Bifurcations of a nongeneric heteroclinic loop with nonhyperbolic equilibria

Fengjie Geng; Junfang Zhao; Deming Zhu; Weipeng Zhang

doi:10.3934/dcdsb.2013.18.133

Article Contents

2013, Volume 18, Issue 1: 133-145. Doi: 10.3934/dcdsb.2013.18.133

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Bifurcations of a nongeneric heteroclinic loop with nonhyperbolic equilibria

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

Received: March 2011

Revised: July 2012

Published: January 2013

Abstract / Introduction Related Papers Cited by

Abstract

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.

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Mathematics Subject Classification: Primary: 34C23, 37G10; Secondary: 34D09.

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