CROSSING LIMIT CYCLES FOR A CLASS OF PIECEWISE LINEAR DIFFERENTIAL CENTERS SEPARATED BY A CONIC

In previous years the study of the version of Hilbert’s 16th problem for piecewise linear differential systems in the plane has increased. There are many papers studying the maximum number of crossing limit cycles when the differential system is defined in two zones separated by a straight line. In particular in [11, 13] it was proved that piecewise linear differential centers separated by a straight line have no crossing limit cycles. However in [14, 15] it was shown that the maximum number of crossing limit cycles of piecewise linear differential centers can change depending of the shape of the discontinuity curve. In this work we study the maximum number of crossing limit cycles of piecewise linear differential centers separated by a conic.


Introduction and statement of the main results
The study of discontinuous piecewise linear differential systems in the plane started with Andronov, Vitt and Khaikin in [1]. After that these systems have been a topic of great interest in the mathematical community because of their applications in various areas. They are used for modeling real phenomena and different modern devices, see for instance the books [4,24] and references therein.
In the qualitative theory of differential systems in the plane a limit cycle is a periodic orbit which is isolated in the set of all periodic orbits of the system. This concept was defined by Poincaré [20,21]. In several papers as [3,10,25] it was shown that the limits cycles model many phenomena of the real world. After these works the non-existence, existence, the maximum number and other properties of the limit cycles have been extensively studied by mathematicians and physicists, and more recently, by biologists, economist and engineers, see for instance [4,17,18,19,26].
As for the general case of planar differential systems one of the main problems for the particular case of the piecewise linear differential centers is to determine the existence and the maximum number of crossing limits cycles that these systems can exhibit. In this paper we study the crossing limit cycles which are periodic orbits isolated in the set of all periodic orbits of the piecewise linear differential centers, which only have isolated points of intersection with the discontinuity curve.
To establish an upper bound for the number of crossing limit cycles for the family of piecewise linear differential systems in the plane separated by a straight line has been the subject of many recent papers, see for instance [2,5,7,23]. In 1990 Lum and Chua [16] conjectured that the continuous piecewise linear systems in the plane separated by one straight line have at most one limit cycle, in 1998 this conjecture was proved by Freire et al [6]. Afterwards in 2010 Han and Zhang [8] conjectured that discontinuous piecewise linear differential systems in the plane separated by a straight line have at most two crossing limit cycles but in 2012 Huan and Yang [9] gave a negative answer to this conjecture through a numerical example with three crossing limit cycles, later on Llibre and Ponce in [12] proved the existence of these three limit cycles analytically, but it is still an open problem to know if 3 is the maximum number of crossing limit cycles that this class of systems can have.
In [11] the problem by Lum and Chua was extended to the class of discontinuous piecewise linear differential systems in the plane separated by a straight line. In particular it was proved that the class of planar discontinuous piecewise linear differential centers has no crossing limit cycles. However, recently in [14,15] were studied planar discontinuous piecewise linear differential centers where the curve of discontinuity is not a straight line. It was shown that the number of crossing limit cycles in these systems is non-zero. For this reason it is interesting to study the role which plays the shape of the discontinuity curve in the number of crossing limit cycles that planar discontinuous piecewise linear differential centers can have.
In this paper we provide an upper bound for the maximum number of crossing limit cycles of the planar discontinuous piecewise linear differential centers separated by a conic Σ.
We do not consider conics of type (p), (CL) or (CE) because they do not separate the plane in connected regions. We observe that we have two options for crossing limit cycles of discontinuous piecewise linear differential centers separated by a conic Σ. First we have the crossing limit cycles such that intersect the discontinuity curve in exactly two points and second we have the crossing limit cycles such that intersect the discontinuity curve Σ in four points; we study these two cases in the following sections. 1.1. Crossing limit cycles intersecting the discontinuity curve Σ in two points. The maximum number of crossing limit cycles of piecewise linear differential centers separated by a conic Σ such that intersect Σ in exactly two points is given in the following theorems.
Theorem 1.1. Consider a planar discontinuous piecewise linear differential centers where Σ is a conic. If Σ is of the type (a) (LV), (PL) or (DL), then there are no crossing limit cycles.
(b) (E), then the maximum number of crossing limit cycles intersecting Σ in two points is two. (a) (P), then the maximum number of crossing limit cycles intersecting Σ in two points is three.
The above Theorem is proved in section 2. In the cases studied up to now, there is no a result determining the maximum number of crossing limit cycles for discontinuous piecewise linear differential centers when Σ is a hyperbola (H). We determine it in the following Theorem Theorem 1.2. Consider a family of planar discontinuous piecewise linear differential centers, F 0 , where Σ is a hyperbola (H). Then the following statement hold: (a) There are systems in F 0 without crossing limit cycles.
