Chaos of the Relativistic Parametrically Forced van der Pol Oscillator

Details Chaos of the Relativistic Parametrically Forced van der Pol Oscillator Chaos of the Relativistic Parametrically Forced van der Pol Oscillator

1997-10-15

arxiv.org/abs/chao-dyn/9710010v2

Phys. Lett. A 243 (1998) 195-204

Physics

High Energy Physics

High Energy Physics - Theory

8 Latex pages including 12 figures. To be published in Phys. Lett. A

Scientific paper

10.1016/S0375-9601(98)00240-0

A manifestly relativistically covariant form of the van der Pol oscillator in 1+1 dimensions is studied. We show that the driven relativistic equations, for which $x$ and $t$ are coupled, relax very quickly to a pair of identical decoupled equations, due to a rapid vanishing of the ``angular momentum'' (the boost in 1+1 dimensions). A similar effect occurs in the damped driven covariant Duffing oscillator previously treated. This effect is an example of entrainment, or synchronization (phase locking), of coupled chaotic systems. The Lyapunov exponents are calculated using the very efficient method of Habib and Ryne. We show a Poincar\'e map that demonstrates this effect and maintains remarkable stability in spite of the inevitable accumulation of computer error in the chaotic region. For our choice of parameters, the positive Lyapunov exponent is about 0.242 almost independently of the integration method.

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