1. Simple Chaotic Systems and Circuits
4/21/2017
J. C. Sprott
Department of Physics
University of Wisconsin - Madison
Presented at
University of Catania
In Catania, Italy
On July 15, 2014
Illustrated with Microsoft PowerPoint 97
500 MHz Pentium Sony laptop PC running Windows 98
Entire presentation available on WWW

2. Outline Abbreviated History Chaotic Equations
4/21/2017
Outline
Abbreviated History
Chaotic Equations
Chaotic Electrical Circuits

3. Abbreviated History Poincaré (1892) Van der Pol (1927) Ueda (1961)
4/21/2017
Abbreviated History
Poincaré (1892)
Van der Pol (1927)
Ueda (1961)
Lorenz (1963)
Knuth (1968)
Rössler (1976)
May (1976)

4. Chaos in Logistic Map

5. Mathematical Models of Dynamical Systems
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Mathematical Models of Dynamical Systems
Logistic Equation (Map):
xn+1 = Axn(1−xn)
Newton’s 2nd Law (ODE):
md2x/dt2 = F(x,dx/dt,t)
Wave Equation (PDE):
2x/t2 = c22x/r2

6. Poincaré-Bendixson Theorem (in 2-D flow)
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Poincaré-Bendixson Theorem (in 2-D flow)
Fixed Point
Limit Cycle y x
Trajectory cannot intersect itself (no chaos)

7. Autonomous Systems d2x/dt2 + Adx/dt – x = Bsint let y = dx/dt
4/21/2017
Autonomous Systems
d2x/dt2 + Adx/dt – x = Bsint
let y = dx/dt
and z = t
dx/dt = y
dy/dt = –x – Ay + Bsin(z)
dz/dt = 

8. Lorenz Equations (1963) dx/dt = Ay – Ax dy/dt = –xz + Bx – y
4/21/2017
Lorenz Equations (1963)
dx/dt = Ay – Ax
dy/dt = –xz + Bx – y
dz/dt = xy – Cz
7 terms, 2 quadratic nonlinearities, 3 parameters

9. Rössler Equations (1976) dx/dt = –y – z dy/dt = x + Ay
4/21/2017
Rössler Equations (1976)
dx/dt = –y – z
dy/dt = x + Ay
dz/dt = B + xz – Cz
7 terms, 1 quadratic nonlinearity, 3 parameters

10. 4/21/2017
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.”

11. Rössler Toroidal Model (1979)
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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

12. 14 additional examples with 6 terms and 1 quadratic nonlinearity
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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)

13. Gottlieb (1996) What is the simplest jerk function that gives chaos?
4/21/2017
Gottlieb (1996)
What is the simplest jerk function that gives chaos?
Displacement: x
Velocity: = dx/dt
Acceleration: = d2x/dt2
Jerk: = d3x/dt3

14. Linz (1997) Lorenz and Rössler systems can be written in jerk form
4/21/2017
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:

15. “Simplest Dissipative Chaotic Flow”
4/21/2017
Sprott (1997)
“Simplest Dissipative Chaotic Flow”
dx/dt = y
dy/dt = z
dz/dt = –az + y2 – x
5 terms, 1 quadratic nonlinearity, 1 parameter

16. 4/21/2017
Bifurcation Diagram

17. 4/21/2017
Return Map

18. 4/21/2017
Zhang and Heidel (1997)
3-D quadratic systems with fewer than 5 terms cannot be chaotic.
They would have no adjustable parameters.

19. Eichhorn, Linz and Hänggi (1998)
4/21/2017
Eichhorn, Linz and Hänggi (1998)
Developed hierarchy of quadratic jerk equations with increasingly many terms:
...

20. Weaker Nonlinearity dx/dt = y dy/dt = z dz/dt = –az + |y|b – x
4/21/2017
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.

21. 4/21/2017
Regions of Chaos

22. Linz and Sprott (1999) dx/dt = y dy/dt = z dz/dt = –az – y + |x| – 1
4/21/2017
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)

23. General Form dx/dt = y dy/dt = z dz/dt = – az – y + G(x)
4/21/2017
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….

24. Universal Chaos Approximator?

25. Operational Amplifiers
4/21/2017
Operational Amplifiers

26. 4/21/2017
First Jerk Circuit
18 components

27. Bifurcation Diagram for First Circuit
4/21/2017
Bifurcation Diagram for First Circuit

28. Strange Attractor for First Circuit
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Strange Attractor for First Circuit
Calculated
Measured

29. Second Jerk Circuit
4/21/2017
15 components

30. 4/21/2017
Chaos Circuit

31. Third Jerk Circuit
4/21/2017
11 components

32. Simpler Jerk Circuit
4/21/2017
9 components

33. Inductor Jerk Circuit
4/21/2017
7 components

34. Delay Lline Oscillator
4/21/2017
6 components

36. 4/21/2017
References
lectures/cktchaos/ (this talk)
haos/abschaos.htm