Numerical Bifurcation Analysis of Homoclinic Orbits Embedded in One-Dimensional Manifolds of Maps
| Autor: | Niels Neirynck, Willy Govaerts, Yuri A. Kuznetsov, Hil Gaétan Ellart Meijer |
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| Jazyk: | angličtina |
| Rok vydání: | 2018 |
| Předmět: |
Computer science
Applied Mathematics Mathematical analysis 010103 numerical & computational mathematics Codimension Fixed point Bifurcation diagram 01 natural sciences 22/4 OA procedure Manifold 010305 fluids & plasmas Bifurcation theory Saddle point 0103 physical sciences Homoclinic orbit 0101 mathematics Software Bifurcation |
| Zdroj: | ACM transactions on mathematical software, 44(3):25. Association for Computing Machinery |
| ISSN: | 0098-3500 |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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