ON THE LIMIT CYCLES OF A CLASS OF DISCONTINUOUS PIECEWISE LINEAR DIFFERENTIAL SYSTEMS

. In this paper we consider discontinuous piecewise linear diﬀerential systems whose discontinuity set is a straight line L which does not pass through the origin. These systems are formed by two linear diﬀerential systems of the form ˙ x = Ax ± b . We study the limit cycles of this class of discontinuous piecewise linear diﬀerential systems. We do this study by analyzing the ﬁxed points of the return map of the system deﬁned on the straight line L . This kind of diﬀerential systems appear in control theory.

1. Introduction and statement of the main results. The theory of discontinuous piecewise differential systems is in constant development due to its applicability in different areas of the knowledge such as ecology, mechanic and electrical engineering, see for instance [6]. However even in the planar case there are important questions unsolved for this class of differential systems as to know the number of their limit cycles.
In 2010 Han and Zhang [4] conjectured that piecewise linear systems with only two regions have at most two limit cycles. In 2012 Huan and Yang [5] investigated the number of limit cycles of planar piecewise linear systems with two regions sharing the same equilibrium. Moreover they provided a numerical example to illustrate the existence of three limit cycles, thus had a negative answer to the conjecture by Han and Zhang. In 2012 Llibre and Ponce [10] provided a rigorous proof of the existence of such three limit cycles. This was the first example that a discontinuous differential piecewise linear systems with two regions can have three limit cycles.
Many others researchers have analyzed the existence of limit cycles for a piecewise linear systems with two regions separated by a straight line. In [3] it is proved that a such piecewise linear system has at most two limit cycles when the singularities are both virtual focus or center. In [1] the authors considered a piecewise linear system separated by a straight line with singularities of type real saddle and proved that this system has at most two limit cycles.
Consider the 2 × 2 real matrix A + and A − , and b + , b − ∈ R 2 . We define the planar piecewise discontinuous linear systemṡ where x = (x 1 , x 2 ) ∈ R 2 . Note that L = h −1 (0) = {x ∈ R 2 ; x 1 = 0} splits the plane in two open regions S + = {x ∈ R 2 ; x 1 > 0} and S − = {x ∈ R 2 ; x 1 < 0}. We say that a limit cycle of system (1) is a crossing limit cycle if it share no points with the sliding set of the system. In [2] Cespedes studied systems (1) satisfying div(X)div(Y ) ≥ 0, i.e. the product of the divergences of the subsystems X and Y is non-negative, and show that such systems have at most two limit cycles. Moreover this author exhibited an example with exactly two crossing limit cycles. In this paper we consider S + = {(x 1 , x 2 ) ∈ R 2 x 1 > 1} and S − = {(x 1 , x 2 ) ∈ R 2 : x 1 < 1} and piecewise linear systems with two zones given byẋ where b ∈ R 2 \ {0}. In this case the discontinuity of system (2) is the straight line L = {(x 1 , x 2 ) ∈ R 2 : x 1 = 1}. Assuming that there are no singularities of system (2) in L we shall study the existence of crossing limit cycles. Set Φ ± (t, x) the flow of the system X and Y , respectively. The flow Φ + is said transversal to L at point p if X(p) is not contained in L. If X(p) ∈ L then the point p is called a contact point of the flow with L. Analogous definitions hold for system Y . We say that p ∈ R 2 is a real singularity of system (2) if p = (x 1 , x 2 ) is such that either x 1 > 1 and X(p) = 0, or x 1 < 1 and Y (p) = 0. On the other hand p is a virtual singularity if p = (x 1 , x 2 ) is such that either x 1 > 1 and Y (p) = 0, or x 1 < 1 and X(p) = 0. It follows that the discontinuous system (2) can be of type virtual-virtual, virtual-real and real-real depending if the singularities of the systems X and Y are virtual or real.
In this work d and t denote the determinant and trace of the matrix A, respectively. Our main results are the following. Theorem 1. Consider the discontinuous piecewise linear differential system (2). If the contact points of system (2) with the straight line L coincide, then system (2) has no limit cycles.
