Simple Driven Conservative Chaotic Flows

J. C. Sprott

Department of Physics, University of Wisconsin, Madison, WI 53706, USA
January 12, 2001
Last revised September 15, 2001

Perhaps the algebraically simplest examples of sinusoidally-driven conservative chaotic flows are of the form:

d2x/dt2 + f(x) = A sin wt

A variant of simulated annealing was used to find the value of w that maximizxes the Lyapunov exponent (base-e)  for A = 1 for various f(x) with the following results:

Equation Lyapunov
d2x/dt2 + sin x = sin 0.50t 0.163
d2x/dt2 + x3 = sin 1.88t 0.097
d2x/dt2 + x5 = sin 2.19t 0.163
d2x/dt2 + x7 = sin 2.32t 0.198
d2x/dt2 + x9 = sin 2.58t 0.230
d2x/dt2 + x11 = sin 2.79t 0.242
d2x/dt2 + x3 - x = sin 1.87t 0.164
d2x/dt2 + x|x|-1/2 = sin 5.57t 0.123
d2x/dt2 + x|x| = sin 1.61t 0.051
d2x/dt2 + x|x|3 = sin 2.00t 0.139
d2x/dt2 + sinh x = sin 1.54t 0.014
d2x/dt2 + tanh x = sin 0.27t 0.009

All cases had initial conditions of dx/dt = x = 0.

Hans Gottlieb (private communication, 14 Dec 2000) has pointed out that the case with f(x) = x3 is especially interesting because it may be the simplest sinusoidally-driven chaotic system.  It is the dissipationless limit of the Ueda oscillator.  It has a single parameter w.  The plot below shows how the Lyapunov exponent for this system varies with w:

A Poincare section in the x-dx/dt plane for wt mod 2pi = 0 is shown below:

More generally, the system

d2x/dt2 + x|x|p-1 = sin wt

has chaotic solutions for a range of p and w.  For example, the case p = 5 appears as follows:

The Lyapunov exponent for arbitrary p and w is shown in a gray scale below for initial conditions dx/dt = x = 0:

Chaos also occurs in nonlinear oscillators driven with square waves rather than sine waves.  For example the system

d2x/dt2 + x3 = sgn(sin wt)

exhibits chaos over a wide range of w as shown below:

This system has the virtue that it may be possible to find an analytic solution.

Ref: J. C. Sprott, Chaos and Time-Series Analysis (Oxford University Press, 2003), pp.192-193.

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