11
Program-12
Reps-01
Reps-02
Reps-03
Reps-04
Reps-05
Reps-06
Reps-07
Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
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Tori-07
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Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
Aufbau Princip for Bounding Tori
Application: Lorenz Dynamics, g=3
![Page 89: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/89.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
Exp’tal-08
Embed-01
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Topology ofOrbits-01
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Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09a
Program-09b
Program-10a
Program-10b
Program-11
Program-12
Reps-01
Reps-02
Reps-03
Reps-04
Reps-05
Reps-06
Reps-07
Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
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Summary-01
Summary-02
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Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
Represerntation Theory Redux
Representation Theory for g > 1
Can we extend the representation theory for strange attractors“with a hole in the middle” (i.e., genus = 1) to higher-genusattractors?
Yes. The results are similar.
Begin as follows:
![Page 90: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/90.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
Exp’tal-08
Embed-01
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Topology ofOrbits-01
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Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09a
Program-09b
Program-10a
Program-10b
Program-11
Program-12
Reps-01
Reps-02
Reps-03
Reps-04
Reps-05
Reps-06
Reps-07
Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
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Tori-20
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
Aufbau Princip for Bounding Tori
Application: Lorenz Dynamics, g=3
![Page 91: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/91.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
Exp’tal-08
Embed-01
Embed-02
Embed-03
Embed-04a
Embed-04b
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05a
Topology ofOrbits-05b
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09a
Program-09b
Program-10a
Program-10b
Program-11
Program-12
Reps-01
Reps-02
Reps-03
Reps-04
Reps-05
Reps-06
Reps-07
Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
Tori-06
Tori-07
Tori-08
Tori-09
Tori-10
Tori-11
Tori-12
Tori-13
Tori-14
Tori-15
Tori-16a
Tori-16b
Tori-16c
Tori-17
Tori-18
Tori-19
Tori-20
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
Embeddings
Embeddings
![Page 92: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/92.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
Exp’tal-08
Embed-01
Embed-02
Embed-03
Embed-04a
Embed-04b
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05a
Topology ofOrbits-05b
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09a
Program-09b
Program-10a
Program-10b
Program-11
Program-12
Reps-01
Reps-02
Reps-03
Reps-04
Reps-05
Reps-06
Reps-07
Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
Tori-06
Tori-07
Tori-08
Tori-09
Tori-10
Tori-11
Tori-12
Tori-13
Tori-14
Tori-15
Tori-16a
Tori-16b
Tori-16c
Tori-17
Tori-18
Tori-19
Tori-20
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
Embeddings
Preparations for Embedding tori into R3
Equivalent to embedding a specific class of directed networksinto R3
![Page 93: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/93.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
Exp’tal-08
Embed-01
Embed-02
Embed-03
Embed-04a
Embed-04b
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05a
Topology ofOrbits-05b
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09a
Program-09b
Program-10a
Program-10b
Program-11
Program-12
Reps-01
Reps-02
Reps-03
Reps-04
Reps-05
Reps-06
Reps-07
Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
Tori-06
Tori-07
Tori-08
Tori-09
Tori-10
Tori-11
Tori-12
Tori-13
Tori-14
Tori-15
Tori-16a
Tori-16b
Tori-16c
Tori-17
Tori-18
Tori-19
Tori-20
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
Extrinsic Embedding of Bounding Tori
Extrinsic Embedding of Intrinsic Tori
A specific simple example.Partial classification by links of homotopy group generators.Nightmare Numbers are Expected.
![Page 94: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/94.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
Exp’tal-08
Embed-01
Embed-02
Embed-03
Embed-04a
Embed-04b
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05a
Topology ofOrbits-05b
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09a
Program-09b
Program-10a
Program-10b
Program-11
Program-12
Reps-01
Reps-02
Reps-03
Reps-04
Reps-05
Reps-06
Reps-07
Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
Tori-06
Tori-07
Tori-08
Tori-09
Tori-10
Tori-11
Tori-12
Tori-13
Tori-14
Tori-15
Tori-16a
Tori-16b
Tori-16c
Tori-17
Tori-18
Tori-19
Tori-20
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
Creating Isotopies
Equivalences by Injection
Obstructions to Isotopy
Index R3 R4 R5
Global Torsion Z⊗3(g−1) Z⊗2(g−1)2 -
Parity Z2 - -Knot Type Gen. KT. - -
In R5 all representations (embeddings) of a genus-g strangeattractor become equivalent under isotopy.
