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Abstract and Applied Analysis
Volume 2014, Article ID 197914, 7 pages
Research Article

Bifurcation of Nongeneric Homoclinic Orbit Accompanied by Pitchfork Bifurcation

School of Science, China University of Geosciences (Beijing), Beijing 100083, China

Received 15 February 2014; Accepted 11 March 2014; Published 7 April 2014

Academic Editor: Yonghui Xia

Copyright © 2014 Fengjie Geng and Song Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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.