Computing Melnikov Curves for Periodically Perturbed Piecewise Smooth Oscillators

Abstract

Curves dividing the parameter plane into regions according to the presence or absence of homoclinic or heteroclinic tangle corresponding to the periodically perturbed saddle of the piecewise smooth oscillator are studied using Melnikov analysis. The analysis is not simplified by choosing the discontinuity plane at a convenient location. Separatrix of the unperturbed system is parametrized exactly in a piecewise manner. Switching times, i.e. parameter values at which the separatrix crosses the discontinuity plane, are obtained. Switching times split the Melnikov integral into various subintegrals which are evaluated either exactly using term-wise integration of the infinite series of the integrand or approximately using a finite-term series approximation of the integrand, the latter being computationally an extensive task. Integral evaluations though approximate, are purely analytical expressions in terms of special functions such as digamma and hypergeometric. Melnikov plots show that the boundary between three regions in the parameter plane differ qualitatively in case of parametric and external excitations, however; adding self-excitation to the external one does not much alter the boundary qualitatively and quantitatively.