Introduction to Chaos Clint Sprott 1/4/2018 Introduction to Chaos Clint Sprott Department of Physics University of Wisconsin – Madison USA Presented at the Utrecht Physics Challenge in Utrecht, Netherlands on May 6, 2017 Entire presentation available on WWW Workshop on Self-Organization

Abbreviated History Kepler (1605) Newton (1687) Poincare (1890) 1/4/2018 Abbreviated History Kepler (1605) Newton (1687) Poincare (1890) Lorenz (1963) Workshop on Self-Organization

Johannes Kepler (1605) Assistant to Tycho Brahe 3 laws of planetary motion Elliptical orbits Repeatable Predictable

Isaac Newton (1687) Invented calculus Derived 3 laws of motion F = ma Proposed law of gravity F = Gm1m2/r 2 Explained Kepler’s laws Got headaches (3-body problem)

3-Body Problem

Simplified Solar System

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

Chaotic Double Pendulum

Sensitive Dependence on I. C.

Double Pendulum Simulation 1/4/2018 Double Pendulum Simulation This models a driven pendulum with friction Looks like those taffy machines at the fair Show silly putty

Equations for Double Pendulum 1/4/2018 Equations for Double Pendulum This models a driven pendulum with friction Looks like those taffy machines at the fair Show silly putty

Edward Lorenz (1963) Meteorologist at MIT Had his own personal computer Rediscovered chaos in a simple system of equations: dx/dt = σ(y - x) dy/dt = -xz + rx - y dz/dt = xy - bz 3 variables (x, y, z) 2 nonlinearities (xz, xy) 3 parameters (σ, r, b)

Lorenz Attractor Strange attractor A fractal object Fractal dim ~ 2.05 Butterfly effect

Butterfly Effect

Sensitive Dependence on Init Cond Initial conditions differ by 0.01%

Conditions for Chaos At least 3 variables (to keep the orbits from intersecting) At least one nonlinearity (to keep the orbits bounded) A source of energy (to keep the system going)

Usual Route to Chaos Stable equilibrium (point attractor) Limit cycle (periodic attractor) Period doubling Strange attractor

Period Doubling  Chaos The simplest chaotic flow! Sprott (1997) dx/dt = y dy/dt= z dz/dt= -az + y2 - x

Chaotic Circuit This models a driven pendulum with friction 1/4/2018 Chaotic Circuit This models a driven pendulum with friction Looks like those taffy machines at the fair Show silly putty

Equations for Chaotic Circuit 1/4/2018 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 This models a driven pendulum with friction Looks like those taffy machines at the fair Show silly putty

Bifurcation Diagram for Chaotic Circuit 1/4/2018 Bifurcation Diagram for Chaotic Circuit

Applications for Chaos Secure communications Meteorology Ecology Economics Sociology Psychology Politics Philosophy

References http://sprott.physics.wisc.edu/ lectures/utrecht.pptx (this talk) http://sprott.physics.wisc.edu/chaostsa/ (my chaos textbook) sprott@physics.wisc.edu (contact me)

Questions Who won King Oscar’s Prize? Johannes Kepler Isaac Newton Henri Poincare Edward Lorenz What is the 3-body problem? Biological survival of the fittest The motion of three bodies with mutual attraction The social interactions of three friends The riddle of a multiple homicide What is a nonlinear system? A system in which effects have multiple causes A system whose whole is not equal to the sum of its parts A system that exhibits chaos A system with many variables What is a strange attractor? An object whose shape is unpredictable An object that attracts another dissimilar object An object that attracts another similar object A fractal produced by a chaotic process What is the butterfly effect? Turbulence produced by fluctuating organisms Irregular oscillations of a dynamical system Behavior of a complex system Sensitive dependence on initial conditions