Parametrically Driven One-Dimensional Sine-Gordon Equation and Chaos

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Abstract
The chaotic behaviour of the parametrically driven one-dimensional sine-Gordon equation with periodic boundary conditions is studied. The initial condition is u(x, 0) = ƒ(x), ut (x, 0) = 0 where ƒ is the breather solution of the one-dimensional sine-Gordon equation at t = 0. We vary the amplitude of the driving force, the frequency of the driving force and the damping constant. For appropriate values of the driving force, frequency and damping constant chaotic behaviour with respect to the time-evolution of w(x = fixed, t) can be found. The space structure u(t = fixed, x) changes with increasing driving force from a zero mode structure to a breather-like structure consisting of a few modes.

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