1. Complex Behavior of Simple Systems
11/10/2018
J. C. Sprott
Department of Physics
University of Wisconsin - Madison
Presented to
International Conference on Complex Systems
in Nashua, NH
on May 23, 2000
Illustrated with Microsoft PowerPoint 97
120 MHz Pentium Toshiba laptop PC running Windows 95
LCD projector?
Entire presentation available on WWW

2. Lorenz Equations (1963) dx/dt = sy - sx dy/dt = -xz + rx - y
11/10/2018
Lorenz Equations (1963)
dx/dt = sy - sx
dy/dt = -xz + rx - y
dz/dt = xy - bz
7 terms, 2 quadratic nonlinearities, 3 parameters

3. Rössler Equations (1976) dx/dt = -y - z dy/dt = x + ay
11/10/2018
Rössler Equations (1976)
dx/dt = -y - z
dy/dt = x + ay
dz/dt = b + xz - cz
7 terms, 1 quadratic nonlinearity, 3 parameters

4. 11/10/2018
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.”

5. Rössler Toroidal Model (1979)
11/10/2018
Rössler Toroidal Model (1979)
“Probably the simplest strange attractor of a 3-D ODE”
(1998)
dx/dt = -y - z
dy/dt = x
dz/dt = ay - ay2 - bz
6 terms, 1 quadratic nonlinearity, 2 parameters

6. 14 more examples with 6 terms and 1 quadratic nonlinearity
11/10/2018
Sprott (1994)
14 more examples with 6 terms and 1 quadratic nonlinearity
5 examples with 5 terms and 2 quadratic nonlinearities

7. Gottlieb (1996) What is the simplest jerk function that gives chaos?
11/10/2018
Gottlieb (1996)
What is the simplest jerk function that gives chaos?
Displacement: x
Velocity: = dx/dt
Acceleration: = d2x/dt2
Jerk: = d3x/dt3

8. “Simplest Dissipative Chaotic Flow”
11/10/2018
Sprott (1997)
“Simplest Dissipative Chaotic Flow”
dx/dt = y
dy/dt = z
dz/dt = -az + y2 - x
5 terms, 1 quadratic nonlinearity, 1 parameter

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Bifurcation Diagram

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Return Map

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Fu and Heidel (1997)
Dissipative quadratic systems with less than 5 terms cannot be chaotic.
They would have no adjustable parameters.

12. Weaker Nonlinearity dx/dt = y dy/dt = z dz/dt = -az + |y|b - x
11/10/2018
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.

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Regions of Chaos

14. Linz and Sprott (1999) dx/dt = y dy/dt = z dz/dt = -az - y + |x| - 1
11/10/2018
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)

15. General Form dx/dt = y dy/dt = z dz/dt = -az - y + G(x)
11/10/2018
General Form
dx/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….

16. 11/10/2018
First Circuit

17. Bifurcation Diagram for First Circuit
11/10/2018
Bifurcation Diagram for First Circuit

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Second Circuit

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Third Circuit

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Chaos Circuit