Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 41, pp. 1–36. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu CROSSING LIMIT CYCLES FOR A CLASS OF PIECEWISE LINEAR DIFFERENTIAL CENTERS SEPARATED BY A CONIC JOHANA JIMENEZ, JAUME LLIBRE, JOÃO C. MEDRADO Abstract. In previous years the study of the version of Hilbert’s 16th prob- lem 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 discontinu- ity curve. In this work we study the maximum number of crossing limit cycles of piecewise linear differential centers separated by a conic. 1. 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 2010 Mathematics Subject Classification. 34C05, 34C07, 37G15. Key words and phrases. Discontinuous piecewise linear differential centers; limit cycles; conics. c 2020 Texas State University. Submitted March 20, 2019. Published May 7, 2020. 1 2 J. JIMENEZ, J. LLIBRE. J. C. MEDRADO EJDE-2020/41 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 sepa- rated by a conic Σ. Using an affine change of coordinates, any conic can be written in one of following nine canonical forms: (p) x2 + y 2 = 0 two complex straight lines intersecting at a real point; (CL) x2 + 1 = 0 two complex parallel straight lines; (CE) x2 + y 2 + 1 = 0 complex ellipse; (DL) x2 = 0 one double real straight line; (PL) x2 − 1 = 0 two real parallel straight lines; (LV) xy = 0 two real straight lines intersecting at a real point; (E) x2 + y 2 − 1 = 0 ellipse; (H) x2 − y 2 − 1 = 0, hyperbola; (P) y − x2 = 0 parabola. 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 differen- tial 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. EJDE-2020/41 CROSSING LIMIT CYCLES 3 (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 differ- ential centers, F0 , where Σ is a hyperbola (H). Then the following statement hold: (a) There are systems in F0 without crossing limit cycles. (b) There are systems in F0 having exactly one crossing limit cycle that inter- sects Σ in two points, see Figure 2. (c) There are systems in F0 having exactly two crossing limit cycles that inter- sect Σ in two points, see Figure 3. (d) For this family of systems F0 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 F1 be the family of planar discontinuous piecewise linear dif- ferential 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: R1 = [(x, y) ∈ R2 : x > 0 and y > 0], R2 = [(x, y) ∈ R2 : x < 0 and y > 0], R3 = [(x, y) ∈ R2 : x < 0 and y < 0], R4 = [(x, y) ∈ R2 : x > 0 and y < 0]. − − Moreover, Σ = Γ+ + 1 ∪ Γ1 ∪ Γ2 ∪ Γ2 , where Γ+ 2 1 = [(x, y) ∈ R : x = 0, y ≥ 0], Γ− 2 1 = [(x, y) ∈ R : x = 0, y ≤ 0], 4 J. JIMENEZ, J. LLIBRE. J. C. MEDRADO EJDE-2020/41 Γ+ 2 2 = [(x, y) ∈ R : y = 0, x ≥ 0], Γ− 2 2 = [(x, y) ∈ R : y = 0, x ≤ 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 F2 be the family of planar discontinuous piecewise linear dif- ferential 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 differ- ential centers F2 . Then the following statement hold. (a) There are systems in F2 with exactly one crossing limit cycle of type 2, see Figure 7. (b) There are systems in F2 with exactly two crossing limit cycles of type 2, see Figure 8. (c) There are systems in F2 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 F2 , the maximum number of crossing limit cycles of type 2 is three. Theorem 1.7. Let F3 be a family of planar discontinuous piecewise linear differ- ential 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 F4 be a family of planar discontinuous piecewise linear differ- ential 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 F5 be a family of planar discontinuous piecewise linear differ- ential 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 disconti- nuity curve Σ simultaneously. Here we study the maximum number of crossing limit cycles of planar discontinuous piecewise linear differential centers that inter- sect 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 EJDE-2020/41 CROSSING LIMIT CYCLES 5 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 F3 , F4 and F5 . Theorem 1.10. The following statements hold. (a) The planar discontinuous piecewise linear differential centers that belong to the family F3 , 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 F4 , 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 F5 , 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 con- sider the planar discontinuous piecewise linear differential centers in the family F2 , 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 dis- continuous piecewise linear differential centers in F2 . In this case we study the maximum number of crossing limit cycles of types 1 and 2 that planar discontin- uous piecewise linear differential centers in the family F2 can have simultaneously. Theorem 1.11. There are planar discontinuous piecewise linear differential centers that belong to the family F2 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 F2 can have at most one crossing limit cycle of type 1 and three crossing limit cycles of type 2 simultaneously. 2. 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 = x2 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. 6 J. JIMENEZ, J. LLIBRE. J. C. MEDRADO EJDE-2020/41 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 discontin- uous piecewise linear differential centers separated by the circle S1 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 R2 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 center 5 1 1 1 ẋ = −x + y − √ − , ẏ = −x + y + √ . 4 2 8 2 This discontinuous piecewise differential system has exactly two crossing limit cy- cles, see Figure 1. Figure 1. The two limit cycles of the discontinuous piecewise linear differential of Example 2.1. 3. Proof of Theorem 1.2 For the systems of the class F0 we have following regions in the plane: R1 = [(x, y) ∈ R2 : x2 − y 2 > 1], which is a region that consist of two connected components, and the region R2 = [(x, y) ∈ R2 : x2 − y 2 < 1]. To have a crossing limit cycle, which intersects the hyperbola x2 − y 2 = 1 in two different points p = (x1 , y1 ) and q = (x2 , y2 ), these points must satisfy the closing equations H1 (x1 , y1 ) = H1 (x2 , y2 ), H2 (x2 , y2 ) = H2 (x1 , y1 ), (3.1) x21 − y12 = 1, x22 − y22 = 1. EJDE-2020/41 CROSSING LIMIT CYCLES 7 Proof of statement (a) of Theorem 1.2. We consider a discontinuous piecewise lin- ear differential system which has the linear center ẋ = −y, ẏ = x, (3.2) in the region R2 , 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 R1 . 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 R1 we consider the linear dif- ferential center √ ẋ = 27 − 5 − 25y, ẏ = −2 + x, (3.3) √ this system has the first integral H1 (x, y) = 4(−4 + x)x + 4y(−54 + 2 5 + 25y). In the region R2 we have the linear differential center √ 3 5 y 3 ẋ = 2 − − , ẏ = − + x, (3.4) 4 4 2 √ which has the first integral H2 (x, y) = 4(−3 + x)x + y(−16 + 6 5 + y). Figure 2. The crossing limit cycle of the discontinuous piecewise linear differential system formed by the centers (3.3) and (3.4). 