TY - JOUR
T1 - Interplay between Normal Forms and Center Manifold Reduction for Homoclinic Predictors near Bogdanov-Takens Bifurcation
AU - Bosschaert, Maikel M.
AU - Kuznetsov, Yuri A.
N1 - Publisher Copyright:
© 2024 Society for Industrial and Applied Mathematics.
PY - 2024
Y1 - 2024
N2 - 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-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 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-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.
AB - 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-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 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-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.
KW - n/a OA procedure
KW - Center manifold reduction
KW - Homoclinic asymptotics
KW - Bogdanov-Takens bifurcation
UR - http://www.scopus.com/inward/record.url?scp=85184064927&partnerID=8YFLogxK
U2 - 10.1137/22M151354X
DO - 10.1137/22M151354X
M3 - Article
AN - SCOPUS:85184064927
SN - 1536-0040
VL - 23
SP - 410
EP - 439
JO - SIAM journal on applied dynamical systems
JF - SIAM journal on applied dynamical systems
IS - 1
ER -