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The Scientific World Journal
Volume 2014, Article ID 585609, 9 pages
http://dx.doi.org/10.1155/2014/585609
Research Article

Bifurcations of Orbit and Inclination Flips Heteroclinic Loop with Nonhyperbolic Equilibria

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

Received 10 October 2013; Accepted 16 January 2014; Published 23 March 2014

Academic Editors: M. Han, Z. Jin, and Y. Xia

Copyright © 2014 Fengjie Geng and Junfang Zhao. 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.

Abstract

The bifurcations of heteroclinic loop with one nonhyperbolic equilibrium and one hyperbolic saddle are considered, where the nonhyperbolic equilibrium is supposed to undergo a transcritical bifurcation; moreover, the heteroclinic loop has an orbit flip and an inclination flip. When the nonhyperbolic equilibrium does not undergo a transcritical bifurcation, we establish the coexistence and noncoexistence of the periodic orbits and homoclinic orbits. While the nonhyperbolic equilibrium undergoes the transcritical bifurcation, we obtain the noncoexistence of the periodic orbits and homoclinic orbits and the existence of two or three heteroclinic orbits.