Bifurcation of Nongeneric Homoclinic Orbit Accompanied by Pitchfork Bifurcation

Fengjie Geng and Song Li

Abstract and Applied Analysis, 2014, vol. 2014, 1-7

Abstract:

The bifurcation of a nongeneric homoclinic orbit (i.e., the orbit comes from the equilibrium along the unstable manifold instead of the center manifold) connecting a nonhyperbolic equilibrium is investigated, and the nonhyperbolic equilibrium undergoes a pitchfork bifurcation. The existence (resp., nonexistence) of a homoclinic orbit and an 1-periodic orbit are established when the pitchfork bifurcation does not happen, while as the nonhyperbolic equilibrium undergoes a pitchfork bifurcation, we obtain the sufficient conditions for the existence of homoclinic orbit and two or three heteroclinic orbits, and so forth. Moreover, we explore the difference between the bifurcation of the nongeneric homoclinic orbit and the generic one.

Date: 2014
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jnlaaa:197914

DOI: 10.1155/2014/197914

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