Simple Chaotic Systems and Circuits

J. C. SprottDepartment of Physics

University of Wisconsin - Madison

Presented at

University of Catania

In Catania, Italy

On July 15, 2014

Outline

Abbreviated History

Chaotic Equations

Chaotic Electrical Circuits

Abbreviated History Poincaré (1892) Van der Pol (1927) Ueda (1961) Lorenz (1963) Knuth (1968) Rössler (1976) May (1976)

Lorenz Equations (1963)

dx/dt = Ay – Ax

dy/dt = –xz + Bx – y

dz/dt = xy – Cz

7 terms, 2 quadratic

nonlinearities, 3 parameters

Rössler Equations (1976)

dx/dt = –y – z

dy/dt = x + Ay

dz/dt = B + xz – Cz

7 terms, 1 quadratic

nonlinearity, 3 parameters

Lorenz Quote (1993)“One other study left me with mixed feelings. Otto Roessler of the University of Tübingen had formulated a system of three differential equations as a model of a chemical reaction. By this time a number of systems of differential equations with chaotic solutions had been discovered, but I felt I still had the distinction of having found the simplest. Roessler changed things by coming along with an even simpler one. His record still stands.”

Rössler Toroidal Model (1979)

dx/dt = –y – z

dy/dt = x

dz/dt = Ay – Ay2 – Bz

6 terms, 1 quadratic

nonlinearity, 2 parameters

“Probably the simplest strange attractor of a 3-D ODE”(1998)

Sprott (1994)

14 additional examples with 6 terms and 1 quadratic nonlinearity

5 examples with 5 terms and 2 quadratic nonlinearities

J. C. Sprott, Phys. Rev. E 50, R647 (1994)

Gottlieb (1996)

What is the simplest jerk function that gives chaos?

Displacement: x

Velocity: = dx/dt

Acceleration: = d2x/dt2

Jerk: = d3x/dt3

x

x

x

)( x,x,xJx

Linz (1997)

Lorenz and Rössler systems can be written in jerk form

Jerk equations for these systems are not very “simple”

Some of the systems found by Sprott have “simple” jerk forms:

b x xxxx –a

Sprott (1997)

dx/dt = y

dy/dt = z

dz/dt = –az + y2 – x

5 terms, 1 quadratic

nonlinearity, 1 parameter

“Simplest Dissipative Chaotic Flow”

xxxax 2

Zhang and Heidel (1997)

3-D quadratic systems with

fewer than 5 terms cannot

be chaotic.

They would have no

adjustable parameters.

Eichhorn, Linz and Hänggi (1998) Developed hierarchy of

quadratic jerk equations with increasingly many terms:

xxxax 2

1–xxbxxax

1–2xxbxax

1–xxcxxbxax 2 ...

Weaker Nonlinearity

dx/dt = y

dy/dt = z

dz/dt = –az + |y|b – x

Seek path in a-b space that gives

chaos as b 1.

xxxaxb

Regions of Chaos

Linz and Sprott (1999)

dx/dt = y

dy/dt = z

dz/dt = –az – y + |x| – 1

6 terms, 1 abs nonlinearity, 2 parameters (but one =1)

1 xxxax

General Formdx/dt = y

dy/dt = z

dz/dt = – az – y + G(x)

G(x) = ±(b|x| – c)

G(x) = ±b(x2/c – c)

G(x) = –b max(x,0) + c

G(x) = ±(bx – c sgn(x))

etc….

)(xGxxax

Universal Chaos Approximator?

Operational Amplifiers

First Jerk Circuit

1 xxxax 18 components

Bifurcation Diagram for First Circuit

Strange Attractor for First Circuit

Calculated Measured

Second Jerk Circuit

CBA xxxx 15 components

Chaos Circuit

Third Jerk Circuit

)sgn(xxxxx A11 components

Simpler Jerk Circuit

)- sgn( xxxxx CBA 9 components

Inductor Jerk Circuit

)- sgn( xxxxx CBA 7 components

Delay Lline Oscillator

xxx - sgn6 components

References

http://sprott.physics.wisc.edu/

lectures/cktchaos/ (this talk)

http://sprott.physics.wisc.edu/

chaos/abschaos.htm

sprott@physics.wisc.edu