![Page 95: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/95.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
Exp’tal-08
Embed-01
Embed-02
Embed-03
Embed-04a
Embed-04b
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05a
Topology ofOrbits-05b
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09a
Program-09b
Program-10a
Program-10b
Program-11
Program-12
Reps-01
Reps-02
Reps-03
Reps-04
Reps-05
Reps-06
Reps-07
Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
Tori-06
Tori-07
Tori-08
Tori-09
Tori-10
Tori-11
Tori-12
Tori-13
Tori-14
Tori-15
Tori-16a
Tori-16b
Tori-16c
Tori-17
Tori-18
Tori-19
Tori-20
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
The Road Ahead
Summary
1 Question Answered ⇒
2 Questions Raised
We must be on the right track !
![Page 96: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/96.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
Exp’tal-08
Embed-01
Embed-02
Embed-03
Embed-04a
Embed-04b
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05a
Topology ofOrbits-05b
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09a
Program-09b
Program-10a
Program-10b
Program-11
Program-12
Reps-01
Reps-02
Reps-03
Reps-04
Reps-05
Reps-06
Reps-07
Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
Tori-06
Tori-07
Tori-08
Tori-09
Tori-10
Tori-11
Tori-12
Tori-13
Tori-14
Tori-15
Tori-16a
Tori-16b
Tori-16c
Tori-17
Tori-18
Tori-19
Tori-20
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
Our Hope
Original Objectives Achieved
There is now a simple, algorithmic procedure for:
Classifying strange attractors
Extracting classification information
from experimental signals.
![Page 97: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/97.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
Exp’tal-08
Embed-01
Embed-02
Embed-03
Embed-04a
Embed-04b
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05a
Topology ofOrbits-05b
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09a
Program-09b
Program-10a
Program-10b
Program-11
Program-12
Reps-01
Reps-02
Reps-03
Reps-04
Reps-05
Reps-06
Reps-07
Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
Tori-06
Tori-07
Tori-08
Tori-09
Tori-10
Tori-11
Tori-12
Tori-13
Tori-14
Tori-15
Tori-16a
Tori-16b
Tori-16c
Tori-17
Tori-18
Tori-19
Tori-20
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
Our Result
Result
There is now a classification theory
for low-dimensional strange attractors.
1 It is topological
2 It has a hierarchy of 4 levels
3 Each is discrete
4 There is rigidity and degrees of freedom
5 It is applicable to R3 only — for now
![Page 98: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/98.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
Exp’tal-08
Embed-01
Embed-02
Embed-03
Embed-04a
Embed-04b
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05a
Topology ofOrbits-05b
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09a
Program-09b
Program-10a
Program-10b
Program-11
Program-12
Reps-01
Reps-02
Reps-03
Reps-04
Reps-05
Reps-06
Reps-07
Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
Tori-06
Tori-07
Tori-08
Tori-09
Tori-10
Tori-11
Tori-12
Tori-13
Tori-14
Tori-15
Tori-16a
Tori-16b
Tori-16c
Tori-17
Tori-18
Tori-19
Tori-20
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
Four Levels of Structure
The Classification Theory has
4 Levels of Structure
1 Basis Sets of Orbits
2 Branched Manifolds
3 Bounding Tori
4 Extrinsic Embeddings
![Page 99: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/99.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
Exp’tal-08
Embed-01
Embed-02
Embed-03
Embed-04a
Embed-04b
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05a
Topology ofOrbits-05b
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09a
Program-09b
Program-10a
Program-10b
Program-11
Program-12
Reps-01
Reps-02
Reps-03
Reps-04
Reps-05
Reps-06
Reps-07
Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
Tori-06
Tori-07
Tori-08
Tori-09
Tori-10
Tori-11
Tori-12
Tori-13
Tori-14
Tori-15
Tori-16a
Tori-16b
Tori-16c
Tori-17
Tori-18
Tori-19
Tori-20
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
Four Levels of Structure
The Classification Theory has
4 Levels of Structure
1 Basis Sets of Orbits
2 Branched Manifolds
3 Bounding Tori
4 Extrinsic Embeddings
![Page 100: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/100.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
Exp’tal-07
Exp’tal-08
Embed-01
Embed-02
Embed-03
Embed-04a
Embed-04b
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05a
Topology ofOrbits-05b
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09a
Program-09b
Program-10a
Program-10b
Program-11
Program-12
Reps-01
Reps-02
Reps-03
Reps-04
Reps-05
Reps-06
Reps-07
Reps-08
Reps-09
Class-01
Class-02
Class-03
Class-04
Class-05
Bases-01
Bases-02
Bases-03
Bases-04
Tori-01
Tori-02
Tori-03
Tori-04
Tori-05
Tori-06
Tori-07
Tori-08
Tori-09
Tori-10
Tori-11
Tori-12
Tori-13
Tori-14
Tori-15
Tori-16a
Tori-16b
Tori-16c
Tori-17
Tori-18
Tori-19
Tori-20
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
Four Levels of Structure
The Classification Theory has
4 Levels of Structure
1 Basis Sets of Orbits
2 Branched Manifolds
3 Bounding Tori
4 Extrinsic Embeddings
