LIMIT CYCLES FOR A CLASS OF CONTINUOUS PIECEWISE LINEAR DIFFERENTIAL SYSTEMS WITH THREE ZONES

Due to the encouraging increase in their applications, control theory [Lefschetz, 1965] and [Narendra et al., 1973], design of electric circuits [Chua et al., 1990], neurobiology [FitzHugh, 1961] and [Nagumo et al., 1962] piecewise linear differential systems were studied early from the point of view of qualitative theory of ordinary differential equations [Andronov et al., 1966]. Nowadays, a lot of papers are being devoted to these differential systems.


Introduction
Due to the encouraging increase in their applications, control theory [Lefschetz, 1965] and [Narendra et al., 1973], design of electric circuits [Chua et al., 1990], neurobiology [FitzHugh, 1961] and [Nagumo et al., 1962] piecewise linear differential systems were studied early from the point of view of qualitative theory of ordinary differential equations [Andronov et al., 1966].Nowadays, a lot of papers are being devoted to these differential systems.
On the other hand, starting from linear theory, in order to capture nonlinear phenomena, a natural step is to consider piecewise linear systems.As local linearizations are widely used to study local behavior, global linearizations (achieved quite naturally by working with models which are piecewise linear) can help to understand the richness of complex phenomena observed in the nonlinear world.
The study of piecewise linear systems can be a difficult task that is not within the scope of traditional nonlinear systems analysis techniques.In particular, a sound bifurcation theory is lacking for such systems due to their nonsmooth character.
In this paper we study the existence of limit cycles for the class of continuous piecewise linear differential systems where x = (x, y) ∈ R 2 , and X is a continuous piecewise linear vector field.We will consider the following situation, that we will name the three-zone case.We have two parallel straight lines L − and L + symmetric with respect to the origin dividing the phase plane in three closed regions: R − , R o and R + with (0, 0) ∈ R o and the regions R − and R + have as boundary the straight lines L − and L + respectively.We will denote by X − the vector field X restrict to R − , by X o the vector field X restricted to R o and by X + the vector field X restrict to R + .
We suppose that the restriction of the vector field to each one of these zones are linear systems with constant coefficients that are glued continuously at the common boundary.In short, system (1) can be written as where {−, o, +} and x ′ = dx dt with t the time.We say that the vector field X has a real equilibrium x * in R i with i ∈ {−, o, +} if x * is an equilibrium of X i and x * ∈ R i .In opposite we will say that X has a virtual equilibrium x * in R i if x * ∈ R c i where R c i denotes the complementary of R i in R 2 .We suppose the following assumptions: (H1) X o has a real equilibrium in the interior of the region R o of focus type.
(H2) The others equilibria (real or virtual) of X − and X + are a center and a focus with different stability with respect to the focus of X o .
We note that if the two equilibria of X − and X + are both centers, or a center and a focus having this focus the same stability than the focus of X o , then the vector field X has no limit cycles, for more details see Proposition 2.4.
As usual a limit cycle of ( 2) is a periodic orbit of (2) isolated in the set of all periodic orbits of (2).A limit cycle is hyperbolic if the integral of the divergent of the system along it is different from zero, for more details see for instance [Dumortier et al., 2006].
Our main result is the following.
More information about the limit cycle of Theorem 1.1 is given in Propositions 4.1 and 4.2, where it is characterized when the limit cycle visit two or three zones.
We observe that Theorem 1.1 is in some situations an extension of Proposition 15 of [Freire et al., 1998].This proposition shows that a piecewise linear differential system with two zones can have at most one limit cycle.More precisely we have that the limit cycle stated in Proposition 15(e) is persistent in the present scenario where we have three zones having a center in R − or in R + .

