Homoclinic Bifurcations in Planar Piecewise-Linear Systems

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.

1. Introduction

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 [1, 2]. In particular, the study of piecewise-linear systems is significantly important because it not only describes some processes such as circuits [3, 4], but also enables us to locally understand the bifurcation phenomena in the nonlinear systems [5].

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. [6] studied discontinuous bifurcations, proved that there exists at most one limit cycle, and proved that if the limit cycle exists then it is either attracting or repelling, which solved the problem proposed in 1991 by Lum and Chua [3]. di Bernardo et al. [1] studied discontinuity induced bifurcations, including boundary equilibrium bifurcations and grazing bifurcations of limit cycles. Concerning the limit cycle problem, Simpson and Meiss [7] investigated the Hopf branch in this kind of systems and mentioned homoclinic loops in [5]. One can refer to [8, 9] for the periodic orbit problems in piecewise-linear systems with multiple nonsmooth boundaries. Bifurcation of limit cycles by perturbing a piecewise-linear Hamiltonian system is considered in [10, 11]. For the study of bifurcations in piecewise-linear systems with some application problems, see also [12].

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 2, we give a brief overview of the concerned system and transform the system into a canonical form. In Section 3, the existence conditions of homoclinic orbits for both nondegenerate and degenerate cases are provided. Then we study the nondegenerate homoclinic bifurcations and the discontinuity induced homoclinic bifurcations in Section 4. Conclusions are drawn in Section 5.

2. Canonical Forms

We start with a planar piecewise-linear system: ̇𝐀𝐱=^𝐿𝐱+𝐛_𝐿,𝑥₁𝐀≤0_𝑅𝐱+𝐛_𝑅,𝑥₁>0, (1) where 𝐱=(𝑥₁,𝑥₂)^𝑇,𝐛_𝐿,𝐛_𝑅∈ℝ², and 𝐀_𝐿,𝐀_𝑅 are 2×2 matrices. By the requirement of continuity on the 𝑥₂-axis, we may assume that the system (1) is of the form ̇𝐀𝐱=^𝐿𝐱+𝐛,𝑥₁𝐀≤0_𝑅𝐱+𝐛,𝑥₁>0, (2) where 𝐀_𝐿=𝑎_𝐿𝑏𝑐_𝐿𝑑,𝐀_𝑅=𝑎_𝑅𝑏𝑐_𝑅𝑑𝑒𝑓,𝐛=. (3) Here we call 𝐀_𝐿𝐱+𝐛(𝐀_𝑅𝐱+𝐛) the left (right) half-system of (2) and the region of the plane 𝐷_𝐿∶𝑥₁<0𝐷_𝑅∶𝑥₁>0 (4) to be the left (right) zone. Suppose that det𝐀_𝑖≠0,𝑖=𝐿,𝑅. An equilibrium 𝐱^∗=(𝑥^∗₁,𝑥^∗₂)^𝑇 of (2), which is of course isolated, is called visible if it is an equilibrium of the left (right) half-system and 𝑥^∗₁≤0 (𝑥^∗₁≥0); otherwise, it is invisible. Let 𝜇=𝑏𝑓−𝑑𝑒. (5) Then it is easy to check that 𝜇=0 if and only if the equilibria of (2) both coincide on the 𝑥₂-axis. We mention that if the equilibrium of one half-system in (2) lies on the 𝑥₂-axis, then by continuity it is also the equilibrium of the other half-system.

A homoclinic orbit has both 𝛼-limit and 𝜔-limit set consisting of one equilibrium, called its limit equilibrium, that with the orbit constitutes the homoclinic loop. A homoclinic orbit is called degenerate if its limit equilibrium lies on the 𝑥₂-axis; otherwise, it is nondegenerate. The following examples for homoclinic orbits describe the two cases.