Theorem 2. Assume that the discontinuous piecewise linear differential system (2) is of type virtual-virtual. If the contact points of the systems X and Y are distinct, then system (2) has at most one limit cycle. There exists necessary and sufficiently conditions for the existence of exactly one limit cycle.
Observe that system (2) satisfy div(X)div(Y ) ≥ 0, but this work provides new results with respect the ones obtained in chapter 2 of [2].
One of the most important tools in the study of the periodic orbits are the Poincaré maps. These maps characterize the behavior of the flows in the neighborhood of periodic orbits. Moreover, there exists a correspondence between the limit cycles of a system and the fixed points of some Poincaré map. In [8] Llibre and Teruel studied Poincaré maps for piecewise linear differential systems in R n . In [9] these authors determined the Poincaré maps and analyzed the existence of crossing limit cycles of planar piecewise linear differential systems defined in three regions separated by two straight lines L + and L − which are symmetric with respect to the origin. In order to prove Theorems 1 and 2 we determine the Poincaré maps of the subsystems X and Y of system (2) with respect to the straight line L, and we use these maps to determine the return map of system (2) in L. After that we analyze the existence of fixed points of this return map.
The paper is organized as follows. In section 2 we provide the basic results that we shall need to prove the mains results. Section 3 is divided in three subsections, in subsection 3.1 and 3.2 we study the existence of the Poincaré maps for the nonhomogeneous systems X and Y , respectively. In subsection 3.3 we define the return map for the discontinuous piecewise linear differential system (2). In section 4 we study the existence of limit cycles for the discontinuous system (2) and prove our main results.
2. Preliminary results. Consider a linear differential systeṁ where x = (x 1 , x 2 ) ∈ R 2 . Given p, v ∈ R 2 let L = {p + λv : λ ∈ R} be a straight line that does not pass through the origin, and let n be a unit vector orthogonal to L such that n T p > 0, then we say that n is oriented in the opposite sense to the origin. Furthermore if nq = 0 the flow of system (3) is transversal to the line L at a point q ∈ L, whereq denotes the vector field of the differential system (3) evaluated at q. Otherwise q is a contact point of the flow with the straight line L.
The following result is proved in Proposition 4.2.7 of [9].
Proposition 1. Consider the differential system (3) with A ∈ GL(R 2 ). Let L be a straight line in the phase plane not passing through the origin, p a contact point of the flow in L, and x(t) the solution of the system such that where S 0 and S are the half-planes bounded by the straight line L, being S 0 the half-plane containing the origin.
A transversal flow to L at a point q ∈ L is said to have outside orientation if n Tq > 0, and it is said to have inside orientation if n Tq < 0. Thus we define the following subsets in L L I = {q ∈ L : n Tq ≤ 0} and L O = {q ∈ L : n Tq ≥ 0}.
If det A = 0 and there exists a contact point p of the flow of system (3) in L, then L I and L O are two half-lines such that the flow over L I and L O has opposite sense and L I ∩ L O = p. This is part of the next result that is proved in Proposition 4.2.5 of [9].
Proposition 2. Consider the differential system (3) with A ∈ GL(R 2 ). Let L be a straight line in the phase plane not passing through the origin.  Poincaré maps of a homogeneous linear system Consider system (3) and two parallel straight lines in the plane L + and L − which are symmetric with respect to the origin. Notice that the lines L + and L − split the plane in three regions S 0 , S + , and S − , where S 0 is the open strip containing the origin and S + and S − the half-planes bounded by L + and L − , respectively. Moreover, we can define the following subsets, Dom ++ = q ∈ L + : ∃t q > 0 such that e Atq q ∈ L + and either e At q ⊂ S + or e At q ⊂ S 0 ∀t ∈ (0, t q ) } ∪ CP + , where CP + and CP − are either empty sets, or consist of contact points of the flow with the lines L + and L − , respectively. We have the following result that is proved in Lemma 4.3.2 of [9].
Proposition 3. Letẋ = Ax be a planar linear differential system with A non identically zero. Consider L + and L − two parallel straight lines symmetric with respect to the origin and let Dom jk be the sets defined in (4). Assume that for some j, k ∈ {+, −} the set Dom jk = ∅. Then there exists a unique contact point p + of the flow in L + , and p − = −p + is the unique contact point of the flow in L − .
