Melnikov's method for non-linear oscillators with non-linear excitations

The response of a non-linear oscillator of the form (x) over tilde + f(A, B, x) = epsilon g(E, mu, w, k, t), where f(A, B, x) is an odd non-linearity and E is small, for A < 0 and B > 0 is considered. The homoclinic orbits for the unperturbed system are obtained by using Jacobian elliptic functions with the generalized harmonic balance method. Also the chaotic limits of this equation are studied with a generalized Melnikov function, M-0(E, mu, (o) over dot, w, k, t(0)), depending on the variable ii. A function R-0(E, mu, w, k) is defined such that there only exists chaotic motion if E/mu > R-0 with k from 0.51 to 0.99. It is demonstrated with Poincare maps in the phase plane that there is good agreement between these predictions and the numerical simulations of the Duffing-Holmes oscillator using the fourth-order Runge-Kutta method of numerical integration. (C) 1998 Academic Press Limited.