Example 1. Nondegenerate homoclinic orbits. Take 𝑏=1,𝑑=0,𝑒=0, and 𝑓=1 in (3).(a)Set 𝑎_𝐿=−0.6333,𝑐_𝐿=0.3667,𝑎_𝑅=0.5, and 𝑐_𝑅=−2. Then (2) is a saddle-focus system with the nondegenerate homoclinic orbit as shown in Figure 1(a).(b)Set 𝑎_𝐿=0,𝑐_𝐿=1,𝑎_𝑅=0, and 𝑐_𝑅=−1. Then (2) is a saddle-center system with the nondegenerate homoclinic orbit as shown in Figure 1(b).

(a)

(b)

Figure 1 

(a) Nondegenerate homoclinic orbit for the saddle-focus system. (b) Nondegenerate homoclinic orbit for the saddle-center system.

Example 2. Degenerate homoclinic orbit. In (3), we set 𝑏=1,𝑑=𝑒=𝑓=0,𝑎_𝐿𝑐=4,_𝐿=−3,𝑎_𝑅=−6,𝑐_𝑅=−3. (6) Then (2) is a node-node system with degenerate homoclinic orbits as shown in Figure 2.

Figure 2 

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 𝑥₂-axis from left to right and some from right to left. That means that the sign of 𝑥̇₁=𝑏𝑥₂+𝑒 for the vector field (2) restricted on the 𝑥₂-axis will vary with 𝑥₂. So we should assume that 𝑏≠0. We mention in passing that the existence of a limit cycle for (2) also implies 𝑏≠0 for the same reason. See also [6].

In order to study homoclinic orbits, it is convenient to simplify system (2) to a canonical form.

Lemma 3. For system (2), let 𝑏≠0. Then there exists a linear transformation such that (2) becomes the following canonical system: ̇𝐟𝐳=^(𝐿)(𝐳)=𝐀_𝐿𝐳+𝜇𝐞₂𝐟,x≤0_(𝑅)(𝐳)=𝐀_𝑅𝐳+𝜇𝐞₂,𝑥>0, (7) where 𝑥𝑦𝐳=,𝐞₂=01,𝐀_𝑖=𝜏_𝑖1−𝛿_𝑖0,𝑖=𝐿,𝑅. (8)𝜏_𝑖 and 𝛿_𝑖(≠0) are, respectively, the trace and determinant of the matrixes 𝐀_𝑖,𝑖=𝐿,𝑅, and 𝜇 is given by (5), that is, 𝜇=𝑏𝑓−𝑑𝑒.

Proof. For (2), 𝑥̇₁=𝑎_𝑖𝑥₁+𝑏𝑥₂𝑥̇+𝑒,₂=𝑐_𝑖𝑥₁+𝑑𝑥₂+𝑓,𝑖=𝐿,𝑅. (9) Set the linear transformation 𝑥=𝑥₁,𝑦=−𝑑𝑥₁+𝑏𝑥₂+𝑒. (10) Equation (9) then becomes 𝑥̇=𝑎_𝑖+𝑑𝑥+𝑦=𝜏_𝑖̇𝑥+𝑦,𝑦=−𝑎_𝑖𝑑+𝑐_𝑖𝑏𝑥+𝑏𝑓−𝑑𝑒=−𝛿_𝑖𝑥+𝜇,𝑖=𝐿,𝑅. (11)

Remark 4. The canonical form (7) was mentioned in Simpson [5] by introducing the observability concept. Here we only need the condition 𝑏≠0 instead.

Remark 5. The equilibria of system (7) are 𝐳^∗_𝑖=𝑥^∗_𝑖𝑦^∗_𝑖=𝜇𝛿_𝑖1−𝜏_𝑖,𝑖=𝐿,𝑅. (12) From (7), it follows that 𝐳^∗_𝐿 (𝐳^∗_𝑅) is visible if and only if 𝜇𝛿_𝐿≤0𝜇𝛿_𝑅≥0. (13) In particular, 𝜇=0 if and only if both equilibria of (7) coincide at the origin.

