Homoclinic orbits embedded in one-dimensional invariant manifolds of maps

Abstract

We describe new methods for initializing the computation of homoclinic orbits for maps in a state space with arbitrary dimension and for detecting their bifurcations. The initialization methods build on known and improved methods for computing onedimensional stable and unstable manifolds. The methods are implemented in MatcontM, a freely available toolbox in Matlab for numerical analysis of bifurcations of fixed points, periodic orbits and connecting orbits of smooth nonlinear maps. The bifurcation analysis of homoclinic connections under variation of one parameter is based on continuation methods and allows to detect all known codimension 1 and 2 bifurcations in 3D maps, including tangencies and generalized tangencies. MatcontM provides a graphical user interface, enabling interactive control for all computations. As the prime new feature we discuss an algorithm for initializing connecting orbits in the important special case where either the stable or unstable manifold is one-dimensional, allowing to compute all homoclinic orbits to saddle points in three-dimensional maps. We illustrate this algorithm in the study of the adaptive control map, a 3D map introduced in 1991 by Frouzakis, Adomaitis and Kevrekidis, to obtain a rather complete bifurcation diagram of the resonance horn in a 1:5 Neimark-Sacker bifurcation point, revealing new features.