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Initialization of homoclinic solutions near Bogdanov-Takens points : Lindstedt-Poincaré compared with regular perturbation method

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Abstract
To continue a branch of homoclinic solutions starting from a Bogdanov--Takens (BT) point in parameter and state space, one needs a predictor based on asymptotics for the bifurcation parameter values and the corresponding small homoclinic orbits in the phase space. We derive two explicit asymptotics for the homoclinic orbits near a generic BT point. A recent generalization of the Lindstedt--Poincaré (L-P) method is applied to approximate a homoclinic solution of a strongly nonlinear autonomous system that results from blowing up the BT normal form. This solution allows us to derive an accurate second-order homoclinic predictor to the homoclinic branch rooted at a generic BT point of an $n$-dimensional ordinary differential equation (ODE). We prove that the method leads to the same homoclinicity conditions as the classical Melnikov technique, the branching method, and the regular perturbation (R-P) method. However, it is known that the R-P method leads to a “parasitic turn” near the saddle point. The new asymptotics based on the L-P method do not have this turn, making them more suitable for numerical implementation. We show how to use these asymptotics to calculate the initial data 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. Two examples with multidimensional state spaces are included.
Keywords
Bogdanov-Takens bifurcation, Lindstedt-Poincaré method, homoclinic orbits, regular perturbation method, MatCont

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Chicago
Al-Hdaibat, Bashir, Willy Govaerts, Yuri A Kuznetsov, and Hil GE Meijer. 2016. “Initialization of Homoclinic Solutions Near Bogdanov-Takens Points : Lindstedt-Poincaré Compared with Regular Perturbation Method.” Siam Journal on Applied Dynamical Systems 15 (2): 952–980.
APA
Al-Hdaibat, B., Govaerts, W., Kuznetsov, Y. A., & Meijer, H. G. (2016). Initialization of homoclinic solutions near Bogdanov-Takens points : Lindstedt-Poincaré compared with regular perturbation method. SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS, 15(2), 952–980.
Vancouver
1.
Al-Hdaibat B, Govaerts W, Kuznetsov YA, Meijer HG. Initialization of homoclinic solutions near Bogdanov-Takens points : Lindstedt-Poincaré compared with regular perturbation method. SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS. 2016;15(2):952–80.
MLA
Al-Hdaibat, Bashir, Willy Govaerts, Yuri A Kuznetsov, et al. “Initialization of Homoclinic Solutions Near Bogdanov-Takens Points : Lindstedt-Poincaré Compared with Regular Perturbation Method.” SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS 15.2 (2016): 952–980. Print.
@article{8023782,
  abstract     = {To continue a branch of homoclinic solutions starting from a Bogdanov--Takens (BT) point in parameter and state space, one needs a predictor based on asymptotics for the bifurcation parameter values and the corresponding small homoclinic orbits in the phase space. We derive two explicit asymptotics for the homoclinic orbits near a generic BT point. A recent generalization of the Lindstedt--Poincar{\'e} (L-P) method is applied to approximate a homoclinic solution of a strongly nonlinear autonomous system that results from blowing up the BT normal form. This solution allows us to derive an accurate second-order homoclinic predictor to the homoclinic branch rooted at a generic BT point of an \$n\$-dimensional ordinary differential equation (ODE). We prove that the method leads to the same homoclinicity conditions as the classical Melnikov technique, the branching method, and the regular perturbation (R-P) method. However, it is known that the R-P method leads to a {\textquotedblleft}parasitic turn{\textquotedblright} near the saddle point. The new asymptotics based on the L-P method do not have this turn, making them more suitable for numerical implementation. We show how to use these asymptotics to calculate the initial data 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. Two examples with multidimensional state spaces are included.},
  author       = {Al-Hdaibat, Bashir and Govaerts, Willy and Kuznetsov, Yuri A and Meijer, Hil GE},
  issn         = {1536-0040},
  journal      = {SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS},
  keyword      = {Bogdanov-Takens bifurcation,Lindstedt-Poincar{\'e} method,homoclinic orbits,regular perturbation method,MatCont},
  language     = {eng},
  number       = {2},
  pages        = {952--980},
  title        = {Initialization of homoclinic solutions near Bogdanov-Takens points : Lindstedt-Poincar{\'e} compared with regular perturbation method},
  url          = {http://dx.doi.org/10.1137/15M1017491},
  volume       = {15},
  year         = {2016},
}

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