Phase space transport. IIsdross/talks/fbp-2003.pdf · Part II Transport using set oriented methods...

C�� Dynamical S��

CA L T EC H

Phase space transport. II

Shane RossControl and Dynamical Systems, Caltech

http://www.cds.caltech.edu/∼shane/

Full Body Problem Workshop

Caltech, November 14-15, 2003

Motivation� Apply transport calculations to asteroid pairs to calcu-

late, e.g., capture & escape rates.

2

Motivation� Apply transport calculations to asteroid pairs to calcu-

late, e.g., capture & escape rates.

Dactyl in orbit about Ida, discovered in 1994 during the Galileo mission.2

Motivation� Movie of simple model of Ida & Dactyl

� Small hard sphere around rotating elliptical asteroid

� Nonmerging collisions modeled as bounces

(work with E. Kanso)

3

In This Talk...

�Part I. Restricted F2BP phase space

� Dependence on energy

� Tube + lobe dynamics

� Modeling collisions

�Part II. Transport using set orientedmethods

� Transport problem described

� Two computational techniques

(a) Invariant manifolds

(b) Almost-invariant sets

� Extensions and future work4

Part I

�Restricted Full Two Body Problem

� Consider two masses: m1 (sphere) & m2 (ellipse)

� m1

m2→ 0

Particle around asteroid

restricted F2BP (RF2BP)

m2m1

� Restricted (as in restricted 3-body problem) simple caseexhibits the basic capture, ejection, collision dynamics(see Koon, Marsden, Ross, Lo, Scheeres [2003])

� Can include bouncing, sticking, ... (see Kanso [2003])

5

Restricted F2BP� Point mass P moving in the x-y plane under the grav-

itational field of a uniformly rotating elliptical body m,without affecting its uniform rotation.

Y

X

xy

t

P

m

Force

The rotating (x-y) and inertial (X-Y ) frames.6

Restricted F2BP� Equations of motion relative to a rotating Cartesian

coordinate frame and appropriately normalized:

x− 2y = −∂U

∂xand y + 2x = −∂U

∂y,

where

U(x, y) = −1

r− 1

2r2 −

3C22

(x2 − y2

)r5

,

and

r =√

x2 + y2.

� Energy integral (Jacobi integral):

E =1

2

(x2 + y2

)+ U(x, y).

7

Restricted F2BP� Gravity field coefficient C22, the ellipticity,

C22 =1

20(1− β2),

varies between 0 and 0.05.

� e.g., Ida: β ≈ 0.43, C22 ≈ 0.04

1

β

8

Phase Space Structure� Energy indicates type of global dynamics.

� Is movement between the exterior and asteroid realmspossible?

E < ES E > ES

9

Phase Space Structure

�Multi-scale dynamics

� Coarse level : tube dynamics between realms

� Fine level : lobe dynamics within realms

10

Phase Space Structure� Slices of energy surface: Poincare sections Ui

� Tube dynamics: evolution between Ui

� Lobe dynamics: evolution on Ui

U2

U1

PO

z0

z1z2

z3

U1

U2

Exit

Collision

Surface

PO

11

Lobe Dynamics� Suppose E < ES; energy surface ME

� Asteroid and exterior realms not connected

� Poincare map in exterior realm: area and orientationpreserving map on M ⊂ R2,

f : M −→ M

whereM = ME ∩ {x = 0, x > 0}

with coordinates (y, py), equiv., (r, pr), on M

12

Lobe Dynamics� Particles are ejected if they lie within lobes enclosed by

the stable and unstable manifolds of a hyperbolic fixedpoint at (+∞, 0)—lobes of ejection.

f

M

f

Position space projection Motion on M

13

Lobe Dynamics

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

r

pr

Numerical simulation using MANGEN

14

Lobe Dynamics

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

r

pr

Curves can be followed to very high accuracy

15

MANGEN Description� Simulations use MANGEN (Coulliete & Lekien)

� Adaptive conditioning of curves based on curvature.

a) b)

c) d)

16

Tube + Lobe Dynamics� Suppose E > ES

� Exterior and asteroid realms connected via tubes

� In exterior realm, some tubes lead to collision

(others lead away from collision)

� Tube + lobe dynamics =

Alternate fates of collision and ejection are intimatelyintermingled.

17

Tube + Lobe Dynamics� Tubes leading to collision with asteroid

M

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

r

pr


18

Tube + Lobe Dynamics� Tubes leading to collision with asteroid

plus tubes coming from collision, e.g., liberated particles

M

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

r

pr


19

Tube + Lobe Dynamics� Escape and re-capture.

20

Collision Modeling� If bouncing is modeled, dynamics is more complicated.