(b) There are systems in F 0 having exactly one crossing limit cycle that intersects Σ in two points, see Figure 2. (c) There are systems in F 0 having exactly two crossing limit cycles that intersect Σ in two points, see Figure 3. (d) For this family of systems F 0 we have that the maximum number of crossing limit cycles that intersect Σ in two points is two.
The above Theorem is proved in section 3.

1.2.
Crossing limit cycles intersecting the discontinuity curve Σ in four points. Here we do not consider the case where the discontinuity curve is the conic (DL), because first in [11,13] it was proved that discontinuous piecewise linear differential systems separated by a straight line have no crossing limit cycles and second because the crossing limit cycles of these discontinuous piecewise linear centers cannot have four points on the discontinuity curve.
In the following theorems we analyze the maximum number of crossing limit cycles for planar discontinuous piecewise linear differential centers with four points on discontinuity curve, where the plane is divided by the curve of discontinuity Σ of the type (PL), (LV),(P),(E) or (H). Theorem 1.3. Let F 1 be the family of planar discontinuous piecewise linear differential systems formed by three linear centers and with Σ of type (PL). Then for this family the maximum number of crossing limit cycles that intersect Σ in four points is one. Moreover there are systems in this class having one crossing limit cycle. Theorem 1.3 for a particular linear center between the two parallel straight lines was done in [13], in section 4 we prove it for any linear center.
If the discontinuity curve Σ is of the type (LV), then we have the following 4 regions in the plane: x > 0 and y > 0], x < 0 and y > 0], . In this case we have two types of crossing limit cycles, namely crossing limit cycles of type 1 which intersect only two branches of Σ in exactly two points in each branch, and crossing limit cycles of type 2 which intersect in a unique point each branch of the set Σ. Theorem 1.4. Let F 2 be the family of planar discontinuous piecewise linear differential systems formed by four linear centers and with Σ of the type (LV). The maximum number of crossing limit cycles type 1 is one. Moreover there are systems in this class having one crossing limit cycle.
The above theorem is proved in Section 5.
Theorem 1.5. Consider a family of planar discontinuous piecewise linear differential centers F 2 . Then the following statement hold.
(a) There are systems in F 2 with exactly one crossing limit cycle of type 2, see Figure 7. (b) There are systems in F 2 with exactly two crossing limit cycles of type 2, see Figure 8. (c) There are systems in F 2 with exactly three crossing limit cycles of type 2, see Figure 9 The above theorem is proved in Section 6. By the calculations made for this case and the illustrated examples in Theorem 1.5 we get the following conjecture Conjecture 1.6. For the family of systems F 2 , the maximum number of crossing limit cycles of type 2 is three. Theorem 1.7. Let F 3 be a family of planar discontinuous piecewise linear differential systems formed by two linear centers and with Σ of type (P). Then for this family the maximum number of crossing limit cycles that intersect Σ in four points is one. Moreover there are systems in this class having one crossing limit cycle.
The above theorem is proved in Section 7. Theorem 1.8. Let F 4 be a family of planar discontinuous piecewise linear differential systems formed by two linear centers and with Σ of type (E). Then for this family the maximum number of crossing limit cycles that intersect Σ in four points is one. Moreover there are systems in this class having one crossing limit cycle.
The above Theorem is proved in Section 8. Theorem 1.9. Let F 5 be a family of planar discontinuous piecewise linear differential systems formed by three linear centers and with Σ of type (H). Then for this family the maximum number of crossing limit cycles that intersect Σ in four points is one. Moreover there are systems in this class having one crossing limit cycle.
The above theorem is proved in Section 9. 1.3. Crossing limit cycles with four and with two points on the discontinuity curve Σ simultaneously. Here we study the maximum number of crossing limit cycles of planar discontinuous piecewise linear differential centers that intersect the discontinuity curve Σ in two and in four points simultaneously.
We do not consider planar discontinuous piecewise linear differential centers with discontinuity curve a conic of type (DL), (PL) and (LV) because as in the proof of Theorem 1.1 they do not have crossing limit cycles that intersect the discontinuity curve in two points. Then we study the maximum number of crossing limit cycles with two and with four points in Σ simultaneously by the families F 3 , F 4 and F 5 .
Theorem 1.10. The following statements hold.
(a) The planar discontinuous piecewise linear differential centers that belong to the family F 3 , can have simultaneous one crossing limit cycle that intersects (P) in two points and one crossing limit cycle that intersects (P) in four points. (b) The planar discontinuous piecewise linear differential centers that belong to the family F 4 , can have simultaneous one crossing limit cycle that intersects (E) in two points and one crossing limit cycle that intersects (E) in four points. (c) The planar discontinuous piecewise linear differential centers that belong to the family F 5 , can have simultaneous one crossing limit cycle that intersects (H) in two points and one crossing limit cycle that intersects (H) in four points.