Provided that Dom jk = ∅, for j, k ∈ {+, −}, we define the Poincaré map Π jk : Dom jk ⊂ L j −→ L k of the linear differential system (3) associated to the lines L j and L k as Π jk (q) = e Atq q. In what follows we present some results on the domains of definition of these Poincaré maps and necessary and sufficient conditions for the existence of these maps.
Notice that when Dom ++ = ∅ and Dom −− = ∅, the contact points p + and p − split L + and L − into respective half-lines L I + , L O + , L I − , and L O − . We have the following results.
Proposition 4. Consider a planar linear differential systemẋ = Ax with A non identically zero. Set L + and L − two parallel straight lines symmetric with respect to the origin and let Dom jk be the sets defined in (4). Suppose that Dom jk = ∅ for every j, k ∈ {+, −}.
(i) If det A > 0, then (ii) If det A < 0, then Proposition 5. Letẋ = Ax be a planar linear differential system with A non identically zero. Consider Observe that, by Proposition 2(b), if det A > 0 then and if det A < 0 then It follows that given any point q on L + or on L − we can associated a unique a ≥ 0, called the coordinate of q. Consider q 1 ∈ L j and q 2 ∈ L k such that Π jk (q 1 ) = q 2 , where j, k ∈ {+, −}. Let a 1 and a 2 be the coordinates of q 1 and q 2 , respectively. Then we define the Poincaré maps π jk by π jk (a 1 ) = a 2 . Thus to know the qualitative behavior of the map π jk is equivalent to know the qualitative behavior of the Poincaré map Π jk .
The next results, proved in Lemma 4.3.5 and Proposition 4.3.7 of [9], respectively, provide some properties of the Poincaré maps π jk . Proposition 6. Consider the linear differential system (3) and let L + and L − be two parallel straight lines which are symmetric with respect to the origin. Suppose that the Poincaré maps π jk with j, k ∈ {+, −} are defined. Then (a) the maps π ++ and π −− coincides.
(b) the maps π +− and π −+ coincides. (c) the Poincaré maps π * jk associated to the flow of the systemẋ = −Ax and to the lines L + and L − are defined, and they satisfy π * jk = π −1 jk . (d) π jk and π −1 jk are analytic functions.
Proposition 7. Consider the linear differential system (3) and let L + and L − be two parallel straight lines which are symmetric with the respect to the origin. Assume that the Poincaré maps π jk with j, k ∈ {+, −} are defined. If M ∈ GL(R 2 ), then the maps π jk are invariant under the change of coordinates y = M x.
Assume that the maps π ++ and π +− are defined. Since these maps are invariant under linear changes of coordinates, see Proposition 7, in what follows we consider A given in its real Jordan normal form. Moreover we denote the eigenvalues of A by λ 1 and λ 2 . In what follows we characterize the behavior of the Poincaré map π ++ depending on t and d. Moreover we characterize the behavior of the composition The next two results are proved in Proposition 4.4.15 and Corollary 4.4.16 of [9], respectively.
Proposition 8. Assume that d < 0, t ≥ 0. Then the eigenvalues of the matrix A satisfy λ 1 > 0 > λ 2 . Let π ++ be the Poincaré map defined by the flow of the linear systemẋ = Ax and associated to the parallel straight lines L + and L − symmetric with respect to the origin. If t = 0, then π ++ is the identity map on the interval the graph of π ++ has a vertical asymptote at a = λ −1 1 . (e) π ++ is implicitly defined by the equation (f ) The qualitative behavior of the graph of π ++ is represented in Figure 1 Figure 1. Qualitative behavior of the Poincaré map π ++ ; (a) t > 0 and (b) t < 0. Proposition 9. Assume that d < 0, t < 0. Then the eigenvalues of the matrix A satisfy λ 1 > 0 > λ 2 . Let π ++ be the Poincaré map defined by the flow of the linear systemẋ = Ax and associated to the parallel straight lines L + and L − symmetric with respect to the origin. If t = 0, then π ++ is the identity map on the interval the graph of π ++ has a horizontal asymptote at a = λ −1 1 when a tends to +∞. (e) π ++ is implicitly defined by equation (10). (f ) The qualitative behavior of the graph of π ++ is represented in Figure 1 Diagonal node: d > 0 and t 2 − 4d > 0.