Normal Form
The aim of this section is to write system (2) in a convenient normal form where the number of parameters are reduced, and consequently the computations of the Poincaré return map will be easier.
Lemma 2.1.Under the assumptions (H1) and (H2) there exists a linear change of coordinates that writes system (2) into the form The dot denotes derivative with respect to a new time s.
Proof.Under the hypotheses, by means of a rotation and a homothecy in the x direction we can write the system in such a way that where b ij , i, j = 1, 2 are the initial entries of the matrix A o and B o = (e 1 , e 2 ).Note that Limit Cycles for a Class of Continuous Piecewise Linear differential systems with Three Zones , and it has equilibrium in x * o = (−b 2 , 0) where |b 2 | < 1.We observe that the straight lines L − and L + are invariant under the change of coordinate (x, y, t) → (u, v, s).
The expressions of the vector fields X − and X + follows by the continuity in L − and L + respectively.
For our purpose we will define a first return map that involves all the vector fields X − , X o and X + in a suitable transversal section.In the case considered in this paper this section will be a segment of the line L − and the next remark and lemma state conditions for the existence of a such map.
Remark 2.2.In [Llibre et al., 2004] and [Llibre et al., 2009] Theorem 4.3.10we see that a necessary and sufficient condition for the existence of Poincaré maps from the straight lines L ± to the straight lines L ± is that there exists a unique contact point of the flow of the linear system with these lines.By contact point we mean a point of the line where the vector field is tangent to it.
Lemma 2.3.In the coordinates given by Lemma 2.1 there is a unique contact point of system (3) with L − and a unique contact point of (3) with L + .These points are respectively p − = (−1, 0) and p + = (1, 0).Moreover the equilibria of X − and X + are virtual.
Proof.The proof of the first part of this lemma follows easily by direct computations.
For the second part note that the eigenvalues of A − are given by the expression [(a 11 + a 1 ) ± (a 11 + a 1 ) 2 − 4(a 11 a 1 + 1 − b 2 + d 2 ) ]/2 and the eigenvalues of A + are given by the expression [(c 11 + a 1 ) ± (c 11 + a 1 ) 2 − 4(c 11 c 1 + 1 + b 2 − f 2 ) ]/2.So in order that the equilibrium points of X − and X + be of center or focus type we must have Now the equilibrium of X − is given by is a virtual equilibrium of the system.The same argument is valid for the equilibrium x * + of X + .
In the rest of the paper for i ∈ {−, o, +} we denote t i the trace of matrix A i , and by d i as been the determinant of the matrix A i .
Denote by Int(Γ) the open region limited by the closed Jordan curve Γ.
The next result is an immediate consequence of the Green's Formula (see for instance Proposition 3 of [Llibre et al., 1996]).
Observe that if we have a focus in R o and a center in R − or R + then a necessary condition for the existence of such Γ is that Γ visit at least the two zones having a focus and that the stability of both foci is different.