The above Theorem is proved in Section 10. In Subsection 1.3 we do not consider the planar discontinuous piecewise linear differential centers in the family F 2 , because they do not have crossing limit cycles that intersect the discontinuity curve (LV) in two points. However in this family there are two types of crossing limit cycles like it was defined in Subsection 1.2. 1.4. Crossing limit cycles of types 1 and 2 simultaneously for planar discontinuous piecewise linear differential centers in F 2 . In this case we study the maximum number of crossing limit cycles of types 1 and 2 that planar discontinuous piecewise linear differential centers in the family F 2 can have simultaneously. Theorem 1.11. There are planar discontinuous piecewise linear differential centers that belong to the family F 2 such that have one crossing limit cycle of type 1 and three crossing limit cycles of type 2 simultaneously.
The above theorem is proved in Section 11. By the illustrated examples in Theorem 1.11 we get the following conjecture Conjecture 1.12. The planar discontinuous piecewise linear differential centers that belong to the family F 2 can have at most one crossing limit cycle of type 1 and three crossing limit cycles of type 2 simultaneously.

Proof of Theorem 1.1
Analyzing the case of discontinuous piecewise linear differential centers with discontinuity curve a conic of the type (LV), (PL) or (DL) the maximum number of crossing limit cycles is equal to the maximum number of crossing limit cycles in discontinuous piecewise linear differential centers in the plane separated by a single straight line which was studied in [11,13]. In these papers it was proved that the discontinuous piecewise linear differential centers separated by one straight line have no crossing limit cycles. This proves the statement (a) of Theorem 1.1.
In [15] the authors considered discontinuous piecewise linear differential centers separated by the parabola y = x 2 and proved that they have at most three crossing limit cycles that intersect Σ in two points, i.e. statement (b) of Theorem 1.1.
With regard to the discontinuous piecewise linear differential systems separated by an ellipse, in the paper [14] the authors shown that the class of planar discontinuous piecewise linear differential centers separated by the circle S 1 has at most two crossing limit cycles. Moreover, there are discontinuous piecewise linear differential centers which reach the upper bound of 2 crossing limit cycles, see Example 2.1. Then we have the statement (c) of Theorem 1.1.
Example 2.1. We consider the discontinuous piecewise linear differential system in R 2 separated by the ellipse (E) and both linear differential centers are defined as follows:ẋ = −2x − 2y − √ 2 − 1,ẏ = 4x + 2y + √ 2, in the unbounded region limited by the ellipse (E), and in the bounded region with boundary the ellipse (E) we have the linear differential centeṙ This discontinuous piecewise differential system has exactly two crossing limit cycles, see Figure 1. For the systems of the class F 0 we have following regions in the plane: which is a region that consist of two connected components, and the region To have a crossing limit cycle, which intersects the hyperbola x 2 − y 2 = 1 in two different points p = (x 1 , y 1 ) and q = (x 2 , y 2 ), these points must satisfy the closing equations H 1 (x 1 , y 1 ) = H 1 (x 2 , y 2 ), H 2 (x 2 , y 2 ) = H 2 (x 1 , y 1 ), Proof of statement (a) of Theorem 1.2. We consider a discontinuous piecewise linear differential system which has the linear centeṙ in the region R 2 , the orbits of this center intersect the hyperbola in two or in four points, when it intersects the hyperbola in exactly two points these are (±1, 0), which are points of tangency between the hyperbola and the solution curves of the center (3.2), then it is impossible that there are crossing periodic orbits independent of the linear differential center that can be considered in the region R 1 . So the orbits which can produce a crossing limit cycle intersect the hyperbola in four points and clearly these orbits cannot be crossing limit cycles with exactly two points on the discontinuity curve (H).
Proof of statement (b) of Theorem 1.2. In the region R 1 we consider the linear differential centerẋ this system has the first integral H 1 (x, y) = 4(−4 + x)x + 4y(−54 + 2 √ 5 + 25y). In the region R 2 we have the linear differential centeṙ which has the first integral H 2 (x, y) = 4(−3 + x)x + y(−16 + 6 √ 5 + y). This discontinuous piecewise differential system formed by the linear differential centers (3.3) and (3.4) has one crossing limit cycle, because the unique real solution (p, q) with p = q of the closing equations given in (3.1), is p = (1, 0) and q = ( √ 5, 2). See the crossing limit cycle of this system in Figure 2.
Proof of statement (c) of Theorem 1.2. In the region R 1 we consider the linear differential centerẋ which has the first integral In the region R 2 we have the linear differential centeṙ this system has the first integral This discontinuous piecewise differential system formed by the linear differential centers (3.5) and (3.6) has two crossing limit cycles, because the unique real solutions (p, q) of system (3.1) are (1, 0, √ 2, 1) and ( √ 2, −1, √ 3, √ 2), therefore the intersection points of the two crossing limit cycles with the hyperbola are the pairs (1, 0), ( √ 2, 1) and ( √ 2, −1), ( √ 3, √ 2). See these two crossing limit cycles in Figure 3.
