Bifurcations of a nongeneric heteroclinic loop with nonhyperbolic equilibria

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January 2013, 18(1): 133-145. doi: 10.3934/dcdsb.2013.18.133

Bifurcations of a nongeneric heteroclinic loop with nonhyperbolic equilibria

Fengjie Geng ^1, , Junfang Zhao ^1, , Deming Zhu ^2, and Weipeng Zhang ^3,

1.	College of Mathematics and Science, China University of Geosciences(Beijing), Beijing, 100083, China, China
2.	Department of Mathematics, East China Normal University, Shanghai, 200241
3.	Department of Mathematics and Statistics, Northeast Normal University, Changchun, Jilin, 130024, China

Received March 2011 Revised July 2012 Published September 2012

Abstract
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In this paper, using the local moving frame approach, we investigate bifurcations of nongeneric heteroclinic loop with a nonhyperbolic equilibrium $p_1$ and a hyperbolic saddle $p_2$, where $p_1$ is assumed to undergo a transcritical bifurcation. Firstly, we establish the persistence of a nongeneric heteroclinic loop, the existence of a homoclinic loop and a periodic orbit when the transcritical bifurcation does not occur. Secondly, bifurcations of a nongeneric heteroclinic loop accompanied with a transcritical bifurcation are discussed. We obtain the existence of heteroclinic orbits, a homoclinic loop, a heteroclinic loop and a periodic orbit. Some bifurcation patterns different from the case of the generic heteroclinic loop accompanied with transcritical bifurcation are revealed. The results achieved here can be extended to higher dimensional systems.

Keywords: nongeneric heteroclinic loop, nonhyperbolic equilibrium., Poincar$\acute{\mbox{e}}$ map, transcritical bifurcation, Local moving frame.

Mathematics Subject Classification: Primary: 34C23, 37G10; Secondary: 34D0.

Citation: Fengjie Geng, Junfang Zhao, Deming Zhu, Weipeng Zhang. Bifurcations of a nongeneric heteroclinic loop with nonhyperbolic equilibria. Discrete and Continuous Dynamical Systems - B, 2013, 18 (1) : 133-145. doi: 10.3934/dcdsb.2013.18.133

References:

[1]	S. N. Chow, B. Deng and J. M. Friedman, Theory and applicationsof a nongeneric heteroclinic loop bifurcation, SIAM J. Appl. Math., 59 (1999), 1303-1321.
[2]	A. R. Champneys, Codimension-one persistence beyond allorders of homoclinic orbits to singular saddle centres in reversible systems, Nonlinearity, 14 (2001), 87-112.
[3]	F. J. Geng, D. Liu and D. M. Zhu, Bifurcations of generic heteroclinic loop accompanied by transcritical bifurcation, International J. Bifurcation and Chaos, 4 (2008), 1069-1083.
[4]	X. B. Liu, X. L. Fu and D. M. Zhu, Homoclinic Bifurcation with non hyperbolic equilibria, Nonlinear Analysis, 66 (2007), 2931-2939. doi: 10.1016/j.na.2006.04.014.
[5]	X. B. Liu and D. M. Zhu, Homoclinic snaking near a heteroclinic cycles inreversible systems, Appl. Math. J. Chinese Univ. Ser. A (in Chinese), 19 (2004), 401-408.
[6]	J. D. M. Rademacher, Homoclinic orbits near heteroclinic cycles with one equilibrium and one periodic orbit, J. Differential Equations, 218 (2005), 390-443. doi: 10.1016/j.jde.2005.03.016.
[7]	J. H. Sun and D. J. Luo, Local and global bifurcations with nonhyperbolic equilibria, Science in China, Series A, 37 (1994), 523-534.
[8]	S. Wiggins, "Introduction to Applied Nonlinear Dynamical Systems and Chaos," Springer-Verlag, New York, 1990.
[9]	D. M. Zhu and Z. H. Xia, Bifurcations of Morse-Smale dynamical systems, Science in China, Series A, 41 (1998), 837-848.

show all references

References:

[1]	S. N. Chow, B. Deng and J. M. Friedman, Theory and applicationsof a nongeneric heteroclinic loop bifurcation, SIAM J. Appl. Math., 59 (1999), 1303-1321.
[2]	A. R. Champneys, Codimension-one persistence beyond allorders of homoclinic orbits to singular saddle centres in reversible systems, Nonlinearity, 14 (2001), 87-112.
[3]	F. J. Geng, D. Liu and D. M. Zhu, Bifurcations of generic heteroclinic loop accompanied by transcritical bifurcation, International J. Bifurcation and Chaos, 4 (2008), 1069-1083.
[4]	X. B. Liu, X. L. Fu and D. M. Zhu, Homoclinic Bifurcation with non hyperbolic equilibria, Nonlinear Analysis, 66 (2007), 2931-2939. doi: 10.1016/j.na.2006.04.014.
[5]	X. B. Liu and D. M. Zhu, Homoclinic snaking near a heteroclinic cycles inreversible systems, Appl. Math. J. Chinese Univ. Ser. A (in Chinese), 19 (2004), 401-408.
[6]	J. D. M. Rademacher, Homoclinic orbits near heteroclinic cycles with one equilibrium and one periodic orbit, J. Differential Equations, 218 (2005), 390-443. doi: 10.1016/j.jde.2005.03.016.
[7]	J. H. Sun and D. J. Luo, Local and global bifurcations with nonhyperbolic equilibria, Science in China, Series A, 37 (1994), 523-534.
[8]	S. Wiggins, "Introduction to Applied Nonlinear Dynamical Systems and Chaos," Springer-Verlag, New York, 1990.
[9]	D. M. Zhu and Z. H. Xia, Bifurcations of Morse-Smale dynamical systems, Science in China, Series A, 41 (1998), 837-848.

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