Homoclinic Bifurcations in Ordinary and Partial Differential Equations

Abstract

The formal analysis of bifurcations from homoclinic orbits in low-dimensional ordinary differential equations is here extended to deal with ordinary differential equations in n dimensions, and to certain partial differential equations in one space variable on the infinite real axis. For ordinary differential equations, results are equivalent to various cases treated by Shil’nikov: depending on the eigenvalues at the fixed point, an infinite number of periodic orbits can bifurcate at the critical parameter value. By contrast, homoclinic bifurcations for partial differential equations can produce an infinite number of quasi-periodic (modulated travelling wave) solutions.