We will use the following lemma which provides a normal form for an arbitrary linear differential center, for a proof see [13].
Lemma 3.1. Through a linear change of variables and a rescaling of the independent variable every center in R 2 can be writteṅ with a = 0 and ω > 0. This system has the first integral Proof of statement (d) of Theorem 1.2. In the region R 1 we consider the arbitrary linear differential center (3.7) which has first integral (3.8). In the region R 2 we consider the arbitrary linear differential centeṙ with A = 0 and Ω > 0. Which has the first integral It is possible to do a rescaling of time in the two above systems. Suppose τ = at in R 1 and s = At in R 2 . These two rescaling change the velocity in which the orbits of systems (3.7) and (3.9) travel, nevertheless they do not change the orbits, therefore they will not change the crossing limit cycles that the discontinuous piecewise linear differential system can have. After these rescalings of the time we can assume without loss of generality that a = A = 1, and the dot in system (3.7) (resp. (3.9)) denotes derivative with respect to the new time τ (resp. s).
We assume that the discontinuous piecewise linear differential system formed by the two linear differential centers (3.7) and (3.9) has three crossing periodic solutions. For this we must impose that the system of equations (3.1) has three pairs of points as solution, namely (p i , q i ), i = 1, 2, 3, since these solutions provide crossing periodic solutions. We consider (3.10) These points are the points where the three crossing periodic solutions intersect the hyperbola (H). Now we consider that the point (p 1 , q 1 ) satisfies system (3.1) and with this condition we obtain the following expression and D has the same expression that d changing (b, c, ω) by (B, C, Ω). We assume that the point (p 2 , q 2 ) satisfies system (3.1) and we get the expression and C has the same expression that c changing (b, ω) by (B, Ω).
Finally we impose that the point (p 3 , q 3 ) satisfies system (3.1) and we get an expression for ω 2 . In this case ω 2 = K/L, where the expression for K is and the expression for Ω 2 is the same than the expression for ω 2 changing b to B. Now we replace d, c, ω 2 in the expression of the first integral H 1 (x, y) and we have and analogously we have Now we analyze if the discontinuous piecewise linear differential system formed by (3.7) and (3.9) has more crossing periodic solutions than the three supposed in (3.10). Taking into account (3.11) and (3.12) the closing equations (3.1) becomes h(x 1 , y 1 , r 1 , r 2 , r 3 , s 1 , s 2 , s 3 ) = h(x 2 , y 2 , r 1 , r 2 , r 3 , s 1 , s 2 , s 3 ), This means, we must solve a system with three equations and four unknowns x 1 , y 1 , x 2 , y 2 , which we know that have at least the three solutions (3.10), so system (3.13) has a continuum of solutions which produce a continuum of crossing periodic solutions, so such systems cannot have crossing limit cycles. Since in statement (c), we have proved that there are systems in F 0 with two crossing limit cycles, it follows that the maximum number of crossing limit cycles that intersect Σ in two points is two. This completes the proof of Theorem 1.2.

Proof of Theorem 1.3
When Σ is of the type (PL), we have following three regions in the plane: We consider a planar discontinuous piecewise differential system separated by two parallel straight lines and formed by three arbitrary linear centers. By Lemma 3.1, we have that these linear centers can be as followṡ (4.1) These linear centers have the first integrals respectively.
We are going to analyze if the discontinuous piecewise linear differential center (4.1) has crossing periodic solutions. Since the orbits in each region R i , for i = 1, 2, 3, are ellipses or pieces of one ellipse, we have that if there is a crossing limit cycle this must intersect each straight line x = ±1 in exactly two points, namely (1, y 1 ), (1, y 2 ) and (−1, y 3 ), (−1, y 4 ), with y 1 > y 2 and y 3 > y 4 . Therefore we must study the solutions of the system or equivalently, we have the system By hypothesis y 1 > y 2 and y 3 > y 4 and therefore system (4.2) is equivalently to the system . As l 1 = 0 and l 3 = 0, we can isolated y 1 and y 4 of the first and the third equations of system (4.3), respectively. Then, we obtain Now replacing these expressions for y 1 and y 4 in the second and fourth equations of (4.3), we have the system of two equations Doing the Groebner basis of the two polynomials E 1 and E 2 with respect to the variables y 2 and y 3 , we obtain the equations We recall that Bézout Theorem states that if a polynomial differential system of equations has finitely many solutions, then the number of its solutions is at most the product of the degrees of the polynomials which appear in the system, see [22]. Then by Bézout Theorem in this case, we have that system (4.4) has at most two solutions. Moreover, from these two solutions (y 1 2 , y 1 3 ) and (y 2 2 , y 2 3 ) of (4.4), we will have two solutions of (4.3) which are of the form (y 1 1 , y 1 2 , y 1 3 , y 1 4 ) and (y 2 1 , y 2 2 , y 2 3 , y 2 4 ), but analyzing system (4.3) we have that if (y 1 1 , y 1 2 , y 1 3 , y 1 4 ) is a solution, then (y 1 2 , y 1 1 , y 1 4 , y 1 3 ) is another solution. Finally due to the fact that y 1 > y 2 and y 3 > y 4 , at most one of these two solutions will be satisfactory. Therefore we have proved that the planar discontinuous piecewise differential systems of the family F 1 , can have at most one crossing limit cycle. Now we verify that this upper bound is reached, for this we present a discontinuous piecewise linear differential system that belongs to the family F 1 and has exactly one crossing limit cycle.