Remark 6. Write Σ=0,𝑦^𝑇∶𝑦∈ℝ,Σ⁺=0,𝑦^𝑇,Σ∶𝑦>0₋=0,𝑦^𝑇∶𝑦<0. (14) We see that, on the 𝑦-axis, Σ, the first component of the vector field (7), becomes 𝑥̇=𝑦. So any orbit of (7) crossing Σ⁺ is from left to right, and crossing Σ⁻ is from right to left. Moreover, for 𝜇≠0, the orbit crossing the origin is tangent to the 𝑦-axis Σ at the origin.

Remark 7. The eigenvalues at the equilibria (12) are 𝜆_1,2=𝜏_𝑖2±14𝜏²_𝑖−𝛿_𝑖,𝑖=𝐿,𝑅. (15) We may then classify each half-system of (7) by 𝜏_𝑖 and 𝛿_𝑖 in the trace-determinant plane as shown in Figure 3. There are 5 types of linear systems: saddle (S), attracting node (AN), repelling node (RN), attracting focus (AF), and repelling focus (RF).

Figure 3 

The trace-determinant plane.

Remark 8. Lemma 3 can be promoted to high-dimensional case; see [5].

To end this section, we introduce two propositions [6].

Proposition 9. System (7) has at most one limit cycle or homoclinic orbit. It is either attracting or repelling.

Proposition 10. If system (7) has a limit cycle or a homoclinic loop, then 𝜏_𝐿𝜏_𝑅≤0.

3. Existence of Homoclinic Orbits

We focus on conditions for system (7) to have homoclinic orbits in this section. The homoclinic orbits can be divided into two classes: nondegenerate homoclinic orbits for visible saddle-focus (or saddle-center) systems and degenerate homoclinic orbits for twofold node systems.

3.1. Nondegenerate Case

Theorem 11. Let (7) have a nondegenerate homoclinic orbit Γ. Then (7) is a saddle-focus (or saddle-center) system with a visible saddle in one zone and a visible focus (or center) in the other zone.

Proof. Without loss of generality, we assume that the limit equilibium of Γ is in the left zone, denoted by 𝐳^∗(𝐿). Due to piecewise-linearity of (7), the orbit Γ starting from 𝐳^∗(𝐿) should cross the positive 𝑦-axis Σ⁺ to enter the right zone and then back to the original 𝐳^∗(𝐿). So 𝐳^∗(𝐿) is a visible saddle in the left zone. By the generalized Poincaré-Bendixon theorem [13], inside the homoclinic loop of Γ there is one equilibrium 𝐳^∗(𝑅), which is a visible focus (or center) in the right zone. So (7) is a saddle-focus (or saddle-center) system.

Remark 12. From Remark 5, it easy to see that the existence of nondegenerate homoclinic orbits implies 𝜇≠0. Moreover, the saddle of (7) as the limit equilibrium of Γ is in the left zone for 𝜇>0 and in the right zone for 𝜇<0.

The following theorem gives conditions for the existence of nondegenerate homoclinic orbits.

Theorem 13. Let system (7) be a saddle-focus (or saddle-center) system with a visible saddle in one zone and a visible focus (or center) in the other zone. Denote by 𝜆₁>0>𝜆₂ the eigenvalues of the saddle half-system and by 𝛼±𝑖𝛽,𝛽>0, the eigenvalues of the focus (or center) half-system. Write 1Λ=2𝜆ln²²𝜆²₁−2𝛼𝜆₁+𝛼²+𝛽²𝜆²₁𝜆²₂−2𝛼𝜆₂+𝛼²+𝛽²−𝛼𝛽𝜆2𝜋−arctan²𝛼−𝛼²+𝛽²𝜆₂𝛽𝛼−arctan²+𝛽²−𝜆₁𝛼𝜆₁𝛽. (16) Then there exists a nondegenerate homoclinic orbit if and only if Λ=0. (17)

