Interplay between normal forms and center manifold reduction for homoclinic predictors near Bogdanov-Takens bifurcation

Autor:	Bosschaert, Maikel M., Kuznetsov, Yuri A.
Rok vydání:	2021
Předmět:	Mathematics - Numerical Analysis Mathematics - Dynamical Systems 37M20 65P30 34C37 34B08 34B15 34B40 34E10
Druh dokumentu:	Working Paper
Popis:	This paper provides for the first time correct third-order homoclinic predictors in $n$-dimensional ODEs near a generic Bogdanov-Takens bifurcation point. To achieve this, higher-order time approximations to the nonlinear time transformation in the Lindstedt-Poincar\'e 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 needed. A detailed comparison is done between applying different normal forms (smooth and orbital), different phase conditions, and different perturbation methods (regular and Lindstedt-Poincar\'e) 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. Comment: 56 pages
Databáze:	arXiv
Externí odkaz:	http://arxiv.org/abs/2109.12570 Zobrazit plný text záznamu View this record from Arxiv