Proof of Theorem 1.4
We consider a planar discontinuous piecewise linear differential system with four zones separated by (LV) and formed by four arbitrary linear centers. By Lemma 3.1 this piecewise linear differential system can be as followṡ If the discontinuous piecewise linear center (5.1) has two crossing limit cycles of type 1, these two crossing limit cycles should be some of Figure 5. Figure 5. Possible cases of two crossing limit cycles of type 1 of discontinuous piecewise linear center (5.1).
We observe that the cases of Figure 5 (b), (c), and (d) are not possible because in these cases the pieces of the ellipses of linear differential centers in the regions R 4 , R 1 and R 2 , respectively would not be nested which contradicts that the linear differential systems in each of these regions are linear centers. Therefore if the discontinuous piecewise linear center (5.1) has two crossing limit cycles of type 1 these could be as in Figure 5 (a). Now we study the conditions in order that the piecewise linear differential center (5.1) has crossing limit cycles of type 1 and we will show that the maximum number of crossing limits cycles of type 1 is one. Without loss of generality we assume that the crossing limit cycles intersect the branches Γ + 1 and Γ + 2 in the points (0, y 1 ), (0, y 2 ) and (x 1 , 0), (x 2 , 0), respectively, where 0 < y 1 < y 2 and 0 < x 1 < x 2 . Then taking into account the first integrals (5.2) for each linear center, these points must satisfy the following equations where l 1 = 4b 2 1 + ω 2 1 , l 2 = 4b 2 2 + ω 2 2 and η = (a 4 c 1 − a 1 c 4 ). Moreover, by hypothesis x 1 < x 2 and y 1 < y 2 , then from the second and the fourth equations of (5.3), we have Substituting these expressions of y 1 and x 2 in the first and third equations of (5.3) we obtain the two equations Doing the Groebner basis of the two polynomials E 1 and E 2 with respect to the variables x 1 and y 2 we get the two equations where The Bézout Theorem (see [22]) applied to system (5.4) says that this system has at most two isolated solutions. Therefore system (5.3) has two solutions which are of the form (x 1 1 , x 1 2 , y 1 1 , y 1 2 ) and (x 2 1 , x 2 2 , y 2 1 , y 2 2 ), but it is possible to prove that if (x 1 , x 2 , y 1 , y 2 ) is a solution of system (5.3), then (x 2 , x 1 , y 2 , y 1 ) is also a solution of this system. Since we must have that x 1 < x 2 and y 1 < y 2 , then system (5.3) has a unique solution, and therefore the discontinuous piecewise linear differential center (5.1) that belongs to the family F 2 can have at most one crossing limit cycle of type 1 intersecting Γ + 1 and Γ + 2 . Now we verify that this upper bound is reached. That is, that there are piecewise linear differential centers in the family F 2 having one crossing limit cycle of type 1. We consider the following discontinuous piecewise linear differential centeṙ In the region R 3 we can consider any linear differential center, because the crossing limit cycle will be formed by parts of the orbits of the centers of the regions R 1 , R 2 and R 4 . Figure 6. The crossing limit cycle of type 1 of discontinuous piecewise linear differential system (5.5) separated by the conic (LV).

Proof of Theorem 1.7
If the discontinuity curve Σ is of the type (P), we have following two regions in the plane: We consider a planar discontinuous piecewise linear differential system formed by two linear arbitrary centers. By Lemma 3.1 these piecewise linear differential centers can be as followṡ y + δ,ẏ = αx + βy + γ, in R 2 . (7.1) These linear differential centers have the first integrals H 1 (x, y) = 4(ax + by) 2 + 8a(cx − dy) + y 2 ω 2 , respectively. After two rescalings of time as in the proof Theorem 1.2 we can assume without loss of generality that a = α = 1.