Poincaré Return Map
We have that system (3) has a unique real equilibrium in R o and, by Lemma 2.3, the two other equilibria are virtual.
In order to study the existence of limit cycles for system (3) we will define a Poincaré return map defined on L − .This Poincaré return map will be defined as the composition of four different Poincaré maps.In what follows we study the qualitative behavior of each one of these maps separately in order to understand the global behavior of the general Poincaré return map.
The next lemma will be useful in the study of the Poincaré return map associated to system (3).The proof of this lemma can be found in [Llibre et al., 2009] Lemma 4.4.10 but in order to have a complete understood of the results we will reproduce the proof here.Proof.Since ∂ϕ ∂x = (1+y 2 )e xy sin x the critical value of ϕ are x k = kπ, where k ∈ Z. From ∂ 2 ϕ ∂x 2 = (1 + y 2 )e xy (cos x + y sin x) and assuming that y o > 0 it follows that ϕ has a local minimum at x k for k even or has a local maximum at x k for k odd.
Note that to study the qualitative behavior of Π − is equivalent to study the qualitative behavior of π − .From now on we will consider the map π − instead of Π − .
From Proposition 4.3.7 in [Llibre et al., 2009] the mapping π − previously defined is invariant under change of coordinates and translation for which the equilibrium x * − remains virtual.So for computing the mapping π − we can suppose that the virtual equilibrium is at the origin and that the matrix A − is given in its real Jordan normal form.
Note that we can define in the same way a map π + associated to the a Poincaré map Π + in L + considering the flow defined by ẋ = A + x + B + and the contact point p + .Proposition 3.2.Consider the vector field X i in R i with i ∈ {−, +}, with a virtual center or focus equilibrium and such that t i ≥ 0. Let π i be the map associated to the Poincaré map Π i : L i → L i defined by the flow of the linear system ẋ = A i x + B i .
) is an asymptote of the graph of π i when a tends to +∞ where Limit Cycles for a Class of Continuous Piecewise Linear differential systems with Three Zones 5 Proof.We will prove the result for i = −.The case i = + follows in the same way.
Let p − be the contact point of the flow with L − and p and q as described above.As q is in the orbit of p in the forward time we have that q = ϕ(s, p) with s ≥ 0. Moreover for computing the map π − we can suppose that the virtual equilibrium is at the origin and that matrix A − is in its real Jordan normal form.
Let p * − be the contact point p − in the coordinates in which A − is in its real Jordan normal form and the virtual equilibrium of X − is at the origin.We denote by ṗ * − = X − (p * − ).So we can write

Now using the fact that ṗ *
where a ≥ 0, π − (a) ≥ 0, s ≥ 0 and the matrix A − is given by A Since p * − = (0, 0) we obtain from equation ( 4) that b = π − (a) is defined by the system and the inequalities a ≥ 0, b ≥ 0 and s ≥ 0.
3.3.Consider the vector field X i in R i with i ∈ {−, +} and with a virtual center or focus equilibrium and such that t i < 0. Let π i be the map associated to the Poincaré map Π i : L i → L i defined by the flow of the linear system ẋ = A i x + B i .
(a) The maps π i satisfy that π ) is an asymptote of the graph of π i when a tends to +∞ where Proof.The proof follows in a similar way to the proof of Proposition 3.
As before we can see the mapping Π o in a different way.Given, q ∈ D * o and r ∈ L + there exist unique b ≥ 0 and c ≥ 0 such that q = p − + b ṗ− and r = p + − c ṗ+ , where ṗ+ = X + (p + ) = (0, b 2 + 1).So the mapping Π o induces a mapping π o given by π o (b) = c.To study the qualitative behavior of Π o is equivalent to study the qualitative behavior of π o .As before we will consider the map π o instead of Π o .
In the same way we can define a first return map Πo : D * o ⊂ L + → L − and the respective πo .The next propositions state the qualitative behavior of these maps.

Proof. The solution of ẋ =
where In order that the mapping π o be defined in 0 and π o (0) = 0, we must have , then π o is also defined in 0 and In the domain of definition of π o we obtain from expression (7) the parametric solution of Using the same arguments than in Proposition 3.2 we can prove the statements (a) and (a. satisfies the same conditions described in (a.1).
Proof.The proof follows using the same ideas of the previous proposition and will be omitted.