The following results are proved in Proposition 4.4.1 and Corollary 4.4.2 of [9], respectively.
Proposition 10. Assume that d > 0, t > 0, and t 2 − 4d > 0. Then the eigenvalues of the matrix A satisfy λ 1 > λ 2 > 0. Let π ++ be the Poincaré map defined by the flow of the linear systemẋ = Ax and associated to the parallel straight lines L + and L − symmetric with respect to the origin. Then (c) if a ∈ (0, λ −1 1 ), then π ++ (a) > 0. (d) the graph of π ++ has a vertical asymptote at a = λ −1 1 . (e) π ++ is implicitly defined by the equation (f ) The qualitative behavior of the graph of π ++ is represented in Figure 2-(a).
In this case A has one null eigenvalue and other equal to t. Hence the matrix A has two different real Jordan normal forms. One being for t = 0 and the other one for t = 0. In any case the behavior of the Poincaré map is defined only in a contact point, see Proposition 4-(iii). Thus the map π ++ is defined only at zero and π ++ (0) = 0. Center and focus: d > 0 and t 2 − 4d < 0.
The results for the centers and foci are proved on sections 4.4 and 4.5 of [9].
The following results are Proposition 4.4.11 and Corollary 4.4.12 of [9], respectively.
Proposition 14. Suppose that d > 0, t ≥ 0, and t 2 − 4d < 0. Then the matrix A has a pair of complex eigenvalues. Let π ++ be the Poincaré map defined by the flow of the linear systemẋ = Ax and associated to the parallel straight lines L + and L − symmetric with respect to the origin. If t = 0 then π ++ is the identity in [0, +∞ ), (e) π ++ is implicitly defined by the equation (13) (f ) The qualitative behavior of the graph of π ++ is represented in Figure 3 Figure 3. Qualitative behavior of the Poincaré map π ++ ; (a) t > 0 and (b) t < 0.
Proposition 16. Consider a matrix A ∈ GL(R 2 ) such that d > 0, t ≥ 0, and t 2 − 4d < 0, and a vector b ∈ R 2 \ {0}. Let π ++ be the Poincaré map defined by the flow of the linear systemẋ = Ax + b and associated to the parallel straight lines L + and L − symmetric with respect to the origin. If t = 0 then π ++ is the identity in (c) if a ∈ (0, +∞), then π ++ (a) > 0.
(b) if a ∈ (a * , +∞), then π ++ (a) > 0 and lim a a * π ++ (a) = +∞. (c) if a ∈ (a * , +∞), then π ++ (a) < 0. (d) the graph of π ++ has an asymptote at b = ae γπ + t(1 + e γπ )/d when a tends to +∞, where γ = t/ √ 4d − t 2 . (e) π ++ is implicitly defined by equation (14). (f ) The qualitative behavior of the graph of map π ++ is represented in Figure  4-(b). 3. Return map of discontinuous differential system. Consider the following discontinuous piecewise linear differential systeṁ Assume that the singularities of system (15) not belong to L. If d = 0. In this section we analyze the Poincaré maps defined by the flow of system (15) and associated to the straight line L and we define the return map for this system.
In what follows we denote by n the unit orthogonal vector to the line L which is oriented in the direction opposite to the origin, and S + = {(x 1 , x 2 ) ∈ R 2 , x 1 > 1} and S − = {(x 1 , x 2 ) ∈ R 2 , x 1 < 1} denote the half-planes bounded by L.
In the study of the Poincaré maps associated to the flow of the linear differential systemsẋ = Ax ± b with respect to the straight line L, we denote by L ± to specify what system we are considering.