In order that the piecewise linear differential centers (7.1) has crossing limit cycles with four point on (P). We must study the solutions of the system: or equivalently We assume that the discontinuous piecewise linear differential centers (7.1) has two crossing periodic solutions. For this we must have that system of equations (7.4) has two real solutions, namely (p 1 , p 2 , p 3 , p 4 ) and (q 1 , q 2 , q 3 , q 4 ), where p i = (k i , k 2 i ) and q i = (L i , L 2 i ), with i = 1, 2, 3, 4. These points are the points where the two crossing periodic solution intersect discontinuity curve (P).
If the point (p 1 , p 2 , p 3 , p 4 ) satisfies system (7.4), by the equation e 1 of (7.4) we obtain the expression by equation e 2 of (7.4) we get the expression by equation e 3 of (7.4) we obtain the expression and by equation e 4 of (7.4) we obtain the expression here we consider l 1 = 4b 2 + ω 2 and l 2 = 4β 2 + Ω 2 . We assume that the point (q 1 , q 2 , q 3 , q 4 ) satisfies system (7.4), then we can obtain the remaining parameters of discontinuous piecewise linear differential center (7.1).
By equation e 1 of (7.4) we obtain ω 2 = S/T , where by equation e 2 of (7.4) we obtain by equations e 3 and e 4 we get that b = β = 0. This implies that the linear differential systems in R 1 and in R 2 are of the forṁ which is a contradiction because by hypothesis each of the linear differential systems considered is a center. Therefore we have proved that the maximum number of crossing limit cycles of the discontinuous piecewise linear differential centers in F 3 is one. Now we verify that this upper bound is reached. That is, that there are piecewise linear differential centers in the family F 3 having one crossing limit cycle. We consider the discontinuous piecewise linear differential system formed by the following linear differential centerṡ These linear differential centers have the first integrals H 1 (x, y) = 64x 2 + x(587 − 128y) + y(−831 + 68y), respectively. Figure 10. The crossing limit cycle of the discontinuous piecewise differential center formed by (7.5) and (7.6) separated by the conic (P).

Proof of Theorem 1.8
When the discontinuity curve Σ is of the type (E), we have following two regions in the plane: R 1 = [(x, y) ∈ R 2 : x 2 + y 2 < 1], and R 2 = [(x, y) ∈ R 2 : x 2 + y 2 > 1]. By Lemma 3.1 a piecewise linear differential center of family F 4 can be consider as (7.1) where the first integrals are given in (7.2). Now we study the conditions in order that a piecewise linear differential center in the family F 4 has crossing limit cycles intersecting the discontinuity curve (E) in exactly four points. Taking into account the first integrals (7.2) a piecewise linear differential center in F 4 has crossing limit cycles if there are points (x i , y i ) for i = 1, 2, 3, 4 satisfying the equations 4 + y 2 4 − 1 = 0, considering l 1 = 4b 2 + ω 2 and l 2 = 4β 2 + Ω 2 , we have the equivalent system Where we consider without generality a = α = 1 as in the proof Theorem 1.2.
We assume that this piecewise linear differential center has two crossing periodic solutions. For this we have that system (8.1) has two pairs of solutions, (p 1 , p 2 , p 3 , p 4 ) and (q 1 , q 2 , q 3 , q 4 ) with p i = p j , and q i = q j , for i = j and i, j = 1, 2, 3, 4. Since these solution points are on the circle (E), then we can consider them in the following way where s i , t i ∈ [0, 2π), for i = 1, 2, 3, 4. Substituting the first solution (p 1 , p 2 , p 3 , p 4 ) with p i as in (8.2) in (8.1) we can determine the parameters d, δ, c, γ of the piecewise linear differential centers (7.1), and obtain Analogously, substituting the second solution (q 1 , q 2 , q 3 , q 4 ) with q i as in (8.2) in (8.1) we get remaining parameters ω, Ω, b, β. Substituting k i , λ i , m i , n i like (8.2) we obtain that b = β = 0, therefore we get that the piecewise linear differential centers is formed by linear differential centerẋ = −y,ẏ = x, in the regions R 1 and R 2 . This is a contradiction because with this linear differential center is not possible to generate crossing limit cycles. Then we proved that the maximum number of crossing limits cycles for piecewise linear differential centers in F 4 is one. Moreover this maximum number is reached, that is, there are piecewise linear differential centers in F 4 such that have one crossing limit cycle with four points on (E), as we see below. Consider the discontinuous piecewise differential center in the family F 4 formed by the following two linear differential centerṡ These linear differential centers have the first integrals H 1 (x, y) = 512x 2 + x(−71 + 89 √ 2 + 320y) + y(107 + 89 √ 2 + 690y), respectively. Then the discontinuous piecewise differential center formed by the linear differential centers (8.3) and (8.4) has one crossing limit cycle, because the unique real solution (p 1 , p 2 , p 3 , p 4 ) of system (8.1) is p 1 = (1, 0), p 2 = (− √ 2/2, 1/ √ 2), p 3 = (−1, 0), and p 4 = (0, −1). See the crossing limit cycle of this system in Figure  11. 9. Proof of Theorem 1.9 If the discontinuity curve Σ is of the type (H) we have following three regions in the plane: We consider a planar discontinuous piecewise linear differential system formed by three linear arbitrary centers. By Lemma 3.1 these linear differential centers can be as followṡ (9.1) These linear differential centers have the first integrals respectively.
