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Title: | Interplay between Normal Forms and Center Manifold Reduction for Homoclinic Predictors near Bogdanov-Takens Bifurcation | Authors: | BOSSCHAERT, Maikel Kuznetsov, Yuri A. |
Issue Date: | 2024 | Publisher: | SIAM PUBLICATIONS | Source: | SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS, 23 (1) , p. 410 -439 | Abstract: | This paper provides for the first time correct third-order homoclinic predictors in n-dimensional ODEs near a generic Bogdanov-Takens bifurcation point, which can be used to start the numerical continuation of the appearing homoclinic orbits. To achieve this, higher-order time approximations to the nonlinear time transformation in the Lindstedt--Poincare'\ method are essential. Moreover, a correct transform between approximations to solutions in the normal form and approximations to solutions on the parameter -dependent center manifold is derived rigorously. A detailed comparison is done between applying different normal forms (smooth and orbital), different phase conditions, and different perturbation methods (regular and Lindstedt--Poincare'\) to approximate the homoclinic solution near Bogdanov-Takens points. Examples demonstrating the correctness of the predictors are given. The new homoclinic predictors are implemented in the open -source MATLAB/GNU Octave continuation package MatCont. | Notes: | Bosschaert, MM (corresponding author), Hasselt Univ, Dept Math, B-3590 Diepenbeek, Belgium. maikel.bosschaert@uhasselt.be; i.a.kouznetsov@uu.nl |
Keywords: | Bogdanov-Takens bifurcation;homoclinic asymptotics;center manifold reduction | Document URI: | http://hdl.handle.net/1942/42694 | ISSN: | 1536-0040 | DOI: | 10.1137/22M151354X | ISI #: | 001171420800002 | Category: | A1 | Type: | Journal Contribution |
Appears in Collections: | Research publications |
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