Simple Chaotic Systems and Circuits 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
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) “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
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)
Gottlieb (1996) What is the simplest jerk function that gives chaos? Displacement: x Velocity: = dx/dt Acceleration: = d2x/dt2 Jerk: = d3x/dt3
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:
Sprott (1997) “Simplest Dissipative Chaotic Flow” dx/dt = y dy/dt = z dz/dt = –az + y2 – x 5 terms, 1 quadratic nonlinearity, 1 parameter
Zhang and Heidel (1997) 3-D quadratic systems with fewer than 5 terms cannot be chaotic. They would have no adjustable parameters.
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)
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….
Strange Attractor for First Circuit Calculated Measured
Fourth Circuit D(x) = –min(x, 0)
Bifurcation Diagram for Fourth Circuit K. Kiers, D. Schmidt, and J. C. Sprott, Am. J. Phys. 72, 503 (2004)
http://sprott.physics.wisc.edu/ lectures/gordon04.ppt (this talk) http://www.css.tayloru.edu/~dsimons/ (circuit #4) email@example.com (to contact me) References