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)
Tycho Brahe
n 3 laws of planetary motion
n Elliptical orbits
n
Newton (1687)
Invented calculus
n Derived 3 laws of motion
F = ma
n Proposed law of gravity
F = Gm1m2/r 2
n Explained Kepler’s laws
n Got headaches (3 body problem)
n
Poincare (1890)
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)
n
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 = md2x/dt2
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 wt
2-D
nonautonomous
3-D
autonomous
dx/dt = v
dv/dt = -sin x – bv + sin z
dz/dt = w
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
n
n
http://sprott.physics.wisc.edu/
lectures/phys311.pptx (this talk)
http://sprott.physics.wisc.edu/chaost
sa/ (my chaos textbook)
sprott@physics.wisc.edu (contact
me)