Limit Cycles having a Focus in R o
In what follows without loss of generality we will suppose that the center of hypothesis (H2) is in R − .Proposition 4.1.Assume that system (3) satisfies assumptions (H1) and (H2).Suppose that the real equilibrium point in R o is between L − and L o where L o is a line parallel to L + through the origin.Then there exists a unique limit cycle of (3), which is hyperbolic.Moreover this limit cycle visits the three regions R − , R o and R + .It is a repeller if t o < 0, and an attractor if t o > 0.
Proof.Suppose that we have a center in X − , and a focus in X o and in X + .By the hypotheses we have 0 ≤ b 2 < 1.
Using the previous notation we have  3).This implies that Note that finding a fixed point of Π is equivalent to find zeroes of the function h.
In a * we have h(a * ) = Π(a * ) − a * > 0. Supposing that h admits a zero and that a s is the smallest zero we must have h ′ (a s ) ≤ 0, or equivalently Π ′ (a s ) ≤ 1.But from the definition of Π we can write  From Propositions 3.2,3.3,3.4,3.5,3.6 and 3.7 it follows that e 2(γo(τo+τo)+γ+τ+) , with τ + ∈ (0, π) increasing with a, and τo + τ o ∈ (0, 2π−τ * ) decreasing with a.As a s and a r are fixed point of Π this implies from (8) that Π ′ (a s ) = 1 and Π ′ (a r ) < 1.So in this case, for a ∈ (a s , a r ) we obtain Π(a) > a.Now from ( 8) it follows that Π ′ (a) < 1 for a ∈ (a s , a r ) and from the Mean Value Theorem we have that implies that Π(a) < a.This is a contradiction and so we have at most a fixed point a s for Π and On the other hand since h ′ (a) = Π ′ (a) − 1 and lim a→∞ h ′ (a) = e 2γ + π − 1 < 0, it follows by the Mean Value Theorem that lim a→∞ h(a) = −∞, and this shows that h admits a zero.Moreover this zero is equivalent to a fixed point of the first return map Π and this fixed point is a hyperbolic attractor because from (9) we have Π ′ (a s ) < 1.Now suppose that γ o < 0 and γ + > 0. In this case we can use the same idea of the previous case and define a first return map Π : where Π(a * ) = a * * and a * * < a * .But now we obtain a unique limit cycle that is a hyperbolic repeller.Now from Proposition 15 of [Freire et al., 1998] it is not possible to have a limit cycle that visit only the regions R o and R + otherwise we would have two hyperbolic attractor limit cycle containing the same equilibrium point what is not possible.
This finish the prove of the theorem.
Proposition 4.2.Assume that system (3) satisfies assumptions (H1) and (H2).Suppose that the real equilibrium point in R o is between L o and L + .Then there exists a unique limit cycle of (3), which is hyperbolic.This limit cycle visits the three regions R − , R o and R + if D(Π) = [0, ∞) and Π(0) > 0 and visits only the regions R o and R + otherwise.
Proof.Suppose that γ o > 0 and γ + < 0. We have −1 < b 2 < 0. In this case the real focus of X o is a repeller and the virtual focus of X + is an attractor.
If 4).This implies that D(Π) = [0, ∞) and by the fact that the real focus is a repeller it is not difficult to see that Π(0) = a * * > 0. In this scenario a similar analysis as the one done in the proof of Proposition 4.1 shows that there exists a unique limit cycle that visits the three regions, and which is hyperbolic.On the other hand, if Here three cases are possible. Case and so Π(a * ) = 0 < a * (see figure 5 (1)).From this figure as the real equilibrium point is repeller it follows that there is a stable limit cycle that visits the regions R o and R + .We observe that for this limit cycle we can use the Proposition 15 of [Freire et al., 1998] and conclude that it is hyperbolic.We can also define a restricted Poincaré return map Π r near this hyperbolic stable limit cycle and we obtain Π ′ r (ā) = e 2(γoτ r o +γ + τ r + ) < 1 where ā is the point associated to the limit cycle, for more de-tails see equation ( 31) of [Freire et al., 1998].This implies that γ o τ r o + γ + τ r + < 0. Now in the points of the domain of definition of the Poincaré return map Π it is easy to see that τ o + τo < τ r o and τ + > τ r + , and as γ o > 0 and γ + < 0 we obtain γ o (τ o +τ o )+γ + τ + < 0. Now if Π admits a fixed point and a s is the smallest fixed point it follows from the fact that Π(a * ) = 0 < a * that we must have Π(a s ) = a s and Π ′ (a s ) ≥ 1. from equation ( 8) we conclude that Π ′ (a s ) = e 2(γo(τos+τos)+γ + τ +s ) < 1, because γ o (τ os +τ os )+γ + τ +s < 0. This contradiction implies that a such fixed point a s does not exist.Case 2: π + (c * ) = d * In this case we have a limit cycle tangent to L − because Π(0) = 0 (see figure 5 (2)).From Proposition 15 of [Freire et al., 1998] this limit cycle is hyperbolic attractor.This implies that Π ′ (0) < 1.Now as the previous case we cannot have a fixed point for Π because again if a s is the smallest fixed point of Π we must have Π ′ (a s ) ≥ 1 and this is a contradiction as in case 1.So in this case only one limit cycle exists, it is hyperbolic and it visits only the regions R o and R + .Case 3: π + (c * ) > d * In this case D(Π) = [0, ∞) and Π(0) = a * * > 0 (see figure 5(3)) and a similar argument as the one given in the proof of Proposition 4.1 shows that there exists a unique limit cycle, which is hyperbolic and it visits the three regions.The situation when γ o < 0 and γ + > 0 can be analyzed in the same way that in the previous case and the details are omitted.
Proof of Theorem 1.1.The proof of Theorem 1.1 follows directly from the proof of Propositions 4.1 and 4.2.
We note that the existence of the hyperbolic limit cycle of Theorem 1.1 does not depend on the sign of γ o + γ + as the result given in Proposition 15 of [Freire et al., 1998].
).So we have π − (0) = 0. Moreover if a = a o , b = b o and s = s o is a solution of (5), then s o is the flight time between the points p = p * − −a ṗ * − and q = p * − + b ṗ * − .Thus β − s is the angle between p and q and consequently β − s ∈ [0, π).Define τ − = β − s and γ − = α − /β − .Solving system (5) with respect to τ − ∈ (0, π) we obtain the following parametric equations for π − (a) = b, a 2. Let p − and p + be the contact point of ẋ = A o x + B o with L − and L + respectively.We can define a Poincaré map Π o : D * o ⊂ L − → L + by Π o (q) = r being the map from points in D o to points in L + defined by the flow of ẋ = A o x + B o in forward time, where D * o is a subset of L − where the mapping Π o is well defined.