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 6= 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 R1 we consider the linear dif- ferential center √ √ √ 289 − 48 2 + 289 3 − 305 6 x 49 ẋ = + √ − y, 768 8 3 192 √ √  p √  (3.5) 32 3 − 289 2 1 + 2 + 3 y ẏ = +x− √ , 768 8 3 8 J. JIMENEZ, J. LLIBRE. J. C. MEDRADO EJDE-2020/41 which has the first integral 1  √ √  q √  √  2 H1 (x, y) = 384x + x 32 3 − 289 2 1 + 2 + 3 − 32 3y 96  q √ √  + y 98y − 3(85057 − 9248 6) + 305 6 − 289 . In the region R2 we have the linear differential center 1 √ √ √  x y 1 √ √  y ẋ = − 3 + 8 2 + 3 − 6 − − , ẏ = − 1 − 5 2 − 3 + x + , (3.6) 8 2 2 8 2 this system has the first integral √ √ √ √ √ H2 (x, y) = 4x2 − x 1 + 5 2 + 3 − 4y + y 3 − 8 2 − 3 + 6 + 2y .   Figure 3. The two limit cycles of the discontinuous piecewise linear differential system formed by the centers (3.5) and (3.6). This discontinuous piecewise differential system formed by the linear differen- tial 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 Fig- ure 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 indepen- dent variable every center in R2 can be written 4b2 + ω 2 ẋ = −bx − y + d, ẏ = ax + by + c, (3.7) 4a with a 6= 0 and ω > 0. This system has the first integral H1 (x, y) = 4(ax + by)2 + 8a(cx − dy) + y 2 ω 2 . (3.8) Proof of statement (d) of Theorem 1.2. In the region R1 we consider the arbitrary linear differential center (3.7) which has first integral (3.8). In the region R2 we consider the arbitrary linear differential center 4B 2 + Ω2 ẋ = −Bx − y + D, ẏ = Ax + By + C, (3.9) 4A EJDE-2020/41 CROSSING LIMIT CYCLES 9 with A 6= 0 and Ω > 0. Which has the first integral H2 (x, y) = 4(Ax + By)2 + 8A(Cx − Dy) + y 2 Ω2 . It is possible to do a rescaling of time in the two above systems. Suppose τ = at in R1 and s = At in R2 . 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 (pi , qi ), i = 1, 2, 3, since these solutions provide crossing periodic solutions. We consider pi = (cosh ri , sinh ri ), qi = (cosh si , sinh si ), for i = 1, 2, 3. (3.10) These points are the points where the three crossing periodic solutions intersect the hyperbola (H). Now we consider that the point (p1 , q1 ) satisfies system (3.1) and with this condition we obtain the following expression 1  d= 4 cosh2 r1 − 4 cosh2 s1 + 8 cosh r1 (c + b sinh r1 ) 8(sinh r1 − sinh s1 )  − 8 cosh s1 (c + b sinh s1 ) + (4b2 + ω 2 )(sinh2 r1 − sinh2 s1 ) , and D has the same expression that d changing (b, c, ω) by (B, C, Ω). We assume that the point (p2 , q2 ) satisfies system (3.1) and we get the expression −1 c= 8(sinh(r1 − r2 ) + sinh(r2 − s1 ) − sinh(r1 − s2 ) + sinh(s1 − s2 ))  × (sinh r2 − sinh s2 ) 4 cosh2 s1 + 4b sinh(2s1 ) − 4 cosh2 r1 − 4b sinh(2r1 )   + (sinh r1 − sinh s1 ) 4 cosh2 r2 − 4 cosh2 s2 + 8b cosh r2 sinh r2 − 8b cosh s2 sinh s2 + (4b2 + ω 2 )(sinh r2 − sinh s2 )(− sinh r1 + sinh r2  − sinh s1 + sinh s2 ) , and C has the same expression that c changing (b, ω) by (B, Ω). Finally we impose that the point (p3 , q3 ) 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  r1 − r2 + s1 − s2  r3 − s3  4 (1 + b2 ) csch sinh 2 2  r1 − r2 − r3 + s1 − s2 − 3s3  r1 − r2 − r3 + s1 − 3s2 − s3  × cosh − cosh 2 2 r1 − r2 − 3r3 + s1 − s2 − s3  r1 − 3r2 − r3 + s1 − s2 − s3  + cosh − cosh 2 2 3r1 + r2 − r3 + s1 + s2 − s3  r1 + 3r2 − r3 + s1 + s2 − s3  − cosh + cosh 2 2 r1 + r2 − r3 + 3s1 + s2 − s3  r1 + r2 − r3 + s1 + 3s2 − s3  − cosh + cosh 2 2 10 J. JIMENEZ, J. LLIBRE. J. C. MEDRADO EJDE-2020/41  3r1 − r2 + r3 + s1 − s2 + s3  r1 − r2 + 3r3 + s1 − s2 + s3  + b2 cosh − cosh 2 2  r1 − r2 − r3 + s1 − s2 − 3s3  r1 − r2 − r3 + s1 − 3s2 − s3  − 2b sinh − sinh 2 2 2r1 − r2 + 2r3 + s1 − s2 + s3  r1 − r3  − sinh sinh 2 2 2r1 + 2r2 − r3 + s1 + s2 − s3  r1 − r2  + 2 cosh sinh 2 2 2r1 − r2 + 2r3 + s1 − s2 + s3  r1 − r3  − 2 cosh sinh 2 2 r2 − r3  r1 − 2r2 − 2r3 + s1 − s2 − s3  + 2 sinh cosh 2 2 r1 + r2 − r3 + 2(s1 + s2 ) − s3  s1 − s2  + 2 cosh sinh 2 2 r1 − r2 + r3 + 2s1 − s2  s1 − s3  − 2 cosh + s3 sinh 2 2  2 r1 − r2 + r3 + 2s1 − s2  s1 − s3  + 2(1 + b ) sinh + s3 sinh 2 2 and the expression for L is r1 − r2 + s1 − s2  r3 − s3  r1 − r2 − r3 + s1 − s2 − 3s3  csch sinh − cosh 2 2 2 r1 − r2 − r3 + s1 − 3s2 − s3  r1 − r2 − 3r3 + s1 − s2 − s3  + cosh − cosh 2 2 r1 − 3r2 − r3 + s1 − s2 − s3  3r1 + r2 − r3 + s1 + s2 − s3  + cosh + cosh 2 2 r1 + 3r2 − r3 + s1 + s2 − s3  r1 + r2 − r3 + 3s1 + s2 − s3  − cosh + cosh 2 2 r1 + r2 − r3 + s1 + 3s2 − s3  3r1 − r2 + r3 + s1 − s2 + s3  − cosh − cosh 2 2 r1 − r2 + 3r3 + s1 − s2 + s3  r1 − r2 + r3 + 2s1 − s2  + cosh − sinh + s3 2 2 s1 − s3  2 sinh , 2 and the expression for Ω2 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 H1 (x, y) and we have H1 (x, y) = 4(x2 − y 2 ) + h(x, y, r1 , r2 , r3 , s1 , s2 , s3 )b, (3.11) and analogously we have H2 (x, y) = 4(x2 − y 2 ) + h(x, y, r1 , r2 , r3 , s1 , s2 , s3 )B. (3.12) 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(x1 , y1 , r1 , r2 , r3 , s1 , s2 , s3 ) = h(x2 , y2 , r1 , r2 , r3 , s1 , s2 , s3 ), x21 − y12 = 1, (3.13) x22 − y22 = 1. EJDE-2020/41 CROSSING LIMIT CYCLES 11 This means, we must solve a system with three equations and four unknowns x1 , y1 , x2 , y2 , 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 F0 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.  4. Proof of Theorem 1.3 When Σ is of the type (PL), we have following three regions in the plane: R1 =[(x, y) ∈ R2 : x < −1], R2 =[(x, y) ∈ R2 : −1 < x < 1], R3 =[(x, y) ∈ R2 : x > 1]. 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 follows 4b2 + ω 2 ẋ = −bx − y + d, ẏ = ax + by + c, in R1 , 4a 2 2 4B + Ω ẋ = −Bx − y + D, ẏ = Ax + By + C, in R2 , (4.1) 4A 4β 2 + λ2 ẋ = −βx − y + δ, ẏ = αx + βy + γ, in R3 . 4α These linear centers have the first integrals H1 (x, y) = 4(ax + by)2 + 8a(cx − dy) + y 2 ω 2 , H2 (x, y) = 4(Ax + By)2 + 8A(Cx − Dy) + y 2 Ω2 , H3 (x, y) = 4(αx + βy)2 + 8α(γx − δy) + y 2 λ2 , 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 Ri , 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, y1 ), (1, y2 ) and (−1, y3 ), (−1, y4 ), with y1 > y2 and y3 > y4 . Therefore we must study the solutions of the system H3 (1, y2 ) = H3 (1, y1 ), H2 (1, y1 ) = H2 (−1, y3 ), H1 (−1, y3 ) = H1 (−1, y4 ), H2 (−1, y4 ) = H2 ( 1, y2 ), 12 J. JIMENEZ, J. LLIBRE. J. C. MEDRADO EJDE-2020/41 or equivalently, we have the system −(y1 − y2 )(8β − 8δ + (4β 2 + λ2 )(y1 + y2 ) = 0, 16C − 8D(y1 − y3 ) + 8B(y1 + y3 ) + (4B 2 + Ω2 )(y12 − y32 ) = 0, (4.2) (y3 − y4 )(−8b − 8d + (4b2 + ω 2 )(y3 + y4 ) = 0, −16C + 8D(y2 − y4 ) − 8B(y2 + y4 ) − (4B 2 + Ω2 )(y22 − y42 ) = 0. By hypothesis y1 > y2 and y3 > y4 and therefore system (4.2) is equivalently to the system γ3 − δ3 + l3 (y1 + y2 ) = 0, η − δ2 (y1 − y3 ) + γ2 (y1 + y3 ) + l2 (y12 − y32 ) = 0, (4.3) −γ1 − δ1 + l1 (y3 + y4 ) = 0, −η + δ2 (y2 − y4 ) − γ2 (y2 + y4 ) − l2 (y22 − y42 ) = 0, where γ1 = 8b, γ2 = 8B, γ3 = 8β, δ1 = 8d, δ2 = 8D, δ3 = 8δ, l1 = 4b2 + ω 2 , l2 = 4B 2 + Ω2 , l3 = 4β 2 + λ2 and η = 16C. As l1 6= 0 and l3 6= 0, we can