2021/09/26 by Maikel M. Bosschaert, Bosschaert, Maikel M., Yuri A. Kuznetsov +1 · 1 citation
Mathematics · Physics and Astronomy · #34B08 #34B15 #34B40 #34C37 #34E10 #37M20 #65P30 #Advanced Chemical Physics Studies #Dynamical Systems (math.DS) #FOS: Mathematics #Fractional Differential Equations Solutions #Numerical Analysis (math.NA) #Spectroscopy and Quantum Chemical Studies
paper · pdf · doi:10.48550/arxiv.2109.12570
openalex publication_date 2021/09/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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-Poincaré 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-Poincaré) 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.