AM  Vol.4 No.2 , February 2013
Resonant Homoclinic Bifurcations with Orbit Flips and Inclination Flips
Author(s) Tiansi Zhang*
ABSTRACT

Homoclinic bifurcation with one orbit flip, two inclination flips and resonance in the tangent directions of homoclinic orbit is considered. By studying the associated successor functions constructed from a local active coordinate system, we prove the existence of double 1-periodic orbit, 1-homoclinic orbit, and also some coexistence conditions of 1-periodic orbit and 1-homoclinic orbit.


Cite this paper
T. Zhang, "Resonant Homoclinic Bifurcations with Orbit Flips and Inclination Flips," Applied Mathematics, Vol. 4 No. 2, 2013, pp. 279-284. doi: 10.4236/am.2013.42042.
References
[1]   A. J. Homburg and B. Krauskopf, “Resonant Homoclinic Flip Bifurcations,” Journal of Dynamics and Differential Equations, Vol. 12, No. 4, 2000, pp. 807-850. doi:10.1023/A:1009046621861

[2]   A. J. Homburg, H. Kokubu and M. Krupa, “The Cusp Horseshoe and Its Bifurcations in the Unfolding of an Inclinication-Flip Homoclinic Orbit,” Ergodic Theory and Dynamical Systems, Vol. 14, No. 4, 1994, pp. 667-693. doi:10.1017/S0143385700008117

[3]   B. E. Oldeman, B. Krauskopf and A. R. Champneys, “Numerical Unfoldings of Codimension-Three Resonant Homoclinic Flip Bifurcations,” Nonlinearity, Vol. 14, No. 3, 2001, pp. 597-621. doi:10.1088/0951-7715/14/3/309

[4]   C. A. Morales and M. J. Pacifico, “Inclination-Flip Homoclinic Orbits Arising from Orbit-Flip,” Nonlinearity, Vol. 14, No. 2, 2001, pp. 379-393. doi:10.1088/0951-7715/14/2/311

[5]   E. Catsigeras and H. Enrich, “Homoclinic Tangencies Near Cascades of Period Doubling Bifurcations,” Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Vol. 15, No. 3, 1998, pp. 255-299. doi:10.1016/S0294-1449(98)80119-4

[6]   H. Kokubu, M. Komuru and H. Oka, “Multiple Homoclinic Bifurcations from Orbit Flip I. Sucessive Homoclinic Doublings,” International Journal of Bifurcation and Chaos, Vol. 6, No. 5, 1996, pp. 833-850. doi:10.1142/S0218127496000461

[7]   J. A. Yorke and K. T. Alligood, “Cascades of Period Doubling Bifurcations: A Prerequisite for Horseshoes,” Bulletin of the American Mathematical Society, Vol. 9, 1983, pp. 319-322. doi:10.1090/S0273-0979-1983-15191-1

[8]   M. V. Shashkov and D. V. Turaev, “An Existence Theorem of Smooth Nonlocal Center Manifolds for Systems Close to a System with a Homoclinic Loop,” Journal of Nonlinear Science, Vol. 9, No. 5, 1999, pp. 525-573. doi:10.1007/s003329900078

[9]   M. Kisaka, H. Kokubu and H. Oka, “Supplement to Homoclinic-Doubling Bifurcation in Vector Fields,” Dynamical Systems, Longman, London, 1993, pp. 92-116.

[10]   M. Kisaka, H. Kokubu and H. Oka, “Bifurcations to NHomoclinic Orbits and N-Periodic Orbits in Vector Fields,” Journal of Dynamics and Differential Equations, Vol. 5, No. 2, 1993, pp. 305-357. doi:10.1007/BF01053164

[11]   F. Geng and D. Zhu, “Bifurcations of Generic Heteroclinic Loop Accompanied by Transcritical Bifurcation,” International Journal of Bifurcation and Chaos, Vol. 18, No. 4, 2008, pp. 1069-1083. doi:10.1142/S0218127408020847

[12]   Q. Lu, Z. Qiao, T. Zhang and D. Zhu, “Heterodimensional Cycle Bifurcation with Orbit-Filp,” International Journal of Bifurcation and Chaos, Vol. 20, No. 2, 2010, pp. 491-508. doi:10.1142/S0218127410025569

[13]   X. Liu, “Homoclinic Flip Bifurcations Accompanied by Transcritical Bifurcation,” Chinese Annals of Mathematics, Series B, Vol. 32, No. 6, 2011, pp. 905-916. doi:10.1007/s11401-011-0675-y

[14]   T. Zhang and D. Zhu, “Homoclinic Bifurcation of Orbit Flip with Resonant Principal Eigenvalues,” Acta Mathematica Sinica, Vol. 22, No. 3, 2006, pp. 855-864.

[15]   T. Zhang and D. Zhu, “Bifurcations of Homoclinic Orbit Connecting Two Nonleading Eigendirections,” International Journal of Bifurcation and Chaos, Vol. 17, No. 3, 2007, pp. 823-836. doi:10.1142/S0218127407017574

 
 
Top