isolated y1 and y4 of the first and the third equations of system (4.3), respectively. Then, we obtain −l3 y2 + γ3 − δ3 −l1 y3 + γ1 + δ1 y1 = , y4 = . l3 l1 Now replacing these expressions for y1 and y4 in the second and fourth equations of (4.3), we have the system of two equations  E1 = l2 (l3 (y2 − y3 ) + ψ3 )(l3 (y2 + y3 ) + ψ3 )  + l3 (l3 (η + (y3 − y2 )γ2 + (y2 + y3 )δ2 ) − ψ2 ψ3 ) /l32 ,  E2 = l2 ψ12 − l1 ψ1 (2l2 y3 + γ2 + δ2 )  − l12 (η + (y2 − y3 )(l2 (y2 + y3 ) + γ2 ) − (y2 + y3 )δ2 ) /l12 . Doing the Groebner basis of the two polynomials E1 and E2 with respect to the variables y2 and y3 , we obtain the equations m0 + m1 y3 + m2 y32 = 0, k0 + k1 y3 + k2 y2 = 0, (4.4) with  1 m0 = 4 2 2l13 l33 ψ22 (l3 ψ1 (γ2 + δ2 ) + l1 (2l3 η − ψ2 ψ3 )) l1 l3  − l12 l2 l32 l32 ψ12 (γ22 − 6γ2 δ2 + δ22 ) + 4l1 l3 ψ2 (2l1 η + ψ1 (γ2 + δ2 ))ψ3  − 5l12 ψ22 ψ32 + 2l1 l22 l3 (−l33 ψ13 (γ2 + δ2 ) + 2l1 l32 ψ12 ψ2 ψ3 − 2l13 ψ2 ψ33 )  2 2 3 2 2 + l1 l3 (2l1 η + ψ1 (γ2 + δ2 ))ψ3 + l2 (l3 ψ1 + l1 ψ3 ) (l3 ψ1 − l1 ψ3 ) , 4l2 ψ1 (−2l1 l3 ψ1 + l1 ψ3 ))(2l1 l3 δ2 − l2 (l3 ψ1 − l1 ψ3 )) m1 = , l13 4l2 (l2 (l3 ψ1 + l1 ψ3 − 2l1 l3 γ2 ))(l2 (l3 ψ1 − l1 ψ3 − 2l1 l2 δ2 )) m2 = , l12 EJDE-2020/41 CROSSING LIMIT CYCLES 13 (l1 l3 (l3 ψ1 (γ2 + δ2 )ψ12 + l12 ψ32 )) (2l3 (l2 ψ1 − l1 (γ2 + δ2 )) k0 = , k1 = , l12 l3 l1 k2 = 2(l3 ψ2 − l2 ψ3 ), where ψ1 = γ1 + δ1 , ψ2 = γ2 − δ2 and ψ3 = γ3 − δ3 . 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 (y21 , y31 ) and (y22 , y32 ) of (4.4), we will have two solutions of (4.3) which are of the form (y11 , y21 , y31 , y41 ) and (y12 , y22 , y32 , y42 ), but analyzing system (4.3) we have that if (y11 , y21 , y31 , y41 ) is a solution, then (y21 , y11 , y41 , y31 ) is another solution. Finally due to the fact that y1 > y2 and y3 > y4 , at most one of these two solutions will be satisfactory. Therefore we have proved that the planar discontinuous piecewise differential systems of the family F1 , can have at most one crossing limit cycle. Now we verify that this upper bound is reached, for this we present a discon- tinuous piecewise linear differential system that belongs to the family F1 and has exactly one crossing limit cycle. We consider the discontinuous piecewise linear differential center 3 x 5 1 y − − y, ẏ = ẋ = − +x+ , in R1 , 16 2 16 16 2 67 x 29 43 y ẋ = − − − y, ẏ = − +x+ , in R2 , (4.5) 500 5 100 1000 5 7 x 13 1 y ẋ = − − y, ẏ = + x + , in R3 . 60 3 36 7 3 These systems have the first integrals H1 (x, y) = 16x2 + y(6 + 5y) + 2x(1 + 8y), y 2 1 H2 (x, y) = 4 x + + y2 + (−43x + 134y), 5 125 8 14 4 H3 (x, y) = x − y + y 2 + (3x + y)2 , 7 15 9 respectively. Figure 4. The crossing limit cycle of the discontinuous piecewise linear differential center (4.5) with three centers separated by the conic (PL). 14 J. JIMENEZ, J. LLIBRE. J. C. MEDRADO EJDE-2020/41 Then the discontinuous piecewise differential system formed by the linear dif- ferential centers formed by the linear differential centers (4.5) has one crossing limit cycle that intersects (PL) in four points, because the unique real solution (y1 , y2 , y3 , y4 ) with y1 > y2 and y3 > y4 of system (4.2) is the point (y1 , y2 , y3 , y4 ) = (3/2, −27/10, 5/2, −1/2). See the crossing limit cycle of this system in Figure 4. This completes the proof of Theorem 1.3. 5. 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 follows 4b21 + ω12 ẋ = −b1 x − y + d1 , ẏ = a1 x + b1 y + c1 , in R1 , 4a1 4b2 + ω22 ẋ = −b2 x − 2 y + d2 , ẏ = a2 x + b2 y + c2 , in R2 , 4a2 (5.1) 4b23 + ω32 ẋ = −b3 x − y + d3 , ẏ = a3 x + b3 y + c3 , in R3 , 4a3 4b2 + ω42 ẋ = −b4 x − 4 y + d4 , ẏ = a4 x + b4 y + c4 , in R4 , 4a4 with ai 6= 0 and ωi > 0 for i = 1, 2, 3, 4. The regions Ri for i = 1, 2, 3, 4 are defined just before the statement of Theorem 1.4. These linear differential centers have the first integrals H1 , H2 , H3 and H4 respectively, where Hi (x, y) = 4(ai x + bi y)2 + 8ai (ci x − di y) + y 2 ωi2 , quadfor i = 1, 2, 3, 4. (5.2) 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. (a) (b) (c) (d) 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 R4 , R1 and R2 , respectively would not be nested which contradicts that the linear differential systems in each of these regions are linear centers. Therefore if the EJDE-2020/41 CROSSING LIMIT CYCLES 15 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 cen- ter (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 as- sume that the crossing limit cycles intersect the branches Γ+ + 1 and Γ2 in the points (0, y1 ), (0, y2 ) and (x1 , 0), (x2 , 0), respectively, where 0 < y1 < y2 and 0 < x1 < x2 . Then taking into account the first integrals (5.2) for each linear center, these points must satisfy the following equations H1 (x2 , 0) = H1 (0, y2 ), H2 (0, y2 ) = H2 (0, y1 ), H1 (0, y1 ) = H1 (x1 , 0), H4 (x1 , 0) = H4 (x2 , 0), equivalently we have 4a21 x22 + 8a1 (c1 x2 + d1 y2 ) − y22 l1 = 0, −(y1 − y2 )(−8a2 d2 + (y1 + y2 )l2 ) = 0, (5.3) −4a21 x21 − 8a1 (c1 x1 + d1 y1 ) + y12 l1 = 0, 4a4 (x1 − x2 )(2c4 + a4 (x1 + x2 )) = 0, where l1 = 4b21 + ω12 , l2 = 4b22 + ω22 and η = (a4 c1 − a1 c4 ). Moreover, by hypothesis x1 < x2 and y1 < y2 , then from the second and the fourth equations of (5.3), we have 8a2 d2 − l2 y2 2c4 + a4 x1 y1 = , x2 = − . l2 a4 Substituting these expressions of y1 and x2 in the first and third equations of (5.3) we obtain the two equations 4a21 (2c4 + a4 x1 )2 − 8a1 a4 (2c1 c4 + a4 c1 x1 − a4 d1 y2 ) − a24 y22 l1 E1 = , a24 l1 (y2 l2 − 8a2 d2 )2  8a2 d1 d2  E2 = 4a21 x21 − 2 − 8a1 d1 y2 − − c1 x1 . l2 l2 Doing the Groebner basis of the two polynomials E1 and E2 with respect to the variables x1 and y2 we get the two equations α0 + α1 y2 + α2 y22 = 0, β0 + β1 x1 + β2 y2 = 0, (5.4) where  a c η 2 (−2a c + a c ) 16a3 a2 d3 l (a d l − 2a d l l ) 1 4 4 1 1 4 2 4 2 1 2 2 1 1 1 1 2 α0 = 4 + a24 l24 1 + (8a2 d2 (a1 η 2 d1 l2 + a2 d2 (2a21 a24 d21 − η 2 l1 )) 2 , l2 8a2 d2  α1 = − 32a22 a24 b21 d22 (2b21 + ω12 ) − 4a22 a24 d22 ω14 + 8a1 a2 a24 d1 d2 l1 l2 + a24 c21 l22 l12 l23  − 2a1 a4 c1 c4 l1 l22 + a21 (−4a24 d21 + c24 l1 l2 , 16 J. JIMENEZ, J. LLIBRE. J. C. MEDRADO EJDE-2020/41  4a2 d2 l2 8a1 a2 d1 d2 l1  α2 = 2a1 a4 c1 c4 l1 − a21 c24 l1 + a24 − c21 l1 + 4a21 d21 + 2 2 2 1 − , l2 l2 a1 c4 η 4a22 a4 d22 l1 4a1 a2 a4 d1 d2 β0 = − + − , a4 l22 l2 a2 a4 d2 l1 β1 = −a1 η, β2 = a1 a4 d1 − . l2 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 (x11 , x12 , y11 , y21 ) and (x21 , x22 , y12 , y22 ), but it is possible to prove that if (x1 , x2 , y1 , y2 ) is a solution of system (5.3), then (x2 , x1 , y2 , y1 ) is also a solution of this system. Since we must have that x1 < x2 and y1 < y2 , then system (5.3) has a unique solution, and therefore the discontinuous piecewise linear differential center (5.1) that belongs to the family F2 can have at most one crossing limit cycle of type 1 intersecting Γ+ + 1 and Γ2 . Now we verify that this upper bound is reached. That is, that there are piecewise linear differential centers in the family F2 having one crossing limit cycle of type 1. We consider the following discontinuous piecewise linear differential center 23177 11 557 1837 11 ẋ = − x− y, ẏ = − + x + y, in R1 , 9000 10 450 1125 10 477 x 53 y ẋ = − − y, ẏ = 1 + x + , in R2 , 64 2 16 2 (5.5) ẋ = −y − β, ẏ = x + α, in R3 , x 17 y ẋ = 2 − − y, ẏ = −2 + x + , in R4 . 