![Page 101: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/101.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
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Program-01
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Summary-01
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Four Levels of Structure
The Classification Theory has
4 Levels of Structure
1 Basis Sets of Orbits
2 Branched Manifolds
3 Bounding Tori
4 Extrinsic Embeddings
![Page 102: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/102.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
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Program-01
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Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Summary-09
Summary-10
Four Levels of Structure
The Classification Theory has
4 Levels of Structure
1 Basis Sets of Orbits
2 Branched Manifolds
3 Bounding Tori
4 Extrinsic Embeddings
![Page 103: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/103.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
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Exp’tal-01
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Program-01
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Summary-01
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Four Levels of Structure
![Page 104: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/104.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
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Exp’tal-01
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Program-01
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Summary-01
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Topological Components
Poetic Organization
LINKS OF PERIODIC ORBITS
organize
BOUNDING TORI
organize
BRANCHED MANIFOLDS
organize
LINKS OF PERIODIC ORBITS
![Page 105: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/105.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
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Exp’tal-01
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Program-01
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Reps-01
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Summary-01
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Answered Questions
There is a Representation Theory for Strange Attractors
There is a complete set of rerpesentation labels for strangeattractors of any genus g.
The labels are complete and discrete.
Representations can become equivalent when immersed inhigher dimension.
All representations (embeddings) of a 3-dimensional strangeattractor become isotopic (equivalent) in R5.
The Universal Representation of an attractor in R5 identifiesmechanism. No embedding artifacts are left.
The topological index in R5 that identifies mechanism remainsto be discovered.
![Page 106: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/106.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
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Program-01
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Summary-01
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Answered Questions
Some Unexpected ResultsPerestroikas of orbits constrained by branched manifolds
Routes to Chaos = Paths through orbit forcing diagram
Perestroikas of branched manifolds constrained bybounding tori
Global Poincare section = union of g − 1 disks
Systematic methods for cover - image relations
Existence of topological indices (cover/image)
Universal image dynamical systems
NLD version of Cartan’s Theorem for Lie Groups
Topological Continuation – Group Continuuation
Cauchy-Riemann symmetries
Quantizing Chaos
![Page 107: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/107.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
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Program-01
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Summary-01
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Unanswered Questions
We hope to find:Robust topological invariants for RN , N > 3
A Birman-Williams type theorem for higher dimensions
An algorithm for irreducible embeddings
Embeddings: better methods and tests
Analog of χ2 test for NLD
Better forcing results: Smale horseshoe, D2 → D2,n×D2 → n×D2 (e.g., Lorenz), DN → DN , N > 2
Representation theory: complete
Singularity Theory: Branched manifolds, splitting points(0 dim.), branch lines (1 dim).
Singularities as obstructions to isotopy
![Page 108: The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia,](https://reader033.fdocuments.us/reader033/viewer/2022050407/5f84741aa8e64648f328a934/html5/thumbnails/108.jpg)
The Topologyof Chaos
RobertGilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
Exp’tal-05
Exp’tal-06
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Program-01
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Summary-01
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Thanks
To my colleagues and friends:
Jorge Tredicce Elia EschenaziHernan G. Solari Gabriel B. MindlinNick Tufillaro Mario NatielloFrancesco Papoff Ricardo Lopez-RuizMarc Lefranc Christophe LetellierTsvetelin D. Tsankov Jacob KatrielDaniel Cross Tim Jones
Thanks also to:NSF PHY 8843235NSF PHY 9987468NSF PHY 0754081