Simple Chaotic Systems and Circuits J. C. Sprott Department of Physics University of Wisconsin - Madison Presented at the University of Augsburg on October 15, 2001

Acknowledgments n George Rowlands - U. Warwick n Stefan Linz - U. Augsburg n Lucas Finco - U. Wisconsin n Tom Lovell - U. Wisconsin n Mikhail Reyfman - U. Wisconsin n Nicos Savva - U. Wisconsin

Outline n Abbreviated History n Chaotic Equations n Chaotic Electrical Circuits

Abbreviated History n Poincaré (1892) n Van der Pol (1927) n Lorenz (1963) n Knuth (1968) n Rössler (1976) n May (1976)

Mathematical Models of Dynamical Systems n Logistic Equation (Map): x n+1 = Ax n (1-x n ) n Newton’s 2 nd Law (ODE): md 2 x/dt 2 = F(x,dx/dt,t) n Wave Equation (PDE):  2 x/  t 2 = c 2  2 x/  r 2

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

Autonomous Systems d 2 x/dt 2 = -x - Adx/dt + Bsin  t let y = dx/dt and z =  t dx/dt = y dy/dt = -x - Ay + Bsin(z) dz/dt = 

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 - Ay 2 - Bz 6 terms, 1 quadratic nonlinearity, 2 parameters “Probably the simplest strange attractor of a 3-D ODE” ( 1998)

Sprott (1994) n 14 examples with 6 terms and 1 quadratic nonlinearity n 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: = d 2 x/dt 2 Jerk: = d 3 x/dt 3

Linz (1997) n Lorenz and Rössler systems can be written in jerk form n Jerk equations for these systems are not very “simple” n Some of the systems found by Sprott have “simple” jerk forms:

Sprott (1997) dx/dt = y dy/dt = z dz/dt = -az + y 2 - x 5 terms, 1 quadratic nonlinearity, 1 parameter “ Simplest Dissipative Chaotic Flow”

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.

Eichhorn, Linz and Hänggi (1998) n Developed hierarchy of quadratic jerk equations with increasingly many terms:...

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.

Regions of Chaos

Linz and Sprott (1999) dx/dt = y dy/dt = z dz/dt = -az - y + |x| terms, 1 abs nonlinearity, 2 parameters (but one =1)

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….

Operational Amplifiers

First Circuit d 3 x/dt 3 = -Ad 2 x/dt 2 - dx/dt + |x| - 1

Bifurcation Diagram for First Circuit

Computer Prediction for First Circuit

Second Circuit

Chaos Circuit

Third Circuit d 3 x/dt 3 = -Ad 2 x/dt 2 - dx/dt + x - sgn(x)

Chua’s Circuit (1984)

Chaotic Inductor-Diode Circuit Testa, Perez, & Jeffries (1982) LdI/dt = V o sin  t - V [C o + I o  T(1-  V)e -  V ] dV/dt = I - I o (1-e -  V )

References n lectures/cktchaos/ (this talk) lectures/cktchaos/ n haos/abschaos.htm haos/abschaos.htm n