Simple Chaotic Systems and Circuits

J. C. SprottDepartment of PhysicsUniversity of Wisconsin - Madison

Presented at the

Gordon Conference on Classical Mechanics and Nonlinear Dynamicson June 16, 2004

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 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: xVelocity: = dx/dtAcceleration: = 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

Bifurcation Diagram

Return Map

Zhang and Heidel (1997)

3-D quadratic systems with fewer than 5 terms cannot be chaotic.

They would have no adjustable parameters.

Linz and Sprott (1999)dx/dt = ydy/dt = zdz/dt = –az – y + |x| – 1

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

1 xxxax

General Formdx/dt = ydy/dt = zdz/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?

First Circuit

1 xxxax

Bifurcation Diagram for First Circuit

Strange Attractor for First Circuit

Calculated Measured

Second Circuit

CBA xxxx

Chaos Circuit

Third Circuit

)sgn(xxxxx A

Fourth Circuit

1)D( xxxx A

D(x) = –min(x, 0)

Bifurcation Diagram for Fourth Circuit

K. Kiers, D. Schmidt, and J. C. Sprott, Am. J. Phys. 72, 503 (2004)

References

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

lectures/gordon04.ppt (this talk)

http://www.css.tayloru.edu/~dsimons/

(circuit #4)

sprott@physics.wisc.edu (to contact me)