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New asymptotics of homoclinic orbits near Bogdanov-Takens bifurcation point

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Abstract
We derive explicit asymptotics for the homoclinic orbits near a generic Bogdanov-Takens (BT) point, with the aim to continue the branch of homoclinic solutions that is rooted in the BT point in parameter and state space. We present accurate second-order homoclinic predictor of the homoclinic bifurcation curve using a generalization of the Poincare-Lindstedt (P-L) method. We show that the P-L method leads to the same homoclinicity conditions as the classical Melnikov technique, the branching method and the regular perturbation method (R-P). The R-P method shows a “parasitic turn” near the saddle point. The new asymptotics based on P-L do not have this turn, making it more suitable for numerical implementation. We show how to use these asymptotics to calculate the initial homoclinic cycle to continue homoclinic orbits in two free parameters. The new homoclinic predictors are implemented in the Matlab continuation package MatCont to initialize the continuation of homoclinic orbits from a BT point. Several examples in the case of multidimensional state spaces are included.

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Chicago
Al-Hdaibat, Bashir, Willy Govaerts, Yuri Kuznetsov, and Hil Meijer. 2015. “New Asymptotics of Homoclinic Orbits Near Bogdanov-Takens Bifurcation Point.” In Applications of Dynamical Systems, Abstracts. Society for Industrial and Applied Mathematics (SIAM).
APA
Al-Hdaibat, B., Govaerts, W., Kuznetsov, Y., & Meijer, H. (2015). New asymptotics of homoclinic orbits near Bogdanov-Takens bifurcation point. Applications of Dynamical Systems, Abstracts. Presented at the 2015 SIAM Conference on Applications of Dynamical Systems, Society for Industrial and Applied Mathematics (SIAM).
Vancouver
1.
Al-Hdaibat B, Govaerts W, Kuznetsov Y, Meijer H. New asymptotics of homoclinic orbits near Bogdanov-Takens bifurcation point. Applications of Dynamical Systems, Abstracts. Society for Industrial and Applied Mathematics (SIAM); 2015.
MLA
Al-Hdaibat, Bashir, Willy Govaerts, Yuri Kuznetsov, et al. “New Asymptotics of Homoclinic Orbits Near Bogdanov-Takens Bifurcation Point.” Applications of Dynamical Systems, Abstracts. Society for Industrial and Applied Mathematics (SIAM), 2015. Print.
@inproceedings{5971057,
  abstract     = {We derive explicit asymptotics for the homoclinic orbits near a generic Bogdanov-Takens (BT) point, with the aim to continue the branch of homoclinic solutions that is rooted in the BT point in parameter and state space. We present accurate second-order homoclinic predictor of the homoclinic bifurcation curve using a generalization of the Poincare-Lindstedt (P-L) method. We show that the P-L method leads to the same homoclinicity conditions as the classical Melnikov technique, the branching method and the regular perturbation method (R-P). The R-P method shows a {\textquotedblleft}parasitic turn{\textquotedblright} near the saddle point. The new asymptotics based on P-L do not have this turn, making it more suitable for numerical implementation. We show how to use these asymptotics to calculate the initial homoclinic cycle to continue homoclinic orbits in two free parameters. The new homoclinic predictors are implemented in the Matlab continuation package MatCont to initialize the continuation of homoclinic orbits from a BT point. Several examples in the case of multidimensional state spaces are included.},
  author       = {Al-Hdaibat, Bashir and Govaerts, Willy and Kuznetsov, Yuri and Meijer, Hil},
  booktitle    = {Applications of Dynamical Systems, Abstracts},
  language     = {eng},
  location     = {Snowbird, UT, USA},
  publisher    = {Society for Industrial and Applied Mathematics (SIAM)},
  title        = {New asymptotics of homoclinic orbits near Bogdanov-Takens bifurcation point},
  year         = {2015},
}