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Analysis of a Shil'nikov Type Homoclinic Bifurcation |
Yan Cong XU1, Xing Bo LIU2 |
1. Department of Mathematics, Hangzhou Normal University, Hangzhou 310036, P. R. China; 2. Department of Mathematics, Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai 200241, P. R. China |
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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.
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Received: 09 April 2015
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Fund:Supported by National NSF (Grant Nos. 11371140, 11671114) and Shanghai Key Laboratory of PMMP |
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[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)
|
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