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Improved homoclinic predictor for Bogdanov–Takens bifurcation

Yu A Kuznetsov, HGE Meijer, Bashir Al-Hdaibat (UGent) and Willy Govaerts (UGent)
(2014) INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS. 24(4).
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Abstract
An improved homoclinic predictor at a generic codim 2 Bogdanov–Takens (BT) bifucation is derived. We use the classical "blow-up" technique to reduce the canonical smooth normal form near a generic BT bifurcation to a perturbed Hamiltonian system. With a simple perturbation method, we derive explicit first- and second-order corrections of the unperturbed homoclinic orbit and parameter value. To obtain the normal form on the center manifold, we apply the standard parameter-dependent center manifold reduction combined with the normalization, that is based on the Fredholm solvability of the homological equation. By systematically solving all linear systems appearing from the homological equation, we remove an ambiguity in the parameter transformation existing in the literature. The actual implementation of the improved predictor in MatCont and numerical examples illustrating its efficiency are discussed.
Keywords
FIELD-ORIENTED CONTROL, MatCont, center manifold, homoclinic orbit, Bogdanov–Takens bifurcation, Normal form, INDUCTION-MOTOR DRIVES, MATCONT

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MLA
Kuznetsov, Yu A., et al. “Improved Homoclinic Predictor for Bogdanov–Takens Bifurcation.” INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS, vol. 24, no. 4, 2014, doi:10.1142/S0218127414500576.
APA
Kuznetsov, Y. A., Meijer, H., Al-Hdaibat, B., & Govaerts, W. (2014). Improved homoclinic predictor for Bogdanov–Takens bifurcation. INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS, 24(4). https://doi.org/10.1142/S0218127414500576
Chicago author-date
Kuznetsov, Yu A, HGE Meijer, Bashir Al-Hdaibat, and Willy Govaerts. 2014. “Improved Homoclinic Predictor for Bogdanov–Takens Bifurcation.” INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS 24 (4). https://doi.org/10.1142/S0218127414500576.
Chicago author-date (all authors)
Kuznetsov, Yu A, HGE Meijer, Bashir Al-Hdaibat, and Willy Govaerts. 2014. “Improved Homoclinic Predictor for Bogdanov–Takens Bifurcation.” INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS 24 (4). doi:10.1142/S0218127414500576.
Vancouver
1.
Kuznetsov YA, Meijer H, Al-Hdaibat B, Govaerts W. Improved homoclinic predictor for Bogdanov–Takens bifurcation. INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS. 2014;24(4).
IEEE
[1]
Y. A. Kuznetsov, H. Meijer, B. Al-Hdaibat, and W. Govaerts, “Improved homoclinic predictor for Bogdanov–Takens bifurcation,” INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS, vol. 24, no. 4, 2014.
@article{4387839,
  abstract     = {{An improved homoclinic predictor at a generic codim 2 Bogdanov–Takens (BT) bifucation is derived. We use the classical "blow-up" technique to reduce the canonical smooth normal form near a generic BT bifurcation to a perturbed Hamiltonian system. With a simple perturbation method, we derive explicit first- and second-order corrections of the unperturbed homoclinic orbit and parameter value. To obtain the normal form on the center manifold, we apply the standard parameter-dependent center manifold reduction combined with the normalization, that is based on the Fredholm solvability of the homological equation. By systematically solving all linear systems appearing from the homological equation, we remove an ambiguity in the parameter transformation existing in the literature. The actual implementation of the improved predictor in MatCont and numerical examples illustrating its efficiency are discussed.}},
  articleno    = {{1450057}},
  author       = {{Kuznetsov, Yu A and Meijer, HGE and Al-Hdaibat, Bashir and Govaerts, Willy}},
  issn         = {{0218-1274}},
  journal      = {{INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS}},
  keywords     = {{FIELD-ORIENTED CONTROL,MatCont,center manifold,homoclinic orbit,Bogdanov–Takens bifurcation,Normal form,INDUCTION-MOTOR DRIVES,MATCONT}},
  language     = {{eng}},
  number       = {{4}},
  pages        = {{12}},
  title        = {{Improved homoclinic predictor for Bogdanov–Takens bifurcation}},
  url          = {{http://doi.org/10.1142/S0218127414500576}},
  volume       = {{24}},
  year         = {{2014}},
}
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