To have a crossing limit cycle, which intersects the discontinuity curve (H) in four different points p i = (x i , y i ), i = 1, 2, 3, 4, these points must satisfy the following equations 3) equivalently, we have where l i = 4b 2 i + ω 2 i , for i = 1, 2, 3. Here we are taking without generality a 1 = a 2 = a 3 = 1 as in the proofs of the previous theorems.
We assume that the discontinuous piecewise linear differential system formed by the three linear differential centers in (9.1) has two crossing periodic solutions. For this we must impose that the system of equations (9.4) has two of real solutions, namely (p 1 , p 2 , p 3 , p 4 ) and (q 1 , q 2 , q 3 , q 4 ). Since these solutions provide crossing periodic solutions and these points are the points where the crossing periodic solutions intersect the hyperbola (H) we can consider p i = (k i , λ i ) = (cosh r i , sinh r i ) and q i = (m i , n i ) = (cosh s i , sinh s i ), (9.5) with r i , s i ∈ R for i = 1, 2, 3, 4. Now we assume that the point (p 1 , p 2 , p 3 , p 4 ) with p i = (k i , λ i ), i = 1, 2, 3, 4 satisfy (9.4), and then we obtain the following expressions for i = 1, 2, 3, and We assume that the point (q 1 , q 2 , q 3 , q 4 ) with q i as in (9.5) satisfies system (9.4). By the first equation in (9.4) we obtain that By the second equation of (9.4) we obtain that ω 2 2 = K/S where By the third equation we obtain that Finally by the fourth equation in (9.4) we have that b 2 = 0. With these expressions for d 2 , c 2 , ω 2 and b 2 we obtain that the linear differential system in the region R 2 isẋ = y,ẏ = x, which is a linear differential system type saddle. This is a contradiction because we are working with centers in each region R i for i = 1, 2, 3. Therefore we have proved that the maximum number of crossing limit cycles for systems in F 5 is one.
Moreover it is possible to show that there are piecewise linear differential centers in F 5 such that have one crossing limit cycle. Indeed, consider the discontinuous piecewise linear differential system in the family F 5 formed by the following linear differential centerṡ  Proof of statement (a) of Theorem 1. 10. In this case we use the notations given in the proof of Theorem 1.7, then we consider the planar discontinuous piecewise linear center (7.1) and the first integrals (7.2). In order that the discontinuous piecewise linear center (7.1) has crossing limit cycles with four points, namely (x 1 , x 2 1 ), (x 2 , x 2 2 ), (x 3 , x 2 3 ), (x 4 , x 2 4 ) and one crossing limit cycle with two points, namely (x 5 , x 2 5 ), (x 6 , x 2 6 ) on (P), we must study the solutions (x 1 , x 2 , x 3 , x 4 , x 5 , x 6 ) of system (7.3) and the equations e 5 = H 1 (x 5 , x 2 5 ) − H 1 (x 6 , x 2 6 ) = 0, e 6 = H 2 (x 6 , x 2 6 ) − H 2 (x 5 , x 2 5 ) = 0, or equivalently systems (7.4) and We assume that systems (7.4) and (10.1) have two real solutions where each real solution provides one crossing limit cycle with four points on (P) and one crossing limit cycle whit two points on (P), but by Theorem 1.7 we have that discontinuous piecewise linear center (7.1) has at most 1 crossing limit cycle with four points on (P), therefore if we have two real solutions of systems (7.4) and (10.1) they are of the form (x 1 , x 2 , x 3 , x 4 , x 5 , x 6 ) = (k 1 , k 2 , k 3 , k 4 , k 5 , k 6 ) and (x 1 , x 2 , x 3 , x 4 , x 5 , x 6 ) = (k 1 , k 2 , k 3 , k 4 , λ 5 , λ 6 ), with k i , λ 5 , λ 6 ∈ R for i = 1, 2, 3, 4, 5, 6. If the point (k 1 , k 2 , k 3 , k 4 , k 5 , k 6 ) satisfies systems (7.4) and (10.1), by the equations e 1 , e 2 , e 3 and e 4 of (7.4) we obtain expressions for the parameters d, δ, c and γ as in the proof of Theorem 1.7, by the equation e 5 of system (10.1) we obtain an expression for ω 2 = S/T with S and T as in the proof of Theorem 1.7 changing L 1 and L 2 by k 5 and k 6 , respectively. By equation e 6 of system (10.1) we obtain Ω 2 = V /W where the expression for V and W are the same expressions that in the proof of Theorem 1.7 changing L 3 by k 5 . We assume that the point (k 1 , k 2 , k 3 , k 4 , λ 5 , λ 6 ) satisfies systems (7.4) and (10.1), then we have e 1 = e 2 = e 3 = e 4 = 0 and by the equations e 5 and e 6 of system (10.1) we obtain b = β = 0. As in the proof of Theorem 1.7 we can conclude that the two linear centers in (7.1) becamė x = 1/2,ẏ = x, which is a contradiction. So systems (7.4) and (10.1) have at most one solution and therefore planar discontinuous piecewise linear centers in F 3 have at most one crossing limit cycle with four point on (P) and one crossing limit cycle with two points on (P) simultaneously. Moreover this upper bound is reached, this is there are systems in F 3 with one crossing limit cycle with four points on (P) and one crossing limit cycle with two points on (P) simultaneously. We consider the discontinuous piecewise linear differential system formed by the linear centersẋ 2) These linear differential centers have the first integrals H 1 (x, y) = 229x 2 + 10y(−245 + 31y) − x(206 + 229y), respectively.

