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.
Nonsmooth dynamical systems are naturally used to model many physical processes, such as impacting, friction, switching, and sliding systems. The study of the nonsmooth dynamical systems has attracted more and more attention in the recent decades. Piecewise-smooth systems, as an important branch of the nonsmooth dynamical systems, involve collision systems, Filippov systems, higher-order discontinuity systems, and so forth [
This paper studies the planar piecewise-linear continuous vector fields with two zones. Without loss of generality, the considered plane is divided into two half-planes by the boundary coinciding with the vertical axis. The system is linear in each of the half-planes and continuous along the vertical axis. In 1998, Freire et al. [
In this paper, we further study the existence problem of homoclinic orbits and homoclinic bifurcations in the above-mentioned 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. For the nondegenerate case, we find that there are two kinds of homoclinic bifurcations: 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. However, for the degenerate case, only discontinuity induced homoclinic bifurcation occurs. Finally, we point out that at least two parameters are needed to unfold all possible homoclinic bifurcations in the mentioned systems.
The rest of the paper is outlined as follows. In Section
We start with a planar piecewise-linear system:
A homoclinic orbit has both
Nondegenerate homoclinic orbits. Take Set Set
(a) Nondegenerate homoclinic orbit for the saddle-focus system. (b) Nondegenerate homoclinic orbit for the saddle-center system.
Degenerate homoclinic orbit. In (
Degenerate homoclinic orbits for the node-node system.
From the above examples, we found that if a homoclinic orbit exists then some of the orbits will cross the
In order to study homoclinic orbits, it is convenient to simplify system (
For system (
For (
The canonical form (
The equilibria of system (
Write
The eigenvalues at the equilibria (
The trace-determinant plane.
Lemma
To end this section, we introduce two propositions [
System (
If system (
We focus on conditions for system (
Let (
Without loss of generality, we assume that the limit equilibium of
From Remark
The following theorem gives conditions for the existence of nondegenerate homoclinic orbits.
Let system (
We may assume that the saddle of (
If the nondegenerate homoclinic orbit
In order to find conditions for the existence of such homoclinic orbit
By using the polar coordinate transformation
We will define a half-return map
As shown in Figure
In general, for any
Next we consider the left zone governed by the saddle half-system
From the discussion of (
For the case
A nondegenerate homoclinic orbit.
The right zone in the polar coordinate system.
Note that (
From
In particular, we consider the saddle-center system, where
On the other hand, it is not hard to check that if
It is easy to see that
Let
For
If
For a continuum of homoclinic loops, see Example
Based on the study of homoclinic orbits in the last section, now the homoclinic bifurcations in (
In this subsection, we discuss bifurcations related to the nondegenerate homoclinic orbits, called generic homoclinic bifurcations. For simplicity, we still assume that the left half-system in (
For the saddle-focus system, When When
We mention here that the system has at most one limit cycle by Proposition
A bifurcation of saddle-focus systems is shown as in Figure
Bifurcation of saddle-focus systems.
Now we discuss the saddle-center system case; that is, For For Varying
A bifurcation of saddle-center systems is shown in Figure
: Bifurcation of saddle-center systems.
In order to summarize the above conclusions we let
A bifurcation diagram.
In this subsection, we study the problem of the discontinuity induced homoclinic bifurcations (DIHBs) related to variation of boundary equilibria. Herein we take
We still assume that the equilibrium of (
Following Example
Discontinuity induced homoclinic bifurcation in the nondegenerate case.
In Figure
As mentioned in Section
Following Example
Discontinuity induced homoclinic bifurcation in the degenerate case.
We have studied in detail the homoclinic bifurcations in the planar piecewise-linear systems with two zones. Firstly, the system is transformed to a canonical form. Then both nondegenerate and degenerate homoclinic orbits are investigated. The necessary and sufficient conditions for the existence of both a nondegenerate homoclinic orbit and a continuum of degenerate homoclinic orbits are established for discussion of the homoclinic bifurcation problem. A generic homoclinic bifurcation diagram is shown as in Figure
This work was supported by the NNSF of China (Grant nos. 11072274 and 11272169).