Some Remarks on the Melnikov Function
Date
2002-02-07
Authors
Battelli, Flaviano
Feckan, Michal
Journal Title
Journal ISSN
Volume Title
Publisher
Southwest Texas State University, Department of Mathematics
Abstract
We study the Melnikov function associated with a periodic perturbation of a differential equation having a homoclinic orbit. Our main interest is the characterization of perturbations that give rise to vanishing or non-vanishing of the Melnikov function. For this purpose we show that, in some cases, the Fourier coefficients of the Melkinov function can be evaluated by means of the calculus of residues. We apply this result, among other things, to the construction of a second-order equation whose Melnikov function vanishes identically for any C1, 2π-periodic perturbation. Then we study the second order Melnikov function of the perturbed equation, and prove it is non-vanishing for a large class of perturbations.
Description
Keywords
Melnikov function, Residues, Fourier coefficients
Citation
Battelli, F., & Feckan, M. (2002). Some remarks on the Melnikov function. <i>Electronic Journal of Differential Equations, 2002</i>(13), pp. 1-29.
Rights
Attribution 4.0 International