(10.4)
We assume that systems (8.1) and (10.4) have two real solutions where each real solution provides one crossing limit cycle with four points on (E) and one crossing limit cycle with two points on (E), like in Theorem 1.8 we proved that discontinuous piecewise linear center (7.1) has at most 1 crossing limit cycle with four points on (E), then we have that if there are two real solutions of systems (8.1) and (10.4) they are of the form (p 1 , p 2 , p 3 , p 4 , p 5 , p 6 ) and (p 1 , p 2 , p 3 , p 4 , q 5 , q 6 ), with p i and q j as (8.2) for i = 1, 2, 3, 4, 5, 6 and j = 5, 6. Substituting the first solution (p 1 , p 2 , p 3 , p 4 , p 5 , p 6 ) in systems (8.1) and (10.4) we obtain from the equations e 1 , e 2 , e 3 and e 4 of (8.1) the same expressions than in the proof of Theorem 1.8 for d, δ, c, γ, and by the equations e 5 and e 6 of system (10.4) we obtain the same expressions than in the proof of Theorem 1.8 for ω and Ω changing (m 1 , n 1 ) by (k 5 , λ 5 ) and (m 2 , n 2 ) by (k 6 , λ 6 ), respectively. We assume that the point (p 1 , p 2 , p 3 , p 4 , q 5 , q 6 ) satisfies systems (8.1) and (10.4), then we have e 1 = e 2 = e 3 = e 4 = 0 and by the equations e 5 and e 6 of system (10.4) we obtain b = β = 0. As in the proof of Theorem 1.8 we obtain that both linear centers in (7.1) becomeẋ = −y,ẏ = x, in contradiction that they have limit cycles. So we can conclude that systems (8.1) and (10.4) have at most one solution and therefore planar discontinuous piecewise linear centers in F 4 have at most one crossing limit cycle with four points on (E) and one crossing limit cycle with two points on (E) simultaneously.
Now we verify that this upper bound is reached, that is there are systems in F 4 with one crossing limit cycle with four points on (E) and one crossing limit cycle with two points on (E) simultaneously. We consider the discontinuous piecewise linear differential system in F 4 formed by the linear centerṡ  Figure 14. The two limit cycles of the discontinuous piecewise linear differential system formed by the centers (10.5) and (10.6).
Considering the first solution (p 1 , p 2 , p 3 , p 4 , p 5 , p 6 ) of systems (9.4) and (10.7) we obtain the same expressions that in the proof of Theorem 1.9 for d 1 , d 2 , d 3 , c 2 , c 1 , ω 2 changing (m 1 , n 1 ) by (k 5 , λ 5 ) and (m 2 , n 2 ) by (k 6 , λ 6 ), respectively. Now we assume that the point (p 1 , p 2 , p 3 , p 4 , q 5 , q 6 ) satisfies systems (9.4) and (10.7), then we have e 1 = e 2 = e 3 = e 4 = 0, and by the equation e 5 of system (10.7) we obtain b 2 = 0 and with this the linear system in the region R 2 becomes a saddle which is a contradiction, because we are working with linear centers in each regions R i for i = 1, 2, 3. Therefore the discontinuous piecewise linear center (9.1) has at most one crossing limit cycle with four points on (H) and one crossing limit cycle with two points on (H) simultaneously. Moreover this upper bound is reached, that is there are piecewise linear differential centers in F 5 such that have one crossing limit cycle with four points on (H) and one crossing limit cycle with two points on (H) simultaneously. Indeed consider the piecewise linear differential system formed by the linear centerṡ