The pseudo-Hopf bifurcation for planar discontinuous piecewise linear differential systems

The creation or destruction of a crossing limit cycle when a sliding segment changes its stability, is known as pseudo-Hopf bifurcation. In this paper, under generic conditions, we find an unfolding for such bifurcation, and we prove the existence and uniqueness of a crossing limit cycle for this family.


Introduction
The study of limit cycles is one of the most important problems in the qualitative theory of ordinary differential equations, however, the proof of their existence is generally very complicated. A large list of papers about the arising of limit cycles in piecewise smooth systems in the plane can be found in the literature of recent years, and in these some techniques has been developed to find them. In smooth systems there is a well known mechanism to search for the occurrence of limit cycles, the Hopf bifurcation theorem, see [13,19]. There are analogous results for piecewise smooth systems, for the case of continuous systems see for example [6,7,27,28], and for the case of discontinuous systems see [1,8,11,12,14,18]. In the discontinuous ones we can have more than one limit cycle, either all crossing cycles or including one sliding cycle, and in fact, the determination of the number of limit cycle has been the subject of several recent papers, see [2][3][4]10,[15][16][17]20,[22][23][24].
When the appearance of more than one limit cycle is considered, often the mechanism to obtain one of them is by the collision of two invisible tangencies. This is, the creation or destruction of one crossing limit cycle occurs when a sliding segment changes its stability, this phenomenon is presented without demonstration in [18] and called pseudo-Hopf bifurcation. The appearance of a crossing limit cycle may occur in cases where there is not sliding segment, see [9,21,26].
In this paper we find an unfolding for the pseudo-Hopf bifurcation for planar discontinuous piecewise linear (DPWL) systems with two zones separated by a straight line. We prove the existence and uniqueness of a crossing limit cycle for all possible dynamic scenarios. It is important to mention that the unfolding found has seven parameters, but at moment that the dynamics on each zone be established, it reduces to five. However, in our result it will not be necessary to establish a priori the dynamics in each zone.
The rest of the paper is organized as follows. In Sect. 2 we define the mathematical concepts used. In Sect. 3 we state the main results. In Sect. 4 we find the unfolding. The existence, uniqueness and stability of the crossing limit cycle is given in Sect. 5. In Sect. 6 we illustrate with two examples the main Theorem. Finally, in Sect. 7 we give the conclusions of this work.

Preliminaries
Consider the planar DPWL system with two zones separated by the straight line where A i are 2 × 2 matrices and b i ∈ R 2 for i = 1, 2.
We distinguish three open regions in the straight line Σ: The sliding region: Σ s = {x ∈ Σ : c T f − (x) > 0 and c T f + (x) < 0}, the escaping region: Σ e = {x ∈ Σ : c T f − (x) < 0 and c T f + (x) > 0}, and the crossing region: Any segment contained in Σ s ∪ Σ e is called a sliding segment. The solutions on Σ s ∪ Σ e can be constructed by the Filippov's convex method, see [5]. Filippov's method takes a simple convex combination f s (x) of the two vector fields A pointx is a boundary equilibrium of (1) if Since the three kinds of regions in Σ are relatively open, their boundaries are the called tangency points: q ∈ Σ such that c T f − (q) = 0 or c T f + (q) = 0 (see [12,18]). That is, points where one of the two vector fields is tangent to Σ. In particular, the boundary equilibria are tangency points, since they are located on the boundary of the sliding region where one of the vector fields vanishes. The simplest tangency is the fold singularity, which is defined as follows.
A point q ∈ Σ is a fold singularity of (1) if either A fold singularity is a point with quadratic tangency with Σ, or is a boundary equilibrium (center or focus). A quadratic tangency point can be classified in visible or invisible as follows: The case where system (1) has a quadratic tangency point for one vector field, and a boundary focus for the other one, at the same point on the switching line is called fold-focus singularity. When system (1) has a double boundary focus at the same point on the switching line, that is, when there is a boundary focus for both sides, this singularity is called the focus-focus singularity. Finally, a fold-fold singularity is when the DPWL system (1) has a double quadratic tangency at the same point on Σ.
For the case of the invisible fold-fold singularity, when the vectors f − (q 0 ) and f + (q 0 ) are antiparallel with q 0 ∈ ∂Σ c , the singularity is called fused-focus in [18]. We are going to call two-fold singularity to the fold-fold, fold-focus or focus-focus singularities.

Statements of the main results
The idea is to unfold the two-fold singularity q 0 in such way that the two fold points, q 1 and q 2 , from f − and f + , respectively, delimit a sliding segment, and when they change their relative position on Σ, after collapse in q 0 , the sliding segment change its stability. As will be proved in this article, for some configurations of the fold points, this change of stability in the sliding segment is accompanied by the birth or destruction of a crossing limit cycle. See Fig. 1. With this idea, we assume that f (x) satisfy the following generic hypothesis: Under Hypothesis (H 0 ), the DPWL system (1) has two fold points q 1 and q 2 . This is clear because of, if we define the straight lines for i = 1, 2. Besides there exist γ 1 and γ 2 with γ 2 = 0 such that The first theorem in this section give us an unfolding for piecewise linear systems that satisfy the generic hypothesis (H 0 ).

