Numerical Bifurcation Analysis of Homoclinic Orbits Embedded in One-Dimensional 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 one-dimensional stable and unstable manifolds. The methods are implemented in M at C ont M, 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 us to detect all known codimension 1 and 2 bifurcations in three-dimensional (3D) maps, including tangencies and generalized tangencies. M at C ont M 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 us to compute all homoclinic orbits to saddle points in 3D 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.

Chenciner bubbles and torus break-up in a periodically forced delay differential equation

“…For example, the investigation in [22] of a three-dimensional map, describing an adaptively controlled system, identified the (equivalent of) the bifurcation curves sne and Hopf connecting the boundary curves of folding resonance tongues; more specifically, the continuation software MatContM was used to find in a two-parameter plane curves of Neimark-Sacker bifurcations of periodic points that connect the respective curves of saddle-node bifurcations of periodic points bounding the respective resonance tongue. Very recently, the results in [22] were extended in [40], again with MatContM, to include the remaining bifurcation curves suggested in [4].…”

Homoclinic Points Calculation Method With Particle Swarm Optimization

“…In a chaotic situation, the selection of the initial conditions is a difficult problem because the value of the objective function is sensitive to the initial conditions. Even in three-dimensional (3D) systems, previous work [15] has approached the homoclinic point problem by assuming a two-dimensional (2D) plane as a stable manifold despite it is actually a 3D surface.…”