2 4 2 In the region R3 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 R1 , R2 and R4 . Figure 6. The crossing limit cycle of type 1 of discontinuous piecewise linear differential system (5.5) separated by the conic (LV). The centers in (5.5) have the first integrals H1 (x, y) = 4500x2 + 44x(−334 + 225y) + y(−23177 + 5570y), 53 H2 (x, y) = 4x2 + 4x(2 + y) + y(−9 + 2y), 8 EJDE-2020/41 CROSSING LIMIT CYCLES 17 H3 (x, y) = (x + α)2 + (y + β)2 , H4 (x, y) = 4(−4 + x)x + 4(−4 + x)y + 17y 2 , in Ri , i = 1, 2, 3, 4, respectively. Then for the discontinuous piecewise linear differ- ential center (5.5) system (5.3) becomes −14696x2 + 4500x22 + (23177 − 5570y2 )y2 = 0, (y1 − y2 )(−9 + 2y1 + 2y2 ) = 0, (5.6) 14696x1 − 4500x21 + y1 (−23177 + 5570y1 ) = 0, (x1 − x2 )(−4 + x1 + x2 ) = 0. Taking into account that the solutions (x1 , x2 , y1 , y2 ) must satisfy x1 < x2 and y1 < y2 , we have that the unique solution of system (5.6) is the point (x1 , x2 , y1 , y2 ) = (1, 3, 1/2, 4). See the crossing limit cycle of type 1 of discontinuous piecewise linear differential system (5.5) in Figure 6. This completes the proof of Theorem 1.4. 6. Proof of Theorem 1.5 Proof of statement (a) of Theorem 1.5. In the region R1 we consider the linear dif- ferential center 13 x y y ẋ = − − − , ẏ = 1 + x + , (6.1) 4 2 2 2 this system has the first integral H1 (x, y) = 2(2x2 + 2x(2 + y) + y(13 + y)). In the region R2 we have the linear differential center 851 x 181 3 y ẋ = − − − y, ẏ = +x+ , (6.2) 3600 3 900 2 3 which has the first integral H2 (x, y) = 4x2 + 4x(9 + 2y)/3 + y(851 + 362y)/450. In the region R3 we have the linear differential center 43 x 5 1 y ẋ = − + − y, ẏ = − + x − , (6.3) 32 4 16 2 4 which has the first integral H3 (x, y) = 4x2 − 3x(2 + y) + y(−43 + 5y)/4. And in the region R4 we have the linear differential center 137 x 25 3 y ẋ = + − y, ẏ = +x− , (6.4) 72 3 144 2 3 which has the first integral H4 (x, y) = 4x(3 + x) − (137 + 24x)y/9 + 25y 2 /36. To have a crossing limit cycle of type 2, which intersects the discontinuity conic (LV) in four different points p1 = (x1 , 0), q1 = (0, y1 ), p2 = (x2 , 0) and q2 = (0, y2 ), with x1 , y1 > 0 and x2 , y2 < 0, these points must satisfy the closing equations e1 = H1 (x1 , 0) − H1 (0, y1 ) = 0, e2 = H2 (0, y1 ) − H2 (x2 , 0) = 0, (6.5) e3 = H3 (x2 , 0) − H3 (0, y2 ) = 0, e4 = H4 (0, y2 ) − H4 (x1 , 0) = 0. Considering the four above linear differential centers (6.1), (6.2), (6.3) and (6.4) and their respective first integrals Hi (x, y), i = 1, 2, 3, 4, we have the following 18 J. JIMENEZ, J. LLIBRE. J. C. MEDRADO EJDE-2020/41 equivalent system 4x1 (2 + x1 ) − 2y1 (13 + y1 ) = 0, 1 −4x2 (3 + x2 ) + y1 (851 + 361y1 ) = 0, 450 1 (6.6) 4(x2 − 1)x2 + (43 − 5y2 )y2 = 0, 4 1 −4x1 (x1 + 3) + y2 (−548 + 25y2 ) = 0, 36 Figure 7. The crossing limit cycle of type 2 of the discontinuous piecewise linear differential system formed by the linear centers (6.1), (6.2), (6.3) and (6.4) separated by (LV). The unique real solution (p1 , q1 , p2 , q2 ) of (6.6) is p1 = (3, 0), q1 = (0, 2), p2 = (−7/2, 0) and q2 = (0, −4), therefore the piecewise differential system formed by the linear differential centers (6.1), (6.2), (6.3) and (6.4) has exactly one crossing limit cycle of type 2. See the crossing limit cycle of this system in Figure 7.  Proof of statement (b) of Theorem 1.5. In the region R1 we consider the linear dif- ferential center 25 x y 11 y ẋ = − + + , ẏ = −x− , (6.7) 8 2 2 2 2 which has the first integral H1 (x, y) = 4x2 + 4x(−11 + y) + y(−25 + 2y). In the region R2 we consider the linear differential center 251 109 293 ẋ = − −x− y, ẏ = − + x + y, (6.8) 400 100 200 this system has the first integral H2 (x, y) = 200x2 +y (251 + 218y)+x(−586+400y). In the region R3 we have the linear differential center 5 x 5 23 y ẋ = + − y, ẏ = +x− , (6.9) 96 4 16 24 4 this system has the first integral H3 (x, y) = 4x2 + x(23/3 − 2y) + 5y(−1 + 3y)/12. In the region R4 we have the linear differential center 73 x 29 31 y ẋ = − + − y, ẏ = − + x − , (6.10) 800 10 400 40 10 this system has the first integral H4 (x, y) = 400x2 − 20x(31 + 4y) + y(73 + 29y). This discontinuous piecewise linear differential center formed by the linear differ- ential centers (6.7), (6.8), (6.9) and (6.10) has two crossing limit cycles of type 2, EJDE-2020/41 CROSSING LIMIT CYCLES 19 because the unique real solutions (pi1 , q1i , pi2 , q2i ), with i = 1, 2 of system (6.5) are p11 = (3/2, 0), q11 = (0, 3), p12 = (−5/2, 0) and q21 = (0, −2) and p21 = (2, 0), q12 = (0, 9/2), p22 = (−4, 0) and q22 = (0, −5). See these two crossing limit cycles in Figure 8.  Figure 8. The two crossing limit cycle of type 2 of the discon- tinuous piecewise linear differential system formed by the linear centers (6.7), (6.8), (6.9) and (6.10) separated by (LV). Proof of statement (c) of Theorem 1.5. In the region R1 we consider the linear dif- ferential center 813 x 300 1207 y ẋ = − − y, ẏ = − +x+ , (6.11) 803 2 803 730 2 which has the first integral H1 (x, y) = 4x2 + x(−4828/365 + 4y) + 24y(−271 + 50y)/803. In the region R2 we have the linear differential center 210061 11 15760 63667 11 ẋ = + x− y, ẏ = + x − y, (6.12) 55055 10 11011 20020 10 this system has the first integral H2 (x, y) = 110110x2 + x(700337 − 242242y) + 4y(−210061 + 39400y). In the region R3 we have the linear differential center 79831 7 3875 421379 7 ẋ = − − x− y, ẏ = + x + y, (6.13) 38904 10 4863 194520 10 this system has the first integral H3 (x, y) = 97260x2 + 5y(79831 + 15500y) + 7x(60197 + 19452y). In the region R4 we have the linear differential center 15513 2 5700 330343 2 ẋ = − + x− y, ẏ = − + x − y, (6.14) 28057 5 28057 280570 5 this system has the first integral H4 (x, y) = 140285x2 +30y(5171+950y)−x(330343+ 112228y). This discontinuous piecewise linear differential center formed by the lin- ear differential centers (6.11), (6.12), (6.13) and (6.14) has three crossing limit cycles of type 2, because the unique real solutions (pi1 , q1i , pi2 , q2i ), with i = 1, 2, 3 of system (6.5) are p11 = (9/5, 0), q11 = (0, 3), p12 = (−7/2, 0) and q21 = (0, −43/10); p21 = (2, 0), q12 = (0, 33/10), p22 = (−39/10, 0) and q22 = (0, −47/10); and p31 = (17/10, 0), q13 = (0, 289/100), p32 = (−33/10, 0) and q23 = (0, −411/100). See these three crossing limit cycles of type 2 in Figure 9.  20 J. JIMENEZ, J. LLIBRE. J. C. MEDRADO EJDE-2020/41 Figure 9. The three crossing limit cycle of type 2 of the discon- tinuous piecewise differential center formed by the centers (6.11), (6.12), (6.13) and (6.14) separated by (LV). 7. Proof of Theorem 1.7 If the discontinuity curve Σ is of the type (P), we have following two regions in the plane: R1 = [(x, y) ∈ R2 : x2 < y], and R2 = [(x, y) ∈ R2 : x2 > y]. 