� Upon bouncing, particle moves to new energy surface

� Work in progress with E. Kanso

−3 −2 −1 0 1 2

−3

−2

−1

0

1

2

3

−35 −30 −25 −20 −15 −10 −5 0 5 10−1.62

−1.6

−1.58

−1.56

−1.54

−1.52

−1.5

−1.48

−1.46

−1.44

−1.42

Inertial frame projection Energy history

21

Part II

�Transport using set oriented methods

� Describe transport of phase points on a k-dimensionalmanifold M

22

Part II



� M could be, e.g., the ocean surface, an energy shell, ora Poincare surface-of-section

22

Part II



� M could be, e.g., the ocean surface, an energy shell, ora Poincare surface-of-section

� We look first at k = 2 for autonomous systems

� Paper: Dellnitz, Junge, Koon, Lekien, Lo, Marsden,Padberg, Preis, R., Thiere [2003]

22

Statement of Problem� Consider a volume- and orientation-preserving map

f : M → M,

on some compact set M ⊂ R2 with volume measure µ.e.g., f may be a discretization of an autonomous flow.

23


f : M → M,


� M is partitioned into regions Ri, i = 1, . . . , NR,such that

M =

NR⋃i=1

Ri and µ(Ri

⋂Rj) = 0 for i 6= j.

23


f : M → M,


� M is partitioned into regions Ri, i = 1, . . . , NR,such that

M =

NR⋃i=1

Ri and µ(Ri

⋂Rj) = 0 for i 6= j.

� Initially, Ri is uniformly covered with species Si.

i.e., Species type indicates where a point was initially.

23

Statement of Problem� Describe the distribution of species Si throughout the

regions Rj at any future iterate n > 0.

24

Transport Quantities� Quantities of interest:

Ti,j(n) ≡ the total amount of species Si contained in

region Rj immediately after the n-th iterate

= µ(f−n(Rj) ∩Ri)

25

Transport Quantities� Quantities of interest:

Ti,j(n) ≡ the total amount of species Si contained in

region Rj immediately after the n-th iterate

= µ(f−n(Rj) ∩Ri)

� Our goal:

Compute the Ti,j(n) up to some nmax

25

Computational Approaches� Compare & combine two computational approaches

26


� 1) Invariant manifolds of fixed points, lobe dynamics;

co-dimension one objects which bound regions, etc.

MANGEN: Manifold Generation, Lekien etc

26


� 1) Invariant manifolds of fixed points, lobe dynamics;

co-dimension one objects which bound regions, etc.

MANGEN: Manifold Generation, Lekien etc

� 2) Set oriented methods, almost-invariant sets;

direct computation of regions

GAIO: Global Analysis of Invariant Objects, Dellnitz etc

26

Particle in 2-Body Field� Example problem: test particles in gravity field of two

masses, m1 and m2, in circular orbit, i.e., the planar,circular restricted three-body problem with m2

m1≈ 10−3.

� Reduce to 2D map via Poincare surface-of-section

xm1

m2

P

(x ,y

,vx

,vy)

y

27

Particle in 2-Body Field◦ Poincare map f : M → M has regular and irregular components. Large

connected irregular component, the “chaotic sea.”

−3 −2.8 −2.6 −2.4 −2.2 −2 −1.8 −1.6 −1.4 −1.2−0.4

−0.2

0

0.2

0.4

X (Nondim.)

X−V

eloc

ity (

Non

dim

.)

28

Transport on M� To understand the transport of points under the Poincare

map f , we consider the invariant manifolds of un-stable fixed points

� Let pi, i = 1, ..., Np, denote a collection of saddle-typehyperbolic fixed points for f .

29

Transport on M◦ Local pieces of unstable and stable manifolds

p2p3

p1

Unstable and stable manifolds in red and green, resp.

30

Transport on M◦ Intersection of unstable and stable manifolds define boundaries.

q2

q1q4

q5

q6

q3

p2p3

p1

31

Transport on M◦ These boundaries divide phase space into regions, Ri, i = 1, . . . , NR

R1

R5

R4

R3

R2

q2

q1q4

q5

q6

q3

p2p3

p1

32

Lobe Dynamics� Local transport: across a boundary

consider small sets bounded by stable & unstable mfds

R1

R2

q

pipj

f -1(q)

L2,1(1)

L1,2(1)

33

Lobe Dynamics� They map from entirely in one region to another under

one iteration of f

L1,2(1) and L2,1(1) are called turnstile lobes

R1

R2

q

pipj

f -1(q)

L2,1(1)

L1,2(1)

f (L1,2(1))

f (L2,1(1))

34

Lobe Dynamics� MANGEN: evolution of a lobe of species S1 into R2

insert S1 into R2 movie

35

Lobe Dynamics� Global transport between regions (Ti,j(n)) is com-

pletely described by the dynamical evolution of lobes.

36

Set Oriented Methods

�Overview

� Partition phase space into loosely coupled regions

Ri, i = 1, . . . , NR,

� Probability is small for a point in a region to leave ina short time under f .

� These almost-invariant sets (AIS’s) define macro-scopic structures preserved by the dynamics.

� The transport, Ti,j(n), between almost-invariant setscan then be determined.

