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

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

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)

Chaos in Logistic Map

Mathematical Models of Dynamical Systems 4/21/2017 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

Poincaré-Bendixson Theorem (in 2-D flow) 4/21/2017 Poincaré-Bendixson Theorem (in 2-D flow) Fixed Point Limit Cycle y x Trajectory cannot intersect itself (no chaos)

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 = 

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

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

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.”

Rössler Toroidal Model (1979) 4/21/2017 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

14 additional examples with 6 terms and 1 quadratic nonlinearity 4/21/2017 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)

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

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:

“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

4/21/2017 Bifurcation Diagram

4/21/2017 Return Map

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.

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

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.

4/21/2017 Regions of Chaos

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)

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….

Universal Chaos Approximator?

Operational Amplifiers 4/21/2017 Operational Amplifiers

4/21/2017 First Jerk Circuit 18 components

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

Strange Attractor for First Circuit 4/21/2017 Strange Attractor for First Circuit Calculated Measured

Second Jerk Circuit 4/21/2017 15 components

4/21/2017 Chaos Circuit

Third Jerk Circuit 4/21/2017 11 components

Simpler Jerk Circuit 4/21/2017 9 components

Inductor Jerk Circuit 4/21/2017 7 components

Delay Lline Oscillator 4/21/2017 6 components

4/21/2017 References http://sprott.physics.wisc.edu/ lectures/cktchaos/ (this talk) http://sprott.physics.wisc.edu/c haos/abschaos.htm sprott@physics.wisc.edu