January 12, 2001

Last revised September 15, 2001

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

*d*^{2}*x*/*dt*^{2} + *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
exponent |

d^{2}x/dt^{2} + sin x =
sin 0.50t |
0.163 |

d^{2}x/dt^{2} + x^{3}
= sin 1.88t |
0.097 |

d^{2}x/dt^{2} + x^{5}
= sin 2.19t |
0.163 |

d^{2}x/dt^{2} + x^{7}
= sin 2.32t |
0.198 |

d^{2}x/dt^{2} + x^{9}
= sin 2.58t |
0.230 |

d^{2}x/dt^{2} + x^{11}
= sin 2.79t |
0.242 |

d^{2}x/dt^{2} + x^{3}
- x = sin 1.87t |
0.164 |

d^{2}x/dt^{2} + x|x|^{-1/2}
= sin 5.57t |
0.123 |

d^{2}x/dt^{2} + x|x|
= sin 1.61t |
0.051 |

d^{2}x/dt^{2} + x|x|^{3}
= sin 2.00t |
0.139 |

d^{2}x/dt^{2} + sinh x
= sin 1.54t |
0.014 |

d^{2}x/dt^{2} + 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*) = *x*^{3} 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

*d*^{2}*x*/*dt*^{2} + *x*|*x| ^{p}*

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

*d*^{2}*x*/*dt*^{2} + *x*^{3}
= 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.