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 follows 4b2 + ω 2 ẋ = −bx − y + d, ẏ = ax + by + c, in R1 , 4a (7.1) 4β 2 + ω 2 ẋ = −βx − y + δ, ẏ = αx + βy + γ, in R2 . 4α These linear differential centers have the first integrals H1 (x, y) = 4(ax + by)2 + 8a(cx − dy) + y 2 ω 2 , (7.2) H2 (x, y) = 4(αx + βy)2 + 8α(γx − δy) + y 2 Ω2 , 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: e1 : H1 (x1 , x21 ) − H1 (x2 , x22 ) = 0, e2 : H2 (x2 , x22 ) − H2 (x3 , x23 ) = 0, (7.3) e3 : H1 (x3 , x23 ) − H1 (x4 , x24 ) = 0, e4 : H2 (x4 , x24 ) − H2 (x1 , x21 ) = 0, EJDE-2020/41 CROSSING LIMIT CYCLES 21 or equivalently e1 : 4x21 (1 + bx1 )2 − 4x22 (1 + bx2 )2 + 8x1 (c − dx1 ) + 8x2 (dx2 − c) + (x41 − x42 )ω 2 = 0, e2 : 4x22 (1 + βx2 )2 + 8x2 (γ − δx2 )2 − 4x23 (1 + βx3 )2 + 8x3 (δx3 − γ) + (x42 − x43 )Ω = 0, (7.4) e3 : 4x23 (1 + bx3 )2 − 4x24 (1 + bx4 )2 + 8x3 (c − dx3 ) + 8x4 (dx4 − c) + (x43 − x44 )ω 2 = 0, e4 : 4x24 (1 + βx4 )2 + 8x1 (δx1 − γ) − 4x21 (1 + βx1 )2 + 8x4 (γ − δx4 ) + (x44 − x41 )Ω2 = 0. 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 (p1 , p2 , p3 , p4 ) and (q1 , q2 , q3 , q4 ), where pi = (ki , ki2 ) and qi = (Li , L2i ), with i = 1, 2, 3, 4. These points are the points where the two crossing periodic solution intersect discontinuity curve (P). If the point (p1 , p2 , p3 , p4 ) satisfies system (7.4), by the equation e1 of (7.4) we obtain the expression 8c + 4(k1 + k2 )(1 + b(k1 + k2 )) + 4b(k12 + k22 ) + (k1 + k2 )(k12 + k22 )l1 d= , 8(k1 + k2 ) by equation e2 of (7.4) we get the expression 8γ + 4(k2 + k3 )(1 + β(k2 + k3 )) + 4β(k22 + k32 ) + (k2 + k3 )(k22 + k32 )l2 δ= , 8(k2 + k3 ) by equation e3 of (7.4) we obtain the expression k1 + k2   (k 2 + k 2 )l 1 2 1 c= (k42 − k32 ) + 1 + (1 + b(k1 + k2 )) 2(k1 + k2 − k3 − k4 )(k3 − k4 ) 4 k 2 + k22   +b 1 + 2b(k43 − k33 ) + (k44 − k34 )l1 k1 + k2 and by equation e4 of (7.4) we obtain the expression 1  γ= 8β(k1 + k4 )k4 − l2 (k22 + k32 − k42 )(k2 + k3 )(k1 + k4 ) 8(k1 − k2 − k3 + k4 )  − 2(k22 + k2 k3 + k32 ) + (k2 + k3 )(k13 l2 + k12 (8β + k4 l2 ) , here we consider l1 = 4b2 + ω 2 and l2 = 4β 2 + Ω2 . We assume that the point (q1 , q2 , q3 , q4 ) satisfies system (7.4), then we can obtain the remaining parameters of discontinuous piecewise linear differential center (7.1). By equation e1 of (7.4) we obtain ω 2 = S/T , where −4b(L1 − L2 )  S= (bk1 + (bk2 + 2))(k3 + k4 − L1 − L2 )k12 k1 + k2 − k3 − k4  + k1 − bk33 + k2 (bk2 + 2)(k3 + k4 − L1 − L2 ) − (k32 + k3 k4 + k42 )(bk4 + 2)  + bL32 + (L21 + L1 L2 + L22 )(bL1 + 2) + k2 (bk2 + 2)(k3 + k4 − L1 − L2 ) 22 J. JIMENEZ, J. LLIBRE. J. C. MEDRADO EJDE-2020/41   + k2 (bL1 + 2)(L21 + L1 L2 + L22 ) − (k32 + k3 k4 + k42 )(bk4 + 2) − bk33  + bk2 L32 + (L2 + L1 ) bk33 + (k3 + k4 )(k4 − L1 )(b(k4 + L1 ) + 2) + k32   (bk4 + 2) − L22 (k3 + k4 )((bL1 + 2) − bL2 ) , and L1 − L2  T = (k13 + k12 k2 + k1 k22 + k23 )(k3 + k4 − L1 − L2 ) k1 + k2 − k3 − k4   + (k1 + k2 ) (L1 + L2 )(L21 + L22 ) − (k3 + k4 )(k32 + k42 )  + (k3 + k4 )(L1 + L2 ) k32 + k42 − L21 − L22 , by equation e2 of (7.4) we obtain Ω2 = V /W , where −4β(L2 − L3 )  V = k1 (βk1 + (βk4 + 2))(k2 + k3 − L2 − L3 ) k1 − k2 − k3 + k4   − 2k1 k22 + k2 (k3 − k4 ) + (k3 − L2 )(k3 − k4 + L2 ) + L3 (k4 − L2 ) − L23   + βk1 − k23 − k22 k3 + k2 (k42 − k32 ) − k33 + k3 k42 + (L2 + L3 )(−k42 + L22 + L23 ) + βk23 (−k4 + L2 + L3 ) − k22 (βk3 + 2)(k4 − L2 − L3 )   − βk2 k32 (k4 − L2 − L3 ) − k43 + (L2 + L3 )(L22 + L23 )   − 2k2 L22 − k42 + k3 (k4 − L2 − L3 ) + L2 L3 + L23  − (k3 − k4 ) (βk32 + k3 (βk4 + 2))(k4 − L2 − L3 )   + β(L2 + L3 )(−k42 + L22 + L23 ) + 2 − k4 (L2 + L3 ) + L22 + L2 L3 + L23 , and L2 − L3  W = k 2 (k2 + k3 − L2 − L3 )(k1 + k4 ) k1 − k2 − k3 + k4 1   + k1 k2 (k42 − k32 ) − k23 − k22 k3 − k33 + k3 k42 + (L2 + L3 ) − k42 + L22 + L23   + L2 (k2 + k3 ) k22 + k32 − k43 − k4 (k2 + k3 )(k22 + k32 − k42 )    + L3 k23 + k22 k3 + k2 k32 − L22 (k2 + k3 − k4 ) + k33 − k43  − (k2 + k3 − k4 )(L32 + L2 L23 + L33 ) , by equations e3 and e4 we get that b = β = 0. This implies that the linear differential systems in R1 and in R2 are of the form 1 ẋ = , ẏ = x, 2 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 F3 is one. EJDE-2020/41 CROSSING LIMIT CYCLES 23 Now we verify that this upper bound is reached. That is, that there are piecewise linear differential centers in the family F3 having one crossing limit cycle. We con- sider the discontinuous piecewise linear differential system formed by the following linear differential centers 831 17 587 ẋ = + x − y, ẏ = + x − y, in R1 , (7.5) 128 16 128 21145 x 5 127 y ẋ = + − y, ẏ = + x − , in R2 . (7.6) 4176 6 18 174 6 These linear differential centers have the first integrals H1 (x, y) = 64x2 + x(587 − 128y) + y(−831 + 68y), 1  y 2 H2 (x, y) = (3048x − 21145y) + 4 x − + y2 , 522 6 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). The discontinuous piecewise differential center formed by the linear differential centers (7.5) and (7.6) has one crossing limit cycle, because the unique real solution (p1 , p2 , p3 , p4 ) of system (7.4) is p1 = (6, 36), p2 = (−5, 25), p3 = (−3/2, 9/4), and p4 = (2, 4). See the crossing limit cycle of this discontinuous piecewise differential center in Figure 10. 8. Proof of Theorem 1.8 When the discontinuity curve Σ is of the type (E), we have following two regions in the plane: R1 = [(x, y) ∈ R2 : x2 + y 2 < 1], and R2 = [(x, y) ∈ R2 : x2 + y 2 > 1]. By Lemma 3.1 a piecewise linear differential center of family F4 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 F4 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 F4 has crossing limit cycles if there are points (xi , yi ) for i = 1, 2, 3, 4 satisfying the equations e1 = H1 (x1 , y1 ) − H1 (x2 , y2 ) = 0, e2 = H2 (x2 , y2 ) − H2 (x3 , y3 ) = 0, e3 = H1 (x3 , y3 ) − H1 (x4 , y4 ) = 0, 24 J. JIMENEZ, J. LLIBRE. J. C. MEDRADO EJDE-2020/41 e4 = H2 (x4 , y4 ) − H2 (x1 , y1 ) = 0, E1 = x21 + y12 − 1 = 0, E2 = x22 + y22 − 1 = 0, E3 = x23 + y32 − 1 = 0, E4 = x24 + y42 − 1 = 0, considering l1 = 4b2 + ω 2 and l2 = 4β 2 + Ω2 , we have the equivalent system e1 =4(x21 − x22 ) + 8(bx1 y1 − bx2 y2 + c(x1 − x2 ) − dy1 + dy2 ) + l1 (y12 − y22 ) = 0, e2 =4(x22 − x23 ) + 8x2 (γ + βy2 ) − 8x3 (γ + βy3 ) + (y2 − y3 )(l2 (y2 + y3 ) − 8δ) = 0, e3 =4(x23 − x24 ) + 8(bx3 y3 − bx4 y4 + c(x3 − x4 ) − dy3 + dy4 ) (8.1) + l1 (y32 − y42 ) = 0, e4 =8δ(y1 − y4 ) + 8x4 (γ + βy4 ) − 4(x21 − x24 ) − 8x1 (γ + βy1 ) + l2 (y42 − y12 ) = 0, E1 = 0, E2 = 0, E3 = 0, E4 = 0. 