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## Simple Chaotic Systems and Circuits

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**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) sprott@physics.wisc.edu (to contact me) References