Multiple Families of Bounded Solutions Near Perturbed Homoclinic Orbits, Application to a Nonlinear Wave Equation

Abstract

We consider a planar functional differential equationwith a perturbed homoclinic orbit. We apply the exponential dichotomy of the variational equation and find a bifurcation function for the corresponding homoclinic bifurcation. By using the Lyapunov– Schmidt reduction method and the Malgrange preparation theorem, we find the roots of the bifurcation function and provide new sufficient conditions for the existence of bounded solutions near the homoclinic orbit.We show that the perturbed system may have multiple families of bounded solutions with chaotic motions bifurcating from the homoclinic orbit. In particular, we find three distinct families of bounded solutions. As far as we know, this is rare for planar systems (see part (iii) of theorem 1.2). Finally, as a numerical simulation, we apply the results to a perturbed wave equation and find a bounded solution near a perturbed homoclinic orbit. The results are more accessible for application in comparison to the former results, and they can be applied to various PDEs and RDEs. In addition, they do not have the limitations of previous results.