Abstract
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.
Highlights
Nonsmooth dynamical systems are naturally used to model many physical processes, such as impacting, friction, switching, and sliding systems
Starting with transforming the system into a canonical form, we found that its homoclinic orbits exist only in two cases: one is the saddle-focus system, which is called nondegenerate homoclinic orbits; the other one has two nodes coinciding on the vertical axis which have opposite stability, which is called degenerate homoclinic system
We study the problem of the discontinuity induced homoclinic bifurcations (DIHBs) related to variation of boundary equilibria
Summary
Nonsmooth dynamical systems are naturally used to model many physical processes, such as impacting, friction, switching, and sliding systems. We further study the existence problem of homoclinic orbits and homoclinic bifurcations in the abovementioned planar piecewise-linear systems. Starting with transforming the system into a canonical form, we found that its homoclinic orbits exist only in two cases: one is the saddle-focus (or saddle-center) system, which is called nondegenerate homoclinic orbits; the other one has two nodes coinciding on the vertical axis which have opposite stability, which is called degenerate homoclinic system For both cases, the necessary and sufficient conditions for the existence of homoclinic orbits are established and used for the study of the homoclinic bifurcation problem. We find that there are two kinds of homoclinic bifurcations: Discrete Dynamics in Nature and Society one is generic, that is, the limit equilibria of the homoclinic orbits digress from the vertical axis; the other is nongeneric, that is, the discontinuity induced homoclinic bifurcation.
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