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 pe- riodic solutions. For this we have that system (8.1) has two pairs of solutions, (p1 , p2 , p3 , p4 ) and (q1 , q2 , q3 , q4 ) with pi 6= pj , and qi 6= qj , for i 6= 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 pi = (ki , λi ), with ki = cos si , λi = sin si , (8.2) qi = (mi , ni ), with mi = cos ti , ni = sin ti , where si , ti ∈ [0, 2π), for i = 1, 2, 3, 4. Substituting the first solution (p1 , p2 , p3 , p4 ) with pi 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 (8c(k1 − k2 ) + 4(k1 − k2 + bλ1 − bλ2 )(k1 + k2 + b(λ1 + λ2 )) + (λ21 − λ22 )ω 2 ) d= , 8(λ1 − λ2 ) 4k 2 − 4k32 + 8k2 (λ2 β + γ) − 8k3 (λ3 β + γ) + (λ22 − λ23 )l2 δ= 2 , 8(λ2 − λ3 ) 1 c= 8((k3 − k4 )(λ1 − λ2 ) − (k1 − k2 )(λ3 − λ4 ))  × 4(λ1 − λ2 )(k42 − k32 − 2bλ3 k3 + 2bk4 λ4 ) + (λ3 − λ4 )(4(k12 − k22 )  + 8b(k1 λ1 − k2 λ2 ) + (λ1 − λ2 )(λ1 + λ2 − λ3 − l4 )l1 ) , 1 γ= 8(−(k1 − k4 )(λ2 − λ3 ) + (k2 − k3 )(λ1 − λ4 ))  × 4(λ1 − λ4 )(k32 − k22 + 2k3 λ3 β − 2k2 λ2 ) + (λ2 − λ3 )(4k12 − 4k42 + 8k1 λ1 β EJDE-2020/41 CROSSING LIMIT CYCLES 25  − 8k4 λ4 β + (λ1 − λ4 )(λ1 − λ2 − λ3 + λ4 )l2 ) . Analogously, substituting the second solution (q1 , q2 , q3 , q4 ) with qi as in (8.2) in (8.1) we get remaining parameters ω, Ω, b, β. Substituting ki , λi , mi , ni like (8.2) we obtain that b = β = 0, therefore we get that the piecewise linear differential centers is formed by linear differential center ẋ = −y, ẏ = x, in the regions R1 and R2 . 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 F4 is one. Moreover this maximum number is reached, that is, there are piecewise linear differential centers in F4 such that have one crossing limit cycle with four points on (E), as we see below. Figure 11. The crossing limit cycle of the discontinuous piecewise differential center formed by the centers (8.3) and (8.4) separated by the conic (E). Consider the discontinuous piecewise differential center in the family F4 formed by the following two linear differential centers √ √ −107 − 89 2 5 345 −71 + 89 2 5 ẋ = − x− y, ẏ = + x + y, in R1 , (8.3) 1024 16 256 1024 16 1 1 ẋ = − − x − 2y, ẏ = + x + y, in R2 . (8.4) 4 4 These linear differential centers have the first integrals √ √ H1 (x, y) = 512x2 + x(−71 + 89 2 + 320y) + y(107 + 89 2 + 690y), H2 (x, y) = x + 2x2 + y + 4xy + 4y 2 , 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 (p1 , p2 , p3 , p4 ) of system (8.1) is p1 = (1, 0), p2 = (− 2/2, 1/ 2), p3 = (−1, 0), and p4 = (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: R1 = [(x, y) ∈ R2 : x2 − y 2 > 1, x > 0], R2 = [(x, y) ∈ R2 : x2 − y 2 < 1] and R3 = [(x, y) ∈ R2 : x2 − y 2 > 1, x < 0]. 26 J. JIMENEZ, J. LLIBRE. J. C. MEDRADO EJDE-2020/41 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 follows 4b2 + ω12 ẋ = −b1 x − 1 y + d1 , ẏ = a1 x + b1 y + c1 , in R1 , 4a1 4b2 + ω22 ẋ = −b2 x − 2 y + d2 , ẏ = a2 x + b2 y + c2 , in R2 , (9.1) 4a2 4b2 + ω32 ẋ = −b3 x − 3 y + d3 , ẏ = a3 x + b3 y + c3 , in R3 . 4a3 These linear differential centers have the first integrals H1 (x, y) = 4(a1 x + b1 y)2 + 8a1 (c1 x − d1 y) + y 2 ω12 , H2 (x, y) = 4(a2 x + b2 y)2 + 8a2 (c2 x − d2 y) + y 2 ω22 , (9.2) 2 2 H3 (x, y) = 4(a3 x + b3 y) + 8a3 (c3 x − d3 y) + y ω32 , respectively. To have a crossing limit cycle, which intersects the discontinuity curve (H) in four different points pi = (xi , yi ), i = 1, 2, 3, 4, these points must satisfy the following equations e1 = H1 (x1 , y1 ) − H1 (x2 , y2 ) = 0, e2 = H2 (x2 , y2 ) − H2 (x3 , y3 ) = 0, e3 = H3 (x3 , y3 ) − H3 (x4 , y4 ) = 0, e4 = H2 (x4 , y4 ) − H2 (x1 , y1 ) = 0, (9.3) E1 = x21 − y12 − 1 = 0, E2 = x22 − y22 − 1 = 0, E3 = x23 − y32 − 1 = 0, E4 = x24 − y42 − 1 = 0, equivalently, we have e1 =4(x21 − x22 ) + 8(c1 (x1 − x2 ) − d1 y1 + b1 x1 y1 + d1 y2 − b1 x2 y2 ) + (y12 − y22 )l1 = 0, e2 =4(x22 − x23 ) + 8(c2 (x2 − x3 ) − d2 y2 + b2 x2 y2 + d2 y3 − b2 x3 y3 ) + (y22 − y32 )l2 = 0, e3 =4(x23 − x24 ) + 8(c3 (x3 − x4 ) − d3 y3 + b3 x3 y3 + d3 y4 − b3 x4 y4 ) (9.4) + (y32 − y42 )l3 = 0, e4 =4(x24 − x21 ) + 8(c2 (x4 − x1 ) − d2 y4 − b2 x1 y1 + d2 y1 + b4 x4 y4 ) + (y42 − y12 )l2 = 0, E1 = 0, E2 = 0, E3 = 0, E4 = 0, where li = 4b2i+ ωi2 , for i = 1, 2, 3. Here we are taking without generality a1 = a2 = a3 = 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, EJDE-2020/41 CROSSING LIMIT CYCLES 27 namely (p1 , p2 , p3 , p4 ) and (q1 , q2 , q3 , q4 ). Since these solutions provide crossing pe- riodic solutions and these points are the points where the crossing periodic solutions intersect the hyperbola (H) we can consider pi = (ki , λi ) = (cosh ri , sinh ri ) and qi = (mi , ni ) = (cosh si , sinh si ), (9.5) with ri , si ∈ R for i = 1, 2, 3, 4. Now we assume that the point (p1 , p2 , p3 , p4 ) with pi = (ki , λi ), i = 1, 2, 3, 4 satisfy (9.4), and then we obtain the following expressions  di = 8ci (ki − ki+1 ) + 4(ki − ki+1 + bi (λi − λi+1 ))(ki + ki+1 + bi (λi + λi+1 ))  + (λ2i − λ2i+1 )ωi2 / 8(λi − λi+1 ) ,  for i = 1, 2, 3, and 1 c2 = 8((k2 − k3 )(λ1 − λ4 ) − (k1 − k4 )(λ2 − λ3 ))  × 4(k32 + 2b2 k3 λ3 − k22 − 2b2 k2 λ2 )(λ4 − λ1 ) + (λ2 − λ3 )(4(k12 − k42 )  + 8b2 (k1 λ1 − k4 λ4 ) + (λ1 − λ4 )(λ1 − λ2 − λ3 + λ4 )l2 ) . We assume that the point (q1 , q2 , q3 , q4 ) with qi as in (9.5) satisfies system (9.4). By the first equation in (9.4) we obtain that 1 c1 = 8((λ1 − λ2 )(m1 − m2 ) − (k1 − k2 )(n1 − n2 ))  × (λ1 − λ2 )(−4(m21 − m22 ) − 8b1 m1 n1 + 8b1 m2 n2 + (λ1 + λ2 )ω12 (n1 − n2 ) − (n21 − n22 )l1 ) 4(k1 − k2 + b1 (λ1 − λ2 ))(k1 + k2 + b1 (λ1 + λ2 ))(n1 − n2 )  + . λ1 − λ2 By the second equation of (9.4) we obtain that ω22 = K/S where 4 K= (k1 − k4 )(λ2 − λ3 ) − (k2 − k3 )(λ1 − λ4 )  × (k32 − k22 ) (λ1 − λ4 )ψ2 − (k1 − k4 )ψ1 + (λ2 − λ3 )  × k12 + b22 (λ1 − λ4 )(λ1 − λ2 − λ3 + λ4 ) − k42 ψ2  + k4 ψ2 − b22 (λ2 + λ3 − n2 − n3 )ψ1 + 2b2 (−λ4 ψ2 + m2 n2 − m3 n3 )   + k1 − ψ2 + b22 (λ2 + λ3 − n2 − n3 )ψ1 + 2b2 (λ1 ψ2 − m2 n2 + m3 n3 )  + k2 (λ1 − λ4 )(ψ2 − b22 (λ1 + λ4 − n2 − n3 )ψ1 ) − (k12 − k42 )ψ1  − 2b2 λ2 (−λ4 ψ2 − (k1 − k4 )ψ1 ) + λ4 ((m2 − k4 )n2 + (k4 − m3 )n3 )λ1 (λ2 ψ2  + (k1 − m2 )n2 + (m3 − k1 )n3 ) + k3 ((λ4 − λ1 )ψ2 + (k12 − k42 )ψ1  + b22 (λ1 − λ4 )(λ1 + λ4 − n2 − n3 )ψ1 + 2b2 λ3 (−λ4 ψ2 − (k1 − k4 )ψ1 )  + λ4 (m2 n2 − k4 ψ1 − m3 n3 ) + λ1 (λ3 ψ2 + k1 ψ1 − m2 n2 + m3 n3 ) , 28 J. JIMENEZ, J. LLIBRE. J. C. MEDRADO EJDE-2020/41 and 1 S= (k1 − k4 )(λ2 − λ3 ) − (k2 − k3 )(λ1 − λ4 )  × λ23 λ4 ψ2 + λ3 λ24 ψ2 + (k1 − k4 )(λ23 n2 − λ3 n22 ) + (k3 − k2 )(λ24 n2 − λ4 n22 ) + λ21 ((k2 − k3 )ψ1 − (λ2 − λ3 )ψ2 ) + λ22 (λ4 ψ2 + (k1 − k4 )ψ1 ) + ((k4 − k1 )λ23 + (k2 − k3 )λ24 )n3 + (k1 λ3 − k4 λ3 − k2 λ4 + k3 λ4 )n23 + λ1 ((λ22 − λ23 )ψ2 − (k2 − k3 )ψ1 (n2 + n3 )) + λ2 (λ24 ψ2  + (k1 − k4 )ψ1 (n2 + n3 )) . By the third equation we obtain that −1 c3 = (((λ3 − λ4 )(4(m24 − m23 ) + 8b3 (m4 n4 − m3 n3 ) 8(λ4 ψ4 − λ3 ψ2 + (k3 − k4 )ψ1 )) 4(k3 − k4 + b3 (λ3 − λ4 ))(k3 + k4 + b3 (λ3 + λ4 ))ψ1 + λ3 − λ4 + (λ3 + λ4 )ω3 ψ1 − (n3 − n24 )l3 )), 2 2 where ψ1 = n2 − n3 , ψ2 = m2 − m3 , ψ3 = n3 − n4 and ψ4 = m3 − m4 . Finally by the fourth equation in (9.4) we have that b2 = 0. With these expres- sions for d2 , c2 , ω2 and b2 we obtain that the linear differential system in the region R2 is ẋ = y, ẏ = x, which is a linear differential system type saddle. This is a contradiction because we are working with centers in each region Ri for i = 1, 2, 3. Therefore we have proved that the maximum number of crossing limit cycles for systems in F5 is one. Moreover it is possible to show that there are piecewise linear differential centers in F5 such that have one crossing limit cycle. Indeed, consider the discontinuous piecewise linear differential system in the family F5 formed by the following linear differential centers √ √ −355 + 64 10 + 80 21 x 29 y ẋ = √ − − y, ẏ = 1 + x + , in R1 , 64(6 + 21) 2 16 2 x 101 y ẋ = K1 − − y, ẏ = K2 + x + , in R2 , 10 100 √ 10 3 √ 337(−3 + 2 3) 3 45 3 3 ẋ = (−11 + 2 3) + √ − x − y, ẏ = − x + y, in R3 . 