Complexity in the bifurcation structure of homoclinic loops to a saddle-focus

Abstract

We report on the study of bifurcations of multi-circuit homoclinic loops in two-parameter families of vector fields in the neighbourhood of a main homoclinic tangency to a saddle-focus with characteristic exponents satisfying the Shil'nikov condition . We prove that one-parameter subfamilies of vector fields transverse to the main homoclinic tangency (1) may be tangent to subfamilies with a triple-circuit homoclinic loop; (2) may have a tangency of an arbitrarily high order to subfamilies with a multi-circuit homoclinic loop. These theorems show the high structural instability of one-parameter subfamilies of vector fields in the neighbourhood of a homoclinic tangency to a Shil'nikov-type saddle-focus. Implications for nonlinear partial differential equations modelling waves in spatially extended systems are briefly discussed.