HOMOCLINIC ORBITS BIFURCATIONS OF ONE- AND TWO-DIMENSIONAL MAPS

Abstract

We study the 1D and 2D maps arising in applications. The bifurcation set exhibiting a box-within-a-box structure for the 1D map is discussed from the homoclinic orbits existence point of view. Some theorems on the bifurcations of nonrough Poincaré homoclinic curves are stated for 2D maps. We introduce the transition of the phase pictures from the closed invariant curve to the complicated homoclinic structures. The latter may be interpreted as a version of torus break-down for a corresponding flow of the 3D vector field.