4 64 5 2 16 2 2 (9.6) Where 1 K1 = p √√ √ √ √ √ √ 200 81 − 12 35(−20 2 + 26 3 − 14 5 + 7 7 − 13 15 + 2 70)  √ √ √ √ √ × 27300 − 62790 2 + 8750 3 − 31356 5 + 15678 7 + 2500 30 √ √ √ √  − 2730 35 − 600 42 + 6279 70 − 420 105  √ √ √ √ √ √ K2 = 3(200 − 800 2 + 5226 3 − 1846 5 + 403 7 − 2613 15 + 240 35 √   √ √ √ √ √ √  + 80 70) / 400(−20 2 + 26 3 − 14 5 + 7 7 − 13 15 + 2 70) . EJDE-2020/41 CROSSING LIMIT CYCLES 29 These linear differential centers have first integrals √ √ √ H1 (x, y) = (355 − 64 10 − 80 21)y + 2(6 + 21)(16x(2 + x) + 16xy + 29y 2 ), 1 H2 (x, y) = p √ √ √ √ √ √ √ 200 27 − 4 35(−20 2 + 26 3 − 14 5 + 7 7 − 13 15 + 2 70) √ √ √ q  2 y 2 × y + x+ + (3 27 − 4 35(200 − 800 2 + 5226 3 √ 10 √ √ √ √ − 1846 5 + 403 7 − 2613 15 + 240 35 + 80 70)x √ √ √ √ √ − 2(8750 + 9100 3 − 20930 6 + 2500 10 − 600 14 − 10452 15 √ √ √ √  + 5226 21 − 420 35 − 910 105 + 2093 210)y) , 1 √ √ H3 (x, y) = 4x2 + 12x(−1 + y) + y(2640 − 480 3 + 1011 5 √ 40 − 674 15 + 450y), respectively. Figure 12. The crossing limit cycle of the discontinuous piecewise linear differential center (9.6) with discontinuity curve the conic (H). √The unique real solution p (p1 , p2 , p3 , p4 ) that √ satisfies (9.4) in this p case is p1 = ( 10, −3), p2 = (−5/2, 21/4), p3 = (−4, 15), and p4 = (−7/2, − 45/4). See the crossing limit cycle of this system in Figure 12. 10. Proof of Theorem 1.10 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 piece- wise linear center (7.1) and the first integrals (7.2). In order that the discon- tinuous piecewise linear center (7.1) has crossing limit cycles with four points, namely (x1 , x21 ), (x2 , x22 ), (x3 , x23 ), (x4 , x24 ) and one crossing limit cycle with two points, namely (x5 , x25 ), (x6 , x26 ) on (P), we must study the solutions (x1 , x2 , x3 , x4 , x5 , x6 ) of system (7.3) and the equations e5 = H1 (x5 , x25 ) − H1 (x6 , x26 ) = 0, e6 = H2 (x6 , x26 ) − H2 (x5 , x25 ) = 0, 30 J. JIMENEZ, J. LLIBRE. J. C. MEDRADO EJDE-2020/41 or equivalently systems (7.4) and e5 =4x25 (1 + bx5 )2 + 8x5 (c − dx5 ) − 4x26 (1 + bx6 )2 + 8x6 (dx6 − c) + (x45 − x46 )ω 2 = 0, (10.1) e6 =4x26 (1 + x6 β)2 − 4x25 (1 + x5 β)2 + 8x5 (x5 δ − γ) + 8x6 (γ − x6 δ) + (x46 − x45 )Ω2 = 0. 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 (x1 , x2 , x3 , x4 , x5 , x6 ) = (k1 , k2 , k3 , k4 , k5 , k6 ) and (x1 , x2 , x3 , x4 , x5 , x6 ) = (k1 , k2 , k3 , k4 , λ5 , λ6 ), with ki , λ5 , λ6 ∈ R for i = 1, 2, 3, 4, 5, 6. If the point (k1 , k2 , k3 , k4 , k5 , k6 ) satisfies systems (7.4) and (10.1), by the equa- tions e1 , e2 , e3 and e4 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 e5 of system (10.1) we obtain an ex- pression for ω 2 = S/T with S and T as in the proof of Theorem 1.7 changing L1 and L2 by k5 and k6 , respectively. By equation e6 of system (10.1) we obtain Ω2 = V /W where the expression for V and W are the same expressions that in the proof of Theorem 1.7 changing L3 by k5 . We assume that the point (k1 , k2 , k3 , k4 , λ5 , λ6 ) satisfies systems (7.4) and (10.1), then we have e1 = e2 = e3 = e4 = 0 and by the equations e5 and e6 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) became ẋ = 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 F3 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 F3 with one crossing limit cycle with four points on (P) and one crossing limit cycle with two points on (P) simultaneously. Figure 13. The two crossing limit cycles of the discontinuous piecewise linear differential system formed by the centers (10.2) and (10.3). EJDE-2020/41 CROSSING LIMIT CYCLES 31 We consider the discontinuous piecewise linear differential system formed by the linear centers 1225 x 310 103 y ẋ = + − y, ẏ = − + x − , in R1 , (10.2) 229 2 229 229 2 6411 x 85 3359 y ẋ = − − y, ẏ = − + x + , in R2 . (10.3) 1424 8 89 712 8 These linear differential centers have the first integrals H1 (x, y) = 229x2 + 10y(−245 + 31y) − x(206 + 229y), 3359 y H2 (x, y) = 4x2 + x −  +y + (−6411 + 680y), 89 178 respectively. The unique real solution of systems (7.4) and (10.1) is (x1 , x2 , x3 , x4 , x5 , x6 ) = (3, −2, −3/2, 1, 2, 12/5), therefore we have one crossing limit cycle that intersects (P) in the points (3, 9), (−2, 4), (−3/2, 9/2) and (1, 1), and one crossing limit cycle that intersects (P) in the points (2, 4) and (12/5, 144/25). See these crossing limit cycles in Figure 13.  Proof of statement (b) of Theorem 1.10. In this case we consider the notation of the proof of Theorem 1.8 and therefore we consider the planar discontinuous piece- wise 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 on (E), namely (x1 , y1 ), (x2 , y2 ), (x3 , y3 ), (x4 , y4 ) and one crossing limit cycle with two points on (E), namely (x5 , y5 ), (x6 , y6 ), we must study the solutions (p1 , p2 , p3 , p4 , p5 , p6 ) of systems (8.1) and e5 =4(x25 − x26 ) + 8(c(x5 − x6 ) − dy5 + bx5 y5 + dy6 − bx6 y6 ) + (y52 − y62 )l1 = 0, e6 =4(x26 − x25 ) + 8(βx6 y6 − βx5 y5 + y5 δ − x5 γ + x6 γ − y6 δ) (10.4) + (y62 − y52 )l2 = 0, E5 = x25 + y52 − 1 = 0, E6 = x26 + y62 − 1 = 0. 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 (p1 , p2 , p3 , p4 , p5 , p6 ) and (p1 , p2 , p3 , p4 , q5 , q6 ), with pi and qj as (8.2) for i = 1, 2, 3, 4, 5, 6 and j = 5, 6. Substituting the first solution (p1 , p2 , p3 , p4 , p5 , p6 ) in systems (8.1) and (10.4) we obtain from the equations e1 , e2 , e3 and e4 of (8.1) the same expressions than in the proof of Theorem 1.8 for d, δ, c, γ, and by the equations e5 and e6 of system (10.4) we obtain the same expressions than in the proof of Theorem 1.8 for ω and Ω changing (m1 , n1 ) by (k5 , λ5 ) and (m2 , n2 ) by (k6 , λ6 ), respectively. We assume that the point (p1 , p2 , p3 , p4 , q5 , q6 ) satisfies systems (8.1) and (10.4), then we have e1 = e2 = e3 = e4 = 0 and by the equations e5 and e6 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) become ẋ = −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 32 J. JIMENEZ, J. LLIBRE. J. C. MEDRADO EJDE-2020/41 planar discontinuous piecewise linear centers in F4 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 F4 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 F4 formed by the linear centers √ √ √ √ √ √ (−6 + 3 2 + 6 + (6 − 4 2 − 6 3)x + 8(−1 + 2 + 2 3)y ẋ = − √ √ , 4(−3 + 2 2 + 3 3) √ √ √ (10.5) −4 + 3 2 + 2 3 + 6 y ẏ = − √ √ + x − , in R1 , 2(−6 + 4 2 + 6 3) 2  √ √ √ √ √ ẋ = − 18 − 93 2 + 4 3 + 33 6 − 230(1 + 3)x + 4(335 − 2 2 √ √  √  + 261 3 + 20 6)y / 920(1 + 3) , (10.6) 1  √ √ √  ẏ = x + 9 + 34 2 − 67 3 − 41 6 − 230y , in R1 . 