Analysis of a Shil'nikov Type Homoclinic Bifurcation

doi:10.1007/s10114-018-5236-9

Abstract
Figure/Table
References
Related Citation

Download: PDF (274 KB) HTML (1 KB)
Export: BibTeX | EndNote (RIS)

Abstract The bifurcation associated with a homoclinic orbit to saddle-focus including a pair of pure imaginary eigenvalues is investigated by using related homoclinic bifurcation theory. It is proved that, in a neighborhood of the homoclinic bifurcation value, there are countably infinite saddle-node bifurcation values, period-doubling bifurcation values and double-pulse homoclinic bifurcation values. Also, accompanied by the Hopf bifurcation, the existence of certain homoclinic connections to the periodic orbit is proved.

Key words： Homoclinic bifurcation Hopf bifurcation Poincaré map

Received: 09 April 2015

MR (2010):	34C23
	34C37
	37C29

Fund:Supported by National NSF (Grant Nos. 11371140, 11671114) and Shanghai Key Laboratory of PMMP

	Service

	E-mail this article
	Add to my bookshelf
	Add to citation manager
	E-mail Alert
	RSS
	Articles by authors
	Yan Cong XU
	Xing Bo LIU

Cite this article:

Yan Cong XU,Xing Bo LIU. Analysis of a Shil'nikov Type Homoclinic Bifurcation[J]. Acta Mathematica Sinica, English Series, 2018, 34(5): 901-910.

URL:

http://123.57.41.99/Jwk_sxxb_en/EN/10.1007/s10114-018-5236-9 OR http://123.57.41.99/Jwk_sxxb_en/EN/Y2018/V34/I5/901

[1]	Algaba, A., Merino, M., Rodrguez-Luis, A. J.:Homoclinic connections near a Belykov point in Chua's equation. Int. J. Bifur. Chaos, 15, 1239-1252(2005)
[2]	Algaba, A., Merino, M., Rodrguez-Luis, A. J.:Analysis of a Belykov homoclinic connection with Z2-symmetry. Nonlinear Dynam., 69, 519-529(2012)
[3]	Belykov, L. A.:The bifurcation set in a system with a homoclinic saddle curve. Math. Z., 28, 910-916(1980)
[4]	Belykov, L. A.:Bifurcation of system with homoclinic curve of a saddle-focus with saddle quantity zero. Math. Z., 36, 838-843(1984)
[5]	Champney, A. R., Rodrguez-Luis, A. J.:The non-transverse Sil'nikov-Hopf bifurcation:uncoupling of homoclinic orbits and homoclinic tangencies. Phys. D, 128, 130-158(1999)
[6]	Chen, F. J., Zhou, L. Q.:Strange attractors in a periodically perturbed Lorenz-Like equation. J. Appl. Analysis Comput., 2, 123-132(2013)
[7]	Deng, B., Sakamoto, K.:Sil'nikov-Hopf bifurcations. J. Differential Equations, 119, 1-23(1995)
[8]	Fernández-Sánchez, F., Freire, E., Rodríguez-Luis, A. J.:Analysis of the T-point-Hopf bifurcation. Phys. D, 237, 292-305(2008)
[9]	Glendinning, P., Sparrow, C.:Local and global behavior near homoclinic orbits. J. Stat. Phys., 35, 645-696(1984)
[10]	Han, M. A., Zhu, H. P.:The loop quantities and bifurcations of homoclinic loops. J. Differential Equations, 234, 339-359(2007)
[11]	Hirschberg, P., Knobloch. E.:Sil'nikov-Hopf bifurcations. Phys. D, 62, 202-216(1993)
[12]	Homburg, A. J., Sandstede, B.:Homoclinic and Heteroclinic Bifurcations in Vector Fields; in:Broer, Henk (ed.) et al., Handbook of Dynamical Systems. 3, Amsterdam:Elsevier, 379-524(2010)
[13]	Knobloch, J., Lloyd David, J. B., Sandstede, B., et al.:Isolas of 2-pulse solutions in homoclinic snaking scenarios. J. Dynam. Differential Equations, 23, 93-114(2011)
[14]	Li, J. B., Jiang, L.:Exact solutions and bifurcations of a modulated equation in a discrete nonlinear electrical transmission line (I). Int. J. Bifur. Chaos., 25, 1550016, 11 pp (2015)
[15]	Liu, X. B., Shi, L. N., Zhang, D. M.:Homoclinic flip bifurcation with a nonhyperbolic equilibrium. Nonlinear Dynam., 69, 655-665(2012)
[16]	Shen, J., Lu, K. N.; Zhang, W. N.:Heteroclinic chaotic behavior driven by a Brownian motion. J. Differential Equations, 255, 4185-4225(2013)
[17]	Shilnikov, L. P.:A case of the existence of a countable number of periodic motions. Sov. Math. Dokl., 6, 163-166(1965)
[18]	Shilnikov, L. P.:A contributation to the problem of the structure of an extended neighborhood of a rough equilibrium state of saddle-focus type. Math. Ussr. Sbornik., 10, 91-102(1970)
[19]	Stephen, S., Sourdis, C.:Heteroclinic orbits in slow-fast Hamiltonian systems with slow manifold bifurcations. J. Dynam. Differential Equations., 22, 629-655(2010)
[20]	Wiggins, S.:Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer-Verlag, New York, 1990
[21]	Xu, Y. C., Zhu, D. M., Liu, X. B.:Bifurcations of multiple homoclinics in general dynamical systems. Discrete Contin. Dyn. Syst., 30(3), 945-963(2011)
[22]	Yang, J. M., Xiong, Y. Q., Han, M. A.:Limit cycle bifurcations near a 2-polycycle or double 2-polycycle of planar systems. Nonlinear Anal., 95, 756-773(2014)
[23]	Yang, Q. G., Chen, Y. M.:Complex dynamics in the unified Lorenz-type system. Int. J. Bifur. Chaos., 24, 1450055, 30 pp (2014)
[24]	Zhu, D. M., Wang, F. J.:Global bifurcation in the shil'nikov phenomenon with weak attractivity. Appl. Math. J. Chinese Univ. Ser. A, 3, 256-265(1994)