A practical use of the Melnikov homoclinic method

Abstract

Using cutoff functions and periodic extensions, we prove that the Melnikov homoclinic method gives a criterium to show that for a finite time interval [−T,T], with T arbitrarily large, the perturbed system is conjugated to a chaotic one for quite general classes of perturbation functions. The method is applied to specific perturbations of the pendulum and of the Gylden’s problem.

Chaotic Motions of the Duffing-Van der Pol Oscillator with External and Parametric Excitations

“…Recently, chaotic motions of many systems, for example, 6 -Rayleigh oscillator [18], Duffing oscillator [19], Gylden's problem [20], and nonsmooth systems [21], have been investigated by the Melnikov method. In this section, we use the Melnikov method to investigate the chaotic motions of system (4).…”