920 Figure 14. The two limit cycles of the discontinuous piecewise linear differential system formed by the centers (10.5) and (10.6). The unique real solution of systems (8.1) and (10.4) in this case is (p1 , p2 , p3 , p4 , p5 , p6 ) with p1 = (cos (π/2), sin (π/2)), p2 = (cos (π), sin (π)), p3 = (cos (3π/2), sin (3π/2)), p4 = (cos (−π/3), sin (−π/3)), p5 = (cos (π/4), sin (π/4)) and p6 = (cos (0), sin (0)). See these crossing limit cycles in Figure 14.  Proof of statement (c) of Theorem 1.10. Here we consider the notation of the proof of Theorem 1.9 and therefore we consider the planar discontinuous piecewise linear center (9.1) and the first integrals (9.2). In order that the discontinuous piece- wise linear center (9.1) has crossing limit cycles with four points on (H), namely (x1 , y1 ), (x2 , y2 ), (x3 , y3 ), (x4 , y4 ) and one crossing limit cycle with two points on (H), namely (x5 , y5 ), (x6 , y6 ), we must study the solutions (p1 , p2 , p3 , p4 , p5 , p6 ) of EJDE-2020/41 CROSSING LIMIT CYCLES 33 systems (9.4) and (10.7) e5 =4(x25 − x26 ) + 8(c2 (x5 − x6 ) − d2 y5 + b2 x5 y5 + d2 y6 − b2 x6 y6 ) + (y52 − y62 )l1 = 0, e6 =4(x26 − x25 ) + 8(b1 x6 y6 − b1 x5 y5 + y5 d1 − x5 c1 + x6 c1 − y6 d1 ) (10.7) + (y62 − y52 )l2 = 0, E5 = x25 − y52 − 1 = 0, E6 = x26 − y62 − 1 = 0. We assume that systems (9.4) and (10.7) have two real solutions where each real solution provides one crossing limit cycle with four points on (H) and one crossing limit cycle with two points on (H). By Theorem 1.9 the discontinuous piecewise linear center (9.1) has at most 1 crossing limit cycle with four points on (H), then we have that if there are two real solutions of systems (9.4) and (10.7) they are of the form (p1 , p2 , p3 , p4 , p5 , p6 ) and (p1 , p2 , p3 , p4 , q5 , q6 ), with pi and qj as (9.5) for i = 1, 2, 3, 4, 5, 6 and j = 5, 6. Considering the first solution (p1 , p2 , p3 , p4 , p5 , p6 ) of systems (9.4) and (10.7) we obtain the same expressions that in the proof of Theorem 1.9 for d1 , d2 , d3 , c2 , c1 , ω2 changing (m1 , n1 ) by (k5 , λ5 ) and (m2 , n2 ) by (k6 , λ6 ), respectively. Now we assume that the point (p1 , p2 , p3 , p4 , q5 , q6 ) satisfies systems (9.4) and (10.7), then we have e1 = e2 = e3 = e4 = 0, and by the equation e5 of system (10.7) we obtain b2 = 0 and with this the linear system in the region R2 becomes a saddle which is a contradiction, because we are working with linear centers in each regions Ri 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 F5 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 centers √ √ √ √ −1215 − 576 2 + 256 7 + 112 13 − 384 15 x 29 ẋ = √ √ √ − − y, 192(2 7 + 13 − α) 2 16 −1  675 √ √ ẏ = q q  − − 64 7 − 28 13 48 − 1 + 2 α + α 7 13 4 (10.8) √ r r r r 2 7 945 13 26 + 288 (23 − 4 30) + 945 + + 144 7 α 2 α α r r 105 195  y + 192 + 96 +x+ in R1 , α α 2 √ √ √ √ √ 1125 + 432 14 + 189 26 + 207 30 + 160 105 + 70 195 x ẋ = − 6ξ 2 √ √ √ √ √ √ 54 2 + 336 7 + 174 13 + 24 15 + 68 210 + 35 390 − η − y, η (10.9) √ √ √ √ √ √ 855 2 + 3516 7 + 1797 13 + 315 15 + 644 210 + 329 390 ẏ = − 6ξ y + x + , in R2 , 2 34 J. JIMENEZ, J. LLIBRE. J. C. MEDRADO EJDE-2020/41 9 73 3 45 3 3 ẋ = − + √ − x − y, ẏ = − + x + y, in R3 , (10.10) 2 8 2 2 16 2 2 √ √ √ √ √ √ √ 30, η √ here√α = 23+4 = 42+48√ 14+24 √ 26+9 30+24 105+12 195 and ξ = 87+ 108 14 + 54 26 + 16 30 + 40 105 + 20 195. The unique real solution√of systems (9.3) √ and (10.7) in √ ) with p1 = (3, − 8),√p2 = this case is (p1 , p2 , p3 , p4 , p5 , p6√ (4, 15), p3 = (−3, 8), p4 = (−1, 0), p5 = (7/6, − 13/6) and p6 = (4/3, 7/3). See these crossing limit cycles in Figure 15.  Figure 15. The two limit cycles of the discontinuous piecewise linear differential center formed by the centers (10.8), (10.9) and (10.10). 11. Proof of Theorem 1.11 Proof. To have a crossing limit cycle of type 1 and one crossing limit cycle of type 2, simultaneously, we must study the real solutions (p1 , q1 , p2 , q2 , p3 , q3 , p4 , q4 ), of systems (5.3) and (6.5) respectively, where pi = (xi , 0) and qi = (0, yi ), with x1 , x2 , x3 , y1 , y2 , y3 > 0 and x4 , y4 < 0. Figure 16. One crossing limit cycle of type 1 and three crossing limit cycles of type 2 of the discontinuous piecewise linear differen- tial system formed by the linear centers (11.1), (11.2), (11.3) and (11.4) separated by (LV). In the region R1 we consider the linear differential center 193 x 58 149 y ẋ = − − y, ẏ = − +x+ , (11.1) 134 3 67 134 3 EJDE-2020/41 CROSSING LIMIT CYCLES 35 this system has the first integral H1 (x, y) = 201x2 + x(134y − 447) + 3y(58y − 193)). In the region R2 we have the linear differential center 9 x 1 y ẋ = − − 2y, ẏ = − + x + , (11.2) 2 2 4 2 which has the first integral H2 (x, y) = 2x(2x − 1) + 4y(x − 9) + 8y 2 . In the region R3 we have the linear differential center √ √ 3 3 ẋ = 1.068079 · · · + x − 1.448022 . . . y, ẏ = −3.860171 · · · + x − y, (11.3) 4 4 which has the first integral H3 (x, y) = x2 + x(−7.720342 · · · − 0.866025..y) + y(−2.136159 · · · + 1.448022 . . . y). And in the region R4 we have the linear dif- ferential center √ √ 51831 − 595 16909 x 6775 − 119 16909 y ẋ = + + y, ẏ = −2 + x − , (11.4) 35912 2 17956 2 2 which√ has the first integral H √ 4 (x, y) = 17956x − 17956x(4 + y) + y(−51831 + 595 16909 + (−6775 + 119 16909)y). The unique real solutions for systems (5.3) and (6.5) are (p1 , q1 , p2 , q2 , p3 , q3 , p4 , q4 ) with p1 = (1, 0), q1 = (0, 1/2), p2 = (3, 0), q2 = (0, 4), p3 = (5, 0), q3 = (0, 6), √p4 = (−4, 0) and q4 = (0, −5); (p1 , q1 , p2 , q2 , l3 , m3 , l4 , m4 ) with l3 = ((149 + 3 16909)/134, 0), m3 = (0, 5), l4 = (−2, 0) and m4 = (0, −3); and (p1 , q1 , p2 , q2 , λ3 , η3 , λ4 , η4 ), where λ3 = ( 4.319114 . . . , 0), η3 = ( 0, 53/10), λ4 = (−2.672755 . . . , 0), η4 = ( 0, −3.703965 . . . ). See these crossing limit cycles of types 1 and 2 in Figure 16.  Acknowledgements The first author is partially supported by CAPES grant 88881.188516/2018-01. The second author is supported by the Ministerio de Ciencia, Innovación y Uni- versidades, Agencia Estatal de Investigación grants MTM2016-77278-P (FEDER) and MDM-2014-0445, the Agéncia de Gestió d’Ajuts Universitaris i de Recerca grant 2017SGR1617, and the H2020 European Research Council grant MSCA- RISE-2017-777911. 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Simpson; Bifurcations in piecewise-Smooth Continuous Systems, World Scientific series on Nonlinear Science A, vol 69, World scientific, Singapure, 2010. [25] B. Van Der Pol; On relaxation-oscillations, The London, Edinburgh and Dublin Phil. Mag and J. of Sci. ,2 (7) (1926), 978–992. [26] Y. Ye; Theory of limit cycles, Translations of Mathematical Monographs American Mathe- matical Society, 1986. Johana Jimenez Universidade Federal do Oeste da Bahia, 46470000 Bom Jesus da Lapa, Bahia, Brazil Email address: jjohanajimenez@gmail.com Jaume Llibre Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain Email address: jllibre@mat.uab.cat João C. Medrado Instituto de Matemática e Estatı́stica, Universidade Federal de Goiás, 74001-970 Goiânia, Goiás, Brazil Email address: medrado@ufg.br, joaocarlosmedrado@gmail.com