On bifurcations of multidimensional diffeomorphisms having a homoclinic tangency to a saddle-node

TL;DR

This paper investigates the complex bifurcation structures of multidimensional diffeomorphisms near homoclinic tangencies to saddle-node fixed points, providing a bifurcation diagram and relating it to known bifurcation phenomena.

Contribution

It introduces a detailed bifurcation analysis for multidimensional systems with homoclinic tangencies to saddle-nodes, extending existing bifurcation theory.

Findings

Bifurcation diagram for single-round periodic orbits near homoclinic tangency.

Relation of results to classical codimension one bifurcations.

Insights into bifurcation structures around saddle-node fixed points.

Abstract

We study main bifurcations of multidimensional diffeomorphisms having a non-transversal homoclinic orbit to a saddle-node fixed point. On a parameter plane we build a bifurcation diagram for single-round periodic orbits lying entirely in a small neighbourhood of the homoclinic orbit. Also a relation of our results to well-known codimension one bifurcations of a saddle fixed point with a quadratic homoclinic tangency and a saddle-node fixed point with a transversal homoclinic orbit is discussed.