37

Almost-Invariant Sets� 1) discretize the phase space into boxes; model boxes

as the vertices and transitions between boxes as edgesof a directed graph

Using the Underlying Graph(Froyland-D. 2003, D.-Preis 2002)

Boxes are verticesCoarse dynamics represented by edges

Use graph theoretic algorithms incombination with the multilevel structure

38

Almost-Invariant Sets� 2) use graph partitioning methods to divide the vertices

of the graph into an optimal number of parts such thateach part is highly coupled within itself and only looselycoupled with other parts




39

Almost-Invariant Sets� 3) by doing so, we can obtain AIS’s and analyze trans-

port between themUsing the Underlying Graph(Froyland-D. 2003, D.-Preis 2002)



40

Almost-Invariant Sets

�Box Formulation

� Create a fine box partition of the phase space

B = {B1, . . . Bq}, where q could be 107+

� Consider a (weighted) q-by-q transition matrix, P ,for our dynamical system, where

Pij =µ(Bi ∩ f−1(Bj))

µ(Bi),

the transition probability from Bi to Bj

� P is an approximation of our dynamical system via afinite state Markov chain.

41

Almost-Invariant Sets

�Graph Formulation and Partitioning

� P has a corresponding graph representation where nodesof the graph correspond to boxes Bi.




42

Almost-Invariant Sets� If Pij > 0, then there is an edge between nodes i and j

in the graph with weight Pij.

� Partitioning into AIS’s becomes a problem of finding aminimal cut of this graph.




43

Almost-Invariant Sets� AIS’s correspond with key dynamical features

More refined methods like MANGEN can pick up details

The phase space is divided into several invariant and almost-invariant sets.44

Almost-Invariant Sets� Using the box formulation and GAIO, the Ti,j(n) can be

computed for large n. Agrees with MANGEN result.

0 5 10 15 20 25 30 35 40 45 500

0.05

0.1

0.15

0.2

0.25

n = Iterate number

Ph

ase

spac

e vo

lum

e

T1,1

(n)T

1,2(n)

45

Almost-Invariant Sets� To speed the computation, box refinements are per-

formed where transport related structures, e.g., lobes,are located.

46

Summary & Conclusion� The merging of statistical and geometric approaches

yields a very powerful tool.

47

Summary & Conclusion� The merging of statistical and geometric approaches

yields a very powerful tool.

� Example problem: restricted 3-body problem.

� Both find the same regions

AIS’s, statistical features, are identified with regions,geometric features

47

Summary & Conclusion� Theoretically, transport between regions determined by

images and pre-images of lobes.

48



� MANGEN computes only region & lobe boundaries;

Stretching of boundaries, memory limits;

nmax ≈ 20

48





nmax ≈ 20

� GAIO uses transition matrix between many boxes;

Resolution limited by max box number; aided by boxrefinement along lobes

nmax ≈ 50

48





nmax ≈ 20

� GAIO uses transition matrix between many boxes;

Resolution limited by max box number; aided by boxrefinement along lobes

nmax ≈ 50

� Same values for Ti,j(n) over common time window.

48

Future Directions� AIS & lobe dynamics in 3D+, e.g., astronomy, chemistry

49


� AIS for time dependent systems? e.g., ocean dynamics

49



� Numerical algorithms are crucial:

GAIO for coarse picture, transport calculations

MANGEN to refine on regions of interest

⇒ important for precision navigation

49



� Numerical algorithms are crucial:

GAIO for coarse picture, transport calculations

MANGEN to refine on regions of interest

⇒ important for precision navigation

� Merge techniques into single package:

Box formulation, graph algorithms

Co-dimension one objects

Adaptive conditioning based on curvature

49

Selected References• Koon, W.S., J.E. Marsden, S.D. Ross, M.W. Lo, & D.J. Scheeres [2003] Geo-

metric mechanics and the dynamics of asteroid pairs. Annals of the New YorkAcademy of Science, to appear.

• Dellnitz, M., O. Junge, W.S. Koon, F. Lekien, M.W. Lo, J.E. Marsden, K. Pad-berg, R. Preis, S. Ross, & B. Thiere [2003], Transport in Dynamical Astronomyand Multibody Problems, preprint.

• Ross, S.D. [2003], Statistical theory of interior-exterior transition and collisionprobabilities for minor bodies in the solar system. International Conference onLibration Point Orbits and Applications, Girona, Spain.

• Jaffe, C., S.D. Ross, M.W. Lo, J. Marsden, D. Farrelly, & T. Uzer [2002] Sta-tistical theory of asteroid escape rates. Physical Review Letters 89, 011101.

• Koon, W.S., M.W. Lo, J.E. Marsden & S.D. Ross [2000] Heteroclinic connectionsbetween periodic orbits and resonance transitions in celestial mechanics. Chaos10(2), 427–469.

For papers, movies, etc., visit the websites:http://www.cds.caltech.edu/∼shane

http://www.nast-group.caltech.edu/50

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