1. Simple Chaotic Systems and Circuits
4/28/2017
J. C. Sprott
Department of Physics
University of Wisconsin - Madison
Presented at the
Gordon Conference on Classical Mechanics and Nonlinear Dynamics
on June 16, 2004
Illustrated with Microsoft PowerPoint 97
500 MHz Pentium Sony laptop PC running Windows 98
Entire presentation available on WWW

2. Lorenz Equations (1963) dx/dt = Ay – Ax dy/dt = –xz + Bx – y
4/28/2017
Lorenz Equations (1963)
dx/dt = Ay – Ax
dy/dt = –xz + Bx – y
dz/dt = xy – Cz
7 terms, 2 quadratic nonlinearities, 3 parameters

3. Rössler Equations (1976) dx/dt = –y – z dy/dt = x + Ay
4/28/2017
Rössler Equations (1976)
dx/dt = –y – z
dy/dt = x + Ay
dz/dt = B + xz – Cz
7 terms, 1 quadratic nonlinearity, 3 parameters

4. 4/28/2017
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)
4/28/2017
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 examples with 6 terms and 1 quadratic nonlinearity
4/28/2017
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)

7. Gottlieb (1996) What is the simplest jerk function that gives chaos?
4/28/2017
Gottlieb (1996)
What is the simplest jerk function that gives chaos?
Displacement: x
Velocity: = dx/dt
Acceleration: = d2x/dt2
Jerk: = d3x/dt3

8. Linz (1997) Lorenz and Rössler systems can be written in jerk form
4/28/2017
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:

9. “Simplest Dissipative Chaotic Flow”
4/28/2017
Sprott (1997)
“Simplest Dissipative Chaotic Flow”
dx/dt = y
dy/dt = z
dz/dt = –az + y2 – x
5 terms, 1 quadratic nonlinearity, 1 parameter

10. 4/28/2017
Bifurcation Diagram

11. 4/28/2017
Return Map

12. 4/28/2017
Zhang and Heidel (1997)
3-D quadratic systems with fewer than 5 terms cannot be chaotic.
They would have no adjustable parameters.

13. Linz and Sprott (1999) dx/dt = y dy/dt = z dz/dt = –az – y + |x| – 1
4/28/2017
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)

14. General Form dx/dt = y dy/dt = z dz/dt = – az – y + G(x)
4/28/2017
General Form
dx/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….

15. Universal Chaos Approximator?

16. 4/28/2017
First Circuit

17. Bifurcation Diagram for First Circuit
4/28/2017
Bifurcation Diagram for First Circuit

18. Strange Attractor for First Circuit
4/28/2017
Strange Attractor for First Circuit
Calculated
Measured

19. Second Circuit
4/28/2017

20. 4/28/2017
Chaos Circuit

21. Third Circuit
4/28/2017

22. Fourth Circuit
D(x) = –min(x, 0)

23. Bifurcation Diagram for Fourth Circuit
K. Kiers, D. Schmidt, and J. C. Sprott, Am. J. Phys. 72, 503 (2004)

25. 4/28/2017
References
lectures/gordon04.ppt (this talk)
(circuit #4)
(to contact me)