3.1. Poincaré maps of linear differential systemẋ = Ax+b. In this subsection we study the Poincaré maps defined by the flow of the linear differential systeṁ and associated to the straight line L + . Since L + does not pass through the origin it can be divided into the subsets L I + = {q ∈ L : n Tq ≤ 0} and L O + = {q ∈ L : n Tq ≥ 0}, whereq = Aq + b. Then the set CP + = L I + ∩ L O + consists of contact points of the flow of system (16) with L + .
Set Φ + (t, q) the flow of system (16) such that Φ + (0, q) = q. Define in L + the subset Assuming that Dom ++ = ∅ we define the Poincaré map of the linear differential system (16) associated to the straight line L + by Suppose that the flow of system (16) has a unique contact point p with L + . We have that Π + (p) = p because p ∈ L I + and p ∈ L O + . Let e + = −A −1 b be the singularity of system (16). Applying the translation y = x − e + we rewrite system (16) asẏ = Ay, and the straight line L + is transformed into the straight line L * + . If the Poincaré map Π + , given in (17), is defined then it induces a Poincaré map Π * ++ associated to the flow of system (18) and to the straight line L * + . Clearly the converse statement is also true and, therefore the behavior of the map Π + can be obtained from the behavior of the map Π * ++ . We have the following result that are proved in Propositions 4.5.1 of [9]. -If det A > 0, then Π * ++ is the Poincaré map Π ++ . -If det A < 0, then Π * ++ is trivial, i.e. the map Π * ++ is only defined in a contact point of the L * + . (ii) Suppose that e + ∈ S + .
-If det A > 0 and t 2 − 4d ≥ 0, then Π * ++ is trivial.  Assuming that det A > 0 Proposition 4(i) implies that there is no Poincaré map Π ++ , associated to the flow of a homogeneous linear system and to two parallel straight lines L + and L − symmetric with respect to the origin, defined on L I + with the image contained on L O + . This implies that either the behavior of Π * ++ is trivial or Π * ++ = Π −+ •Π −− •Π +− . Notice that in the last case the orbits have to surround the origin. It follows that Π * ++ is trivial for t 2 −4d ≥ 0 and Π * Assuming that det A < 0 we have that Π * ++ coincides either with the Poincaré map Π ++ or with the composition Π −+ • Π −− • Π +− , see Proposition 4(ii). But, in the last case we need that the orbits surround the origin and this is a contradiction with det A < 0. Therefore, Π * ++ is the map Π ++ .
By Proposition 19 the behavior of the map Π * ++ depends on whether e + ∈ S − , or e + ∈ S + and t 2 − 4d < 0. In order to distinguish between these situations we will denote Π ++ the map Π −+ • Π −− • Π +− . Consequently we reduce the study of the Poincaré map Π + associated to the flow of system (16) and to the straight line L + to study the Poincaré maps Π ++ and Π ++ defined by the flow of the homogeneous linear system (18) and associated to the lines L * + and L * − . Therefore, to know the qualitative behavior of the Poincaré map Π + defined by system (16) is equivalent to know the behavior of Poincaré maps π ++ and π ++ = π −+ • π −− • π +− .

3.2.
The Poincaré maps of the linear differential systemẋ = Ax − b. In this subsection we consider the linear differential equatioṅ which is defined in the region S − and we study the Poincaré map defined by the flow Set Φ − (t, q) the flow of system (19) such that Φ − (0, q) = q. Then we define in L − the following subset Suppose that Dom −− = ∅ then we can define the Poincaré map of the linear differential system (19) associated to the straight line L − by (20) Suppose that the flow of system (19) has a unique contact point p with L − . We have that Π − (p) = p because p ∈ L I − and p ∈ L O − . Let e − = A −1 b be the singularity of system (19). Notice that the translation -If det A > 0 and t 2 − 4d < 0, then Π * ++ coincides with the composition By Proposition 21 the behavior of the map Π * ++ depends on whether e − ∈ S − , or e − ∈ S + . In order to distinguish between these situation we will denote the map We reduce the study of the Poincaré map Π − associated to the flow of system (19) and to the straight line L − to study the Poincaré maps Π ++ and Π ++ defined by the flow of the homogeneous linear system (21) and associated to two parallel lines L * − and L * + . Then to know the qualitative behavior of the Poincaré map Π − , defined by system (19), is equivalent to know the behavior of of Poincaré maps π ++ and π ++ = π −+ • π −− • π +− . Remark 1. By Proposition 6 we have that π ++ = π −− and π ++ = π −− . In order to not confuse the Poincaré maps π jk , where j, k ∈ {+, −}, associated to system (19), with the maps studied on the previous section, we will use the notation π −− and π −− .

