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.
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.
| Original language | English |
|---|---|
| Title of host publication | Proceedings of the 9th European Nonlinear Dynamics Conference |
| Editors | Gábor Stépán, Gábor Csernák |
| Place of Publication | Budapest |
| Publisher | CongressLIne Ltd. |
| Number of pages | 2 |
| ISBN (Electronic) | 978-963-12-9168-1 |
| Publication status | Published - 2017 |
| Event | 9th European Nonlinear Dynamics Conference - Budapest University of Technology end Economics, Budapest, Hungary Duration: 25 Jun 2017 → 30 Jun 2017 Conference number: 9 |
Conference
| Conference | 9th European Nonlinear Dynamics Conference |
|---|---|
| Abbreviated title | ENOC 2017 |
| Country/Territory | Hungary |
| City | Budapest |
| Period | 25/06/17 → 30/06/17 |