Proof. We may assume that the saddle of (7) is in the left zone and the focus (or center) is in the right zone, denoted by 𝐳^∗(𝐿) and 𝐳^∗(𝑅), respectively. So one has 𝛿_𝐿<0,𝛿_𝑅>14𝜏²_𝑅,𝜇>0. (18) By using (15), the eigenvalues and eigenvectors of the saddle half-system 𝐀_𝐿 are, respectively, 𝜆_1,2=𝜏_𝐿2±14𝜏²_𝐿−𝛿_𝐿,𝜐_1,2=1−𝜏_𝐿2±14𝜏²_𝐿−𝛿_𝐿=1−𝜆_2,1, (19)(20) and the eigenvalues of the focus (or center) half-system 𝐀_𝑅 are 𝛼±𝑖𝛽, where 𝜏𝛼=^𝑅2,𝛽=𝛿_𝑅−14𝜏²_𝑅. (21)
If the nondegenerate homoclinic orbit Γ exists, Γ will start with the unstable manifold of the saddle 𝐳^∗(𝐿) to intersect Σ⁺ at a point, say, 𝐶₁=(0,𝑐₁)^𝑇,𝑐₁>0, then along an orbit of the right half focus (or center) system in the right zone to intersect Σ⁻ at another point, say, 𝐶₂=(0,𝑐₂)^𝑇,𝑐₂<0, and will finally return along the stable manifold of 𝐳^∗(𝐿) in the left zone, as shown in Figure 4.
In order to find conditions for the existence of such homoclinic orbit Γ, we consider the right zone governed by the focus (or center) half-system 𝐟^𝑅(𝐳) in (7). Firstly, we transform the right half-system by applying the coordinate transformation 𝑥𝑦=𝑇⁻¹𝑥𝑦, (22) where 1𝑇=𝛽10−𝛼𝛽. (23) Note that (22) is 𝑥=𝛽𝑥,𝑦=𝛼𝑥+𝑦, (24) so (22) is 𝑦-axis invariant, and the right half-system of (7) becomes ̇𝑥̇𝑦=𝑇⁻¹0𝜇+𝑇⁻¹𝜏_𝑅1−𝛿_𝑅0𝑇𝑇⁻¹𝑥𝑦=0𝜇+𝑥𝑦.𝛼𝛽−𝛽𝛼 (25) The equilibrium is 𝑧^∗(𝑅)=𝑥₀𝑦₀=𝛽𝜇𝛼²+𝛽²−𝛼𝜇𝛼²+𝛽². (26)
By using the polar coordinate transformation 𝑟=𝑥−𝑥₀²+𝑦−𝑦₀²,𝜃=arctan𝑦−𝑦₀𝑥−𝑥₀. (27) Equation (25) then becomes ̇̇𝑟=𝛼𝑟,𝜃=−𝛽. (28) Rewrite (28) as 𝑑𝑟𝛼𝑑𝜃=−𝛽𝑟, (29) which has the solution −𝛼𝛽𝜃(𝑡)=ln𝑟(𝑡)+𝛾, (30) where 𝛾 is a constant.
We will define a half-return map 𝒫 from Σ⁺ to Σ⁻ by 𝒫𝑦₁=𝑦₂ (31) such that for 𝑌₁=(0,𝑦₁)∈Σ⁺ there exists an orbit from 𝑌₁=(0,𝑦₁) to 𝑌₂=(0,𝑦₂)∈Σ⁻ within the right zone. In fact, we let the points 𝑌₁ and 𝑌₂ in the polar coordinate (27) be given by (𝑟(𝑡₁)𝜃(𝑡₁)) and (𝑟(𝑡₂),𝜃(𝑡₂)), respectively. From (30), it follows that 𝑟𝑡ln²𝑟𝑡₁=𝛼𝛽𝜃𝑡₁−𝜃𝑡₂. (32)