3.3.
Return map of the discontinuous system (15). In this subsection we describe the return map Π defined by the flow of the discontinuous piecewise linear differential system (15) and associated to the straight line L as the composition of the Poincaré maps Π + and Π − defined in subsections 3.1 and 3.2.
Notice that the half-lines L I + , L O + , L I − , and L O − are contained in the straight line L. Furthermore, when the Poincaré map Π + is defined we have that its domain is Dom ++ ⊂ L O + and its image is contained in L I + . Moreover, when the Poincaré map Π − is defined we have that its domain is Dom −− ⊂ L I − and its image is contained in L O − . Then, consider the subset L I q) is the solution of the discontinuous system (15). Observe that if d = 0 then Dom = ∅ and therefore the return map is not defined. In what follows we assume that A ∈ GL(R 2 ).
When Dom = ∅ we define the return map associated to the flow of the discontinuous system(15) and to the straight line L by In what follows we refer Π + and Π − the restriction of the Poincaré maps define in subsections 3.1 and 3.2 to subsets L O + ∩ L O − and L I + ∩ L I − , respectively. Theorem 3. Consider the discontinuous piecewise linear differential system (15). Assume that the return map Π associated to the flow of system (15) and to the straight line L is defined.
-If detA < 0, then the return map Π is not defined. (ii) Suppose that e + ∈ S − and e − ∈ S − .
-If det A < 0 or det A > 0 and t 2 − 4d ≥ 0 then return map Π is not defined. (iii) Suppose that e + ∈ S + and e − ∈ S − .
-If det A < 0 or det A > 0 and t 2 − 4d < 0, then Π = Π − • Π + . In what follows we assume that the return map of the discontinuous system (15) is defined. That is Dom = ∅.
Let p + and p − be the contact point of the systemẋ = Ax ± b, respectively. By definition of L * O + and L * I − and from Figure 5 we have the following: • If det A > 0, e + ∈ S − and e − ∈ S + , then • If det A > 0, e + ∈ S − and e − ∈ S − , then • If det A > 0, e + ∈ S + and e − ∈ S − , then • If det A < 0, e + ∈ S + and e − ∈ S − , then Since the behavior of maps Π + and Π − are determined by the Poincaré maps π ++ , π −− , π ++ and π −− , we will study the behavior of the map Π via the Poincaré maps π jk , where j, k ∈ {+, −}. Note that the behavior of those maps were studied in subsection 2.1.
We will use π ± to identify which of the linear systemsẋ = Ax±b defines the map π jk and we denote p ± the respective contact points with respect to the straight line L. Thus either π + = π ++ or π + = π ++ , and either π − = π −− or π − = π −− . Given q ∈ Dom, let r and r 0 be the coordinates of q and Π(q) on the half-line L O + , respectively. We define the return map π as π(r) = r 0 , that is the return map transforms the coordinate of q into the coordinate of Π(q).
Consider p ∈ L I + ∩ L I − and q ∈ L O + ∩ L O − , set r and s the coordinates of p on the half-lines L I + and L I − , respectively, and m and n the coordinates of q on the half-lines L O + and L O − , respectively. By previous definition of L I ± we have that • if det A > 0, e + ∈ S − and e − ∈ S + then p = p + + rṗ + = p − − sṗ − and q = p + − mṗ + = p − + nṗ − .
