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Computational and Mathematical Methods in Medicine
Volume 2013, Article ID 582820, 9 pages
http://dx.doi.org/10.1155/2013/582820
Research Article

Bifurcations and Stability of Nondegenerated Homoclinic Loops for Higher Dimensional Systems

1Science College, Linyi University, Linyi, Shandong 276005, China
2School of Mathematics Science, Shandong Normal University, Jinan 250014, China

Received 10 August 2013; Revised 23 September 2013; Accepted 23 September 2013

Academic Editor: Jinde Cao

Copyright © 2013 Yinlai Jin et al. 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

By using the foundational solutions of the linear variational equation of the unperturbed system along the homoclinic orbit as the local current coordinates system of the system in the small neighborhood of the homoclinic orbit, we discuss the bifurcation problems of nondegenerated homoclinic loops. Under the nonresonant condition, existence, uniqueness, and incoexistence of 1-homoclinic loop and 1-periodic orbit, the inexistence of -homoclinic loop and -periodic orbit is obtained. Under the resonant condition, we study the existence of 1-homoclinic loop, 1-periodic orbit, 2-fold 1-periodic orbit, and two 1-periodic orbits; the coexistence of 1-homoclinic loop and 1-periodic orbit. Moreover, we give the corresponding existence fields and bifurcation surfaces. At last, we study the stability of the homoclinic loop for the two cases of non-resonant and resonant, and we obtain the corresponding criterions.