As shown in Figure 5, one has 𝑟₁=𝑟𝑡₁,𝑟₂=𝑟𝑡₂;𝜋−𝜃₁=𝜃𝑡₁,−𝜋+𝜃₂=𝜃𝑡₂, (33) where 𝑟₁=𝑥²₀+𝑦₁−𝑦₀²,𝑟₂=𝑥²₀+𝑦₀−𝑦₂²,tan𝜃₁=𝑦₁−𝑦₀𝑥₀,tan𝜃₂=𝑦₀−𝑦₂𝑥₀. (34) Then (32) becomes 𝑟ln²𝑟₁𝛼=−𝛽𝜃₁+𝜃₂−2𝜋. (35)
In general, for any 𝑦₁>𝑦₀>𝑦₂, let (0,𝑦₁) and (0,𝑦₂) be still given by (𝑟₁,𝜃₁) and (𝑟₂,𝜃₂), respectively, as shown in Figure 5. We mention that (𝑟₁,𝜃₁) and (𝑟₂,𝜃₂) are determined uniquely by 𝑦₁ and 𝑦₂ due to 0<𝜃₁,𝜃₂<𝜋/2. Write Λ=Λ𝑦₁,𝑦₂𝑟;𝜉=ln²𝑟₁+𝛼𝛽𝜃₁+𝜃₂−2𝜋, (36) where the parameter 𝜉=(𝛼,𝛽). If 𝛼 and 𝛽 are fixed, we often omit the parameter 𝜉. From the above discussion, we see that Λ𝑦₁,𝑦₂=0 (37) if and only if (31) holds. That means that Λ𝑦₁,𝒫𝑦₁=0. (38) We point out that, by using (34), formula (36) becomes 1Λ=2𝑥ln²⁰+𝑦₀−𝑦₂²𝑥²₀+𝑦₁−𝑦₀²+𝛼𝛽𝑦arctan¹−𝑦₀𝑥₀𝑦+arctan⁰−𝑦₂𝑥₀−2𝜋. (39)
Next we consider the left zone governed by the saddle half-system 𝐟^𝐿(𝐳) in (7). If the nondegenerate homoclinic orbit Γ exists, then, by using (20), it is not hard to check that the intersection points 𝐶₁=(0,𝑐₁)^𝑇 and 𝐶₂=(0,𝑐₂)^𝑇 are given by 𝑐_1,2=𝜇𝛿_𝐿−𝜏_𝐿2∓14𝜏²_𝐿−𝛿_𝐿=−𝜇𝜆_1,2𝛿_𝐿=−𝜇𝜆_1,2𝜆₁𝜆₂𝜇=−𝜆_2,1. (40) So 𝐶₁∈Σ⁺ and 𝐶₂∈Σ⁻. We mention that, from (24), one has 𝐶_1,2=𝐶_1,2.
From the discussion of (37) and (38), it follows that the existence of the homoclinic orbit Γ is equivalent to 𝒫(𝑐₁)=𝑐₂. Substituting for (39) by 𝑦₁=𝑐₁ and 𝑦₂=𝑐₂, we have 1Λ=2𝑥ln²⁰+𝑦₀−𝑐₂²𝑥²₀+𝑐₁−𝑦₀²+𝛼𝛽𝑐arctan¹−𝑦₀𝑥₀𝑦+arctan⁰−𝑐₂𝑥₀−2𝜋. (41) By using (26), (34), and (40), a further simplification yields Λ𝑐₁,𝑐₂=1;𝛼,𝛽2𝜆ln²²𝜆²₁−2𝛼𝜆₁+𝛼²+𝛽²𝜆²₁𝜆²₂−2𝛼𝜆₂+𝛼²+𝛽²−𝛼𝛽𝜆2𝜋−arctan²𝛼−𝛼²+𝛽²𝜆₂𝛽𝛼−arctan²+𝛽²−𝜆₁𝛼𝜆₁𝛽, (42) which is just formula (16). The result is then obtained.
For the case 𝜇<0, the conclusion is similar; just exchange 𝐿 and 𝑅.

