Complex Behavior of Simple Systems

J. C. SprottDepartment of Physics

University of Wisconsin - Madison

Presented to

International Conference on Complex Systems

in Nashua, NH

on May 23, 2000

Lorenz Equations (1963)

dx/dt = y - x

dy/dt = -xz + rx - y

dz/dt = xy - bz

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 more examples with 6 terms and 1 quadratic nonlinearity

5 examples with 5 terms and 2 quadratic nonlinearities

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

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

Fu and Heidel (1997)

Dissipative quadratic

systems with less than 5

terms cannot be chaotic.

They would have no

adjustable parameters.

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) = -bmax(x,0) + c

G(x) = ±(bx - csgn(x))etc….

)(xGxxax

First Circuit

1 xxxax

Bifurcation Diagram for First Circuit

Second Circuit

Third Circuit

)sgn(xxxxax

Chaos Circuit