Proposition 3. 4 .
Consider the vector field X o in R o with t o > 0. Let π o be the map associated to the Poincaré map Π o : D * o ⊂ L − → L + defined by the of the linear system ẋ = A o x + B o from the straight line L − to the straight line L + .(a) If 0 < b 2 < 1 then the map π o satisfies that π o : [b * , ∞) → [c * , ∞), b * , c * ≥ 0 with π o (b * ) = c * and lim b→∞ π o (b) = +∞.Moreover b * = 0 if and only if e γoπ ≥ 1 + b 2 1 − b 2 and c * = 0 if and only if and lim b→∞ π o (b) = +∞.(b.1) π ′ o (b) the same conditions described in (a.1).
1).Statement (b) can be obtained in the same way.Proposition 3.5.Consider the vector field X o in R o with t o < 0. Let π o be the map associated to the Poincaré map Π o : D * o ⊂ L − → L + defined by the flow of the linear system ẋ = A o x + B o from the straight line L − to the straight line L + .(a) If −1 < b 2 < 0 then π o satisfies that π o : [b * , ∞) → [c * , ∞), b * , c * ≥ 0 with π o (b * ) = c * and lim b→∞ π o (b) = +∞.Moreover b * = 0 if and only if e γoπ ≥ 1 + b 2 1 − b 2 and c * = 0 if and only

Fig. 3 .
Fig. 3.The flow of the three zones vector field with the real equilibrium between L − and L o where A * = p − − a * X − (p − ) and A * * = p − − a * * X − (p − ).Define the displacement function a) where b = π − (a) = a, c = π o (b) and d = πo (c).

Fig. 4 .
Fig. 4. The flow of the three zones vector field with the real equilibrium between L o and L + where A * * = p − − a * * X − (p − ).