Figure 4 

A nondegenerate homoclinic orbit.

Figure 5 

The right zone in the polar coordinate system.

Remark 14. Note that (42) is independent of 𝜇(≠0), so we may directly set 𝜇=1. In fact, from (12), (26), and (40), we find that 𝜇(>0) only affects the size of trajectories.

Remark 15. From 0<𝜃₁+𝜃₂<𝜋, in (36), we find in the second term of (42) that 𝜆𝜋<2𝜋−arctan²𝛼−𝛼²+𝛽²𝜆₂𝛽𝛼−arctan²+𝛽²−𝜆₁𝛼𝜆₁𝛽<2𝜋. (43) On the other hand, in the first term of (42), one has Δ≡𝜆²₂𝜆²₁−2𝛼𝜆₁+𝛼²+𝛽²−𝜆²₁𝜆²₂−2𝛼𝜆₂+𝛼²+𝛽²=𝜆₁−𝜆₂2𝛼𝜆₁𝜆₂−𝜏_𝐿𝛼²+𝛽², (44) where 𝜏_𝐿=𝜆₁+𝜆₂. From (43) and (44), we find that the existence of a nondegenerate homoclinic loop implies 𝜏_𝐿𝜏_𝑅<0, as stated in Proposition 10.

Remark 16. In particular, we consider the saddle-center system, where 𝛼=0. From (42), one has 1Λ=2𝜆ln²²𝜆²₁+𝛽²𝜆²₁𝜆²₂+𝛽². (45) From Δ=𝜆²₂𝜆²₁+𝛽²−𝜆²₁𝜆²₂+𝛽²=𝜏_𝐿𝜆₂−𝜆₁𝛽², (46) it is easy to see that Λ<0,=0 and >0 are equivalent to 𝜏_𝐿<0,=0 and >0, respectively.
On the other hand, it is not hard to check that if Λ=0 and 𝜏_𝐿=0, then 𝛼=0. In this case, all the trajectories inside the homoclinic loop are closed around the center.

Remark 17. It is easy to see that 𝑦₂=𝒫(𝑦₁) is a decreasing function of 𝑦₁. From (38), Λ(𝑦₁,𝒫(𝑦₁))=0, we have 𝜕Λ𝜕𝑦₁+𝜕Λ𝜕𝑦₂𝒫^′𝑦₁=0. (47) So 𝜕Λ/𝜕𝑦₁ and 𝜕Λ/𝜕𝑦₂ have the same sign.

Let 𝑦₃<𝑦₂ and (0,𝑦₃) be given by (𝑟₃,𝜃₃); then 𝑟₃>𝑟₂,𝜃₃>𝜃₂. Let 𝛼≥0. From (38) and (36), it follows that Λ(𝑦₁,𝑦₃)>Λ(𝑦₁,𝑦₂). Moreover, by (38), one has (1)Λ(𝑐₁,𝑐₂)<0 if and only if 𝒫(𝑐₁)<𝑐₂, (2)Λ(𝑐₁,𝑐₂)=0 if and only if 𝒫(𝑐₁)=𝑐₂, (3)Λ(𝑐₁,𝑐₂)>0 if and only if 𝒫(𝑐₁)>𝑐₂.

3.2. Degenerate Case

For 𝜇=0, the origin is the only equilibrium of both half-systems. If the system has twofold node with opposite stability, then, as pointed in Freire et al. [6], there exists a continuum of homoclinic loops, consisting of degenerate homoclinic orbits. Conversely, the existence of degenerate homoclinic orbits implies that the equilibrium is neither saddle nor focus (or nor center). So one has the following result.

