Introduction to Chaos Clint Sprott Department of Physics University of Wisconsin - Madison Presented to Physics 311 at University of Wisconsin in Madison, WI on October 31, 2014

Abbreviated History n Kepler (1605) n Newton (1687) n Poincare (1890) n Lorenz (1963)

Kepler (1605) n Tycho Brahe n 3 laws of planetary motion n Elliptical orbits

Newton (1687) n Invented calculus n Derived 3 laws of motion F = ma n Proposed law of gravity 1 F = Gm 1 m 2 /r 2 n Explained Kepler’s laws n Got headaches (3 body problem)

Poincare (1890) n 200 years later! n King Oscar (Sweden, 1887) n Prize won – 200 pages n No analytic solution exists! n Sensitive dependence on initial conditions (Lyapunov exponent) n Chaos! (Li & Yorke, 1975)

3-Body Problem

Chaos n Sensitive dependence on initial conditions (positive Lyapunov exp) n Aperiodic (never repeats) n Topologically mixing n Dense periodic orbits

Simple Pendulum F = ma -mg sin x = md 2 x/dt 2 dx/dt = v dv/dt = -g sin x  dv/dt = -x (for g = 1, x << 1) Dynamical system Flow in 2-D phase space

Phase Space Plot for Pendulum

Features of Pendulum Flow n Stable (O) & unstable (X) equilibria n Linear and nonlinear regions n Conservative / time-reversible n Trajectories cannot intersect

Pendulum with Friction dx/dt = v dv/dt = -sin x – bv

Features of Pendulum Flow n Dissipative (cf: conservative) n Attractors (cf: repellors) n Poincare-Bendixson theorem n No chaos in 2-D autonomous system

Damped Driven Pendulum dx/dt = v dv/dt = -sin x – bv + sin  t 2-D 3-D nonautonomousautonomous dx/dt = v dv/dt = -sin x – bv + sin z dz/dt = 

New Features in 3-D Flows n More complicated trajectories n Limit cycles (2-D attractors) n Strange attractors (fractals) n Chaos!

Stretching and Folding

Chaotic Circuit

Equations for Chaotic Circuit dx/dt = y dy/dt = z dz/dt = az – by + c(sgn x – x) Jerk system Period doubling route to chaos

Bifurcation Diagram for Chaotic Circuit

Invitation I sometimes work on publishable research with bright undergraduates who are crack computer programmers with an interest in chaos. If interested, contact me.

References n lectures/phys311.pptx (this talk) lectures/phys311.pptx n sa/ (my chaos textbook) sa/ n (contact me)

Props n Hard copy of slides n Driven chaotic pendulum n Ball point pen n Silly putty n Chaotic circuit / speaker