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Discrete Dynamics in Nature and Society
Volume 2013, Article ID 732321, 9 pages
Research Article

Homoclinic Bifurcations in Planar Piecewise-Linear Systems

1Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China
2Central University of Finance and Economics, School of Applied Mathematics, Beijing 100084, China

Received 25 January 2013; Accepted 29 August 2013

Academic Editor: Rob Sturman

Copyright © 2013 Bin Xu 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.


The problem of homoclinic bifurcations in planar continuous piecewise-linear systems with two zones is studied. This is accomplished by investigating the existence of homoclinic orbits in the systems. The systems with homoclinic orbits can be divided into two cases: the visible saddle-focus (or saddle-center) case and the case of twofold nodes with opposite stability. Necessary and sufficient conditions for the existence of homoclinic orbits are provided for further study of homoclinic bifurcations. Two kinds of homoclinic bifurcations are discussed: one is generically related to nondegenerate homoclinic orbits; the other is the discontinuity induced homoclinic bifurcations related to the boundary. The results show that at least two parameters are needed to unfold all possible homoclinc bifurcations in the systems.