Thus we can define the maps • if det A > 0, e + ∈ S − and e − ∈ S − , then Thus we can define the maps • if det A > 0, e + ∈ S + and e − ∈ S − , then Thus we can define the maps • if det A < 0, e + ∈ S + and e − ∈ S − , then Thus we can define the maps where are the coordinates of the contact points p − and p + with respect to the straight lines L + and L − , respectively. On the other hand, if p − belongs to L O + then the maps f i and h i satisfy with r * and s * given in (32). Finally, assuming that p − = p + we get Observe that f i and h i , i = 1, 2, 3, 4, are increasing linear maps that tends to infinity when r and s tends to infinity, respectively. Moreover, we have the following results. • p − ∈ L I + then f i (r) < h −1 i (r) for every r ∈ [ r * , ∞). Proof. Consider e + ∈ S − and e − ∈ S − . In this case we have Therefore p − = p + implies that h −1 2 (r) = f 2 (r), see (28). On the other hand, if p − = p + , taking r ∈ [ r * , ∞) we have that p 1 = p + +rṗ + ∈ L I + and p 2 = p + −rṗ + ∈ L O + are symmetric points with respect to the contact point p + . Furthermore The others cases can be proved in a similar way.
The following result provides the return map π as compositions of the Poincaré maps π + , π − and the maps f i and h i , i = 1, 2.
Assume that e + ∈ S − and e − ∈ S + . By Theorem 3-(i) the return map of the discontinuous system (15) is given by Π = Π − • Π + for d > 0 and it is not defined when d < 0. Furthermore Proposition 19 implies that the behavior of the Poincaré map Π + is equivalent to behavior of Π ++ for d > 0 and Π + is trivial for d < 0. Proposition 21 and remark 1 imply that the behavior of the Poincaré map Π − is equivalent to the behavior of Π −− for d > 0 and Π − is trivial for d < 0. For the Poincaré maps π jk , j, k ∈ {+, −} we have that π + = π ++ and π − = π −− for d > 0 and π − , π + are defined only at zero for d < 0. Since e + ∈ S − , e − ∈ S + and d > 0, we have that the maps f 1 and h 1 are defined, see equation (27). Therefore the return map is given by π(r) = h 1 (π −− (f 1 (π ++ (r)))) for d > 0 and it is not defined for d < 0. We have proved statement (1).
The others cases can be proved in a similar way and the result follows.

4.
Existence of limit cycle for the discontinuous system (15). From the definition of the return map π we have that the existence of periodic orbits for system (15) is equivalent to the existence of a fixed point of π. Moreover any limit cycle is associated to an isolated fixed point of π.
Theorem 6. Consider the discontinuous piecewise linear differential system (15). If one of the statements below is satisfied, then system (15) has no limit cycles.
Proof. On these assumptions the return map π is not defined, see Theorem 5. Then system (15) has no limit cycles.
In the rest of this paper we assume that the return map π for the discontinuous piecewise linear differential system (2) is defined.
Suppose that p − = p + , that is the contact points of system (16) and (19) coincide. Then we have the following results.
Theorem 8. Consider the discontinuous piecewise linear differential system (15), assume that p − = p + and the return map π is defined.
• If t = 0 then π has not fixed points.
• If t = 0 then π is the identity map.
Proof. According to Theorem 5 we should consider the following cases: (i) e + ∈ S − and e − ∈ S + , (ii) e + ∈ S − and e − ∈ S − , and (iii) e + ∈ S + and e − ∈ S − . We shall prove the case (i), the others cases can be proved in a similar way, observing that the behavior of the maps π ++ is characterized in Propositions 16 and 17.
Consequently if t = 0 then either g(r) > 0, or g(r) < 0. Therefore the return map has not fixed points, see Lemma 7.
Proof of Theorem 1. By Theorem 8 we conclude that either system (2) has no closed orbits, or it contains a continuum of closed orbits. Therefore system (2) has no limit cycle.
In what follows we assume that p − = p + .
If e + ∈ S − and e − ∈ S − we have that Finally assume that e + ∈ S + and e − ∈ S − , thus 4.1. Virtual-virtual case. In this subsection we assume that the singularities of the discontinuous piecewise linear differential system (15) are such that e + ∈ S − and e − ∈ S + and we analyze the existence of limit cycles.
Theorem 9. Suppose that d > 0 and p − ∈ L O + . If t ≥ 0, then π has not fixed points. If t < 0, then π has one fixed point.