Theorem 18. If 𝜇=0, then the existence of homoclinic orbits for (7) is equivalent to 𝜏_𝐿𝜏_𝑅<0,0<𝛿_𝑖<14𝜏²_𝑖,𝑖=𝐿,𝑅. (48) Under the conditions (48), the continuum of homoclinic loops is above the 𝑥-axis for 𝜏_𝐿>0 and 𝜏_𝑅<0 and under the 𝑥-axis for 𝜏_𝐿<0 and 𝜏_𝑅>0.

Remark 19. For a continuum of homoclinic loops, see Example 2. In fact, by calculating the eigenvalue (19) with the corresponding eigenvectors (20), one may find that all the homoclinic loops lie between an unstable manifold and a stable manifold.

4. Homoclinic Bifurcations

Based on the study of homoclinic orbits in the last section, now the homoclinic bifurcations in (7) are discussed. We first provide the generic homoclinic bifurcation related to nondegenerate homoclinic orbits and then study the problem of discontinuity induced homoclinic bifurcations. The latter will be divided into two cases to study: nondegenerate and the degenerate.

4.1. Generic Homoclinic Bifurcations

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 (7) is saddle, so 𝜇>0. Two kinds of homoclinic bifurcations are studied: for the saddle-focus system and for the saddle-center system.

For the saddle-focus system, 𝜏_𝑅=2𝛼≠0. As we saw in Theorem 13 and Remark 15, the existence of a nondegenerate homoclinic loop depends on Λ=0 and 𝜏_𝐿𝜏_𝑅<0. Varying the parameters related to Λ slightly, the homoclinic loop may disappear.(1)When 𝜏_𝑅>0, the right zone is governed by an RF system. From Remark 17 and the analysis of the phase portraits, we find that, for Λ(𝑐₁,𝑐₂)<0, one has 𝒫(𝑐₁)<𝑐₂ and there is no periodic orbits in the system and that, for Λ(𝑐₁,𝑐₂)>0, one has 𝒫(𝑐₁)>𝑐₂ and there is an attracting limit cycle around the focus. (2)When 𝜏_𝑅<0, in the right zone is an AF system. For 𝒫(𝑐₁)>𝑐₂, there is no periodic orbits in the system. For 𝒫(𝑐₁)<𝑐₂, there is a repelling limit cycle around the focus.

We mention here that the system has at most one limit cycle by Proposition 9.

Example 20. A bifurcation of saddle-focus systems is shown as in Figure 6, where the parameters in (8) are 𝜇=1,𝜏_𝐿=−0.6333,𝛿_𝐿=−0.3667, and 𝛿_𝑅=2, and we take 𝜏_𝑅=0.4 for (a), 0.5 for (b), and 0.6 for (c). There is a limit cycle in Figure 6(a).

(a)

(b)

(c)

Figure 6 

Bifurcation of saddle-focus systems.

Now we discuss the saddle-center system case; that is, 𝜏_𝑅=0. (1)For 𝜏_𝑅=0 vary 𝜏_𝐿 and 𝛿_𝐿, there exists a limit cycle in the right zone tangent to the 𝑦-axis. It is the unique limit cycle. (2)For 𝜏_𝐿 and 𝛿_𝐿 fixed vary 𝜏_𝑅, in the right zone is a focus system, no limit cycle in this case. (3)Varying 𝜏_𝐿,𝛿_𝐿, and 𝜏_𝑅, it turns into the saddle-focus system case.

Example 21. A bifurcation of saddle-center systems is shown in Figure 7, where the parameters are 𝜇=1,𝛿_𝐿=−1,𝜏_𝑅=0, and 𝛿_𝑅=1, and take 𝜏_𝐿<0 for (a), 𝜏_𝐿=0 for (b), and 𝜏_𝐿>0 for (c).

(a)

(b)

(c)

Figure 7 

: Bifurcation of saddle-center systems.