Theorem 1 Under hypothesis (H 0 ) the change of coordinates
where Q 1 = c T c T A 1 and , transforms the differential system (1) into the differential systemẏ where Theorem 1 is proved in Sect. 4.
Remark 1 (a) If q 2 → q 1 then b → 0, i.e. at b = 0, the fold points collapse at q 0 . (b) If r j = 0 then the fold point q j is a boundary equilibrium point, which must be a boundary focus, with eigenvalues α j ± iβ j for j = 1, 2. (c) If r 1 > 0 then q 1 is an invisible fold point. (d) If r 2 < 0 then q 2 is an invisible fold point.
The following corollary establishes the Σequivalence of the change of coordinates (2), see [12]. .
Then Fig. 2 Change of coordinates (2) and Remark 2 The change of coordinates (2) classifies all the DPWL systems (1) that satisfy (H 0 ) into two classes: those systems that have a sliding segment (γ 2 > 0) and those that have a crossing segment (γ 2 < 0).
From now on we will assume that γ 2 > 0. Figure 2 shows the effect of the orthogonalization of the change of coordinates (2). From the unfolding (3), for b = 0, we find nine different scenarios in which the two-fold singularity can be unfolded in a such way that is it possible to observe a change of stability in a sliding segment. See Fig. 3. The main theorem of the paper establishes that the unfolding (3) undergoes the pseudo-Hopf bifurcation only at four cases (r 1 ≥ 0 and r 2 ≤ 0). Next lemma is a technical result that we need to prove the main theorem.

Lemma 1 Consider a smooth real function H
Proof The hypothesis imply that H has a local extremum value at z = 0. Without loss of generality, we assume a minimum. Then, there exist ε 1 , Fig. 3 The two-fold singularity for the unfolding (3)

Corollary 2 Consider a smooth complex function H
Proof Just observe that the hypothesis imply that the real function θ has a local extremum value at z = 0. Then, from Lemma 1, there exists a function h : Theorem 2 (Pseudo-Hopf bifurcation theorem) Suppose that the DPWL system (1) satisfy (H 0 ) with γ 2 > 0. If r 1 ≥ 0 and r 2 ≤ 0, then for each b sufficiently small with bΛ 0 < 0, system (1) has a unique crossing limit cycle. If Λ 0 < 0 the limit cycle is stable, while if Theorem 2 is proved in Sect. 5.

Remark 3
It is not necessary to calculate the change of coordinates (2), nor the unfolding (3) to use Theorem 2, it is enough to calculate the expressions given in (4) from the original DPWL system (1).

Proof of Theorem 2
Consider the unfolding (3), i.e., We call φ t and ψ t the flow for y 1 < 0 and y 1 > 0, respectively. To prove the existence of a crossing limit cycle we are going to findq 1 = 0 u with u > 0 and v < 0, and times t 1 , t 2 , such that the system In this case Regardless of the value of λ i , real or complex, Similarly Remark 5 Observe that G 1 (0, 0) = 0 and, for b = 0, thenλ i =δ i for i = 1, 2, and That is, the curve G 2 (u, v) = 0 can be obtained by the reflection of the curve G 1 (u, v) = 0, with respect to the straight line v = −u followed by the translation Fig. 6. Then, it is sufficient to solve G 1 (u, v) = 0.

Improper nodes
Assume λ 1 = λ 2 and δ 1 = δ 2 . In this case , . That is , . That is As in the previous section

Assume
Similarly to the previous cases (10), and Lemma 1 is satisfied for H (z) = (1 +λ 1 z)e −λ 1 z , with the same expressions for h 1 and h 2 .
. For each ε > 0 we define the Poincaré map P : From the previous cases we know that, β 2 + 1). Then the Poincaré map is given by Again we observe that the function such that G(v, g(v)) = 0 for each v < 0 sufficiently small. In other words, for each b sufficiently small with bΛ 0 < 0, there exists v < 0 such that P(v, g(v)) = v. That is the unfolding (3) has a crossing limit cycle. Finally to determine the stability of the limit cycle we observe that ∂ ∂v In this case the stability of the limit cycle only depends on the stability of the boundary focus.

5.3.2
Assume r 1 = 0 and r 2 < 0 Then, the Poincaré map is given by where g 0 (b) = −2b + O(|b| 2 ). Again observe that the function such that G(v, g(v)) = 0 for each v < 0 sufficiently small. In other words, for each b sufficiently small with bΛ 0 < 0, there exists v < 0 such that P(v, g(v)) = v. That is, the unfolding (3) has a crossing limit cycle. Finally, to determine the stability of the limit cycle observe that ∂ ∂v P(v, b) = e As in the previous case, the stability of the boundary focus determines the stability of the limit cycle. This completes the proof of Theorem 2. (c) Fig. 7 Destruction of the stable crossing limit cycle due to the change of stability of the sliding segment: a stable crossing limit cycle and escaping segment for μ = 1 2 , b fused-focus for μ = 1 and c stable sliding segment for μ = 3