In order to summarize the above conclusions we let 𝜇>0, 𝛽>0, and 𝛿_𝐿<0 be fixed and consider the (𝜏_𝑅,𝜏_𝐿)-plane, here 𝜏_𝑅=2𝛼 and 𝜏_𝐿=𝜆₁+𝜆₂. Figure 8 is the homoclinic bifurcation diagram for the nondegenerate case. The line Λ=0 divides the plane into two parts: upper is for Λ>0 and bellow is for Λ<0. All the trajectories are clockwise. From Figure 8, we can find that the parameters 𝜏_𝑅 and 𝜏_𝐿 unfold all possible generic homoclinic bifurcation behaviors.

Figure 8 

A bifurcation diagram.

4.2. Discontinuity Induced Homoclinic Bifurcations

In this subsection, we study the problem of the discontinuity induced homoclinic bifurcations (DIHBs) related to variation of boundary equilibria. Herein we take 𝜇 as a bifurcation parameter and discuss two kinds of DIHBs: the nondegenerate case and the degenerate case.

4.2.1. Nondegenerate Case

We still assume that the equilibrium of (7) in the left zone is a visible saddle for 𝜇>0, and then the right zone has a visible focus (or center). It is obvious that whenever 𝜇=0 both of the visible equilibria will coincide on the nonsmooth boundary 𝑦-axis and become invisible as 𝜇<0, which is a nonsmooth fold boundary equilibrium bifurcation; see also [1]. From the discussion in Section 3, the homoclinic orbits exist for Λ=0 and shrink to an equilibrium gradually as the scale parameter 𝜇 decreases from positive to 0 then disappear as 𝜇<0. This is the discontinuity induced homoclinic bifurcation in the nondegenerate case.

Example 22. Following Example 1(a), set 𝜏_𝐿=−0.6333,𝛿_𝐿=−0.3667,𝜏_𝑅=0.5, and 𝛿_𝑅=2 in (7) and (8). Take 𝜇 near zero. Figures 9(a), 9(b), and 9(c) show a discontinuity induced homoclinic bifurcation of the system for 𝜇>0, 𝜇=0, and 𝜇<0, respectively.

(a)

(b)

(c)

Figure 9 

Discontinuity induced homoclinic bifurcation in the nondegenerate case.

In Figure 9, we find that, for 𝜇>0, the system is saddle-focus with a nondegenerate homoclinic orbit as shown in (a), that, for 𝜇=0, two equilibria coincide at the origin and the homoclinic orbit becomes a broken line as in (b), and that for 𝜇<0 both equilibria become invisible and the homoclinic orbits disappear as in (c).

4.2.2. Degenerate Case

As mentioned in Section 3, under the conditions of Theorem 18 there exists a continuum of homoclinic loops when 𝜇=0. Varying the parameter 𝜇, the system (7) has a type of boundary equilibrium bifurcation, called persistent-node bifurcation. Here as the bifurcation parameter 𝜇 crosses the critical value 0, the visible node of one half-system in (7) becomes invisible, and the other invisible node becomes visible, while all homoclinic orbits disappear for 𝜇≠0. This is a discontinuity induced homoclinic bifurcation in the degenerate case.

Example 23. Following Example 2, for (7) and (8), set 𝜏_𝐿=4,𝛿_𝐿=3,𝜏_𝑅=−6, and 𝛿_𝑅=3. We take 𝜇 near zero. Figures 10(a), 10(b), and 10(c) show the discontinuity induced homoclinic bifurcation for degenerate case for 𝜇=−0.1, 𝜇=0, and 𝜇=0.1, respectively.

(a)

(b)

(c)

Figure 10 

Discontinuity induced homoclinic bifurcation in the degenerate case.

5. Conclusions

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 8. We have found that there are two kinds of DIHBs, one is nondegenerate and the other is degenerate. Nonsmooth fold boundary equilibrium bifurcation occurs accompanying DIHB in nondegenerate case, and persistent-node boundary equilibrium bifurcation occurs accompanying DIHB in degenerate case. To sum up, at least two parameters are needed to unfold all possible homoclinic bifurcations.

Acknowledgment

This work was supported by the NNSF of China (Grant nos. 11072274 and 11272169).