Electronic Journal of Qualitative Theory of Differential Equations 2020, No. 19, 1–48; https://doi.org/10.14232/ejqtde.2020.1.19 www.math.u-szeged.hu/ejqtde/ Crossing limit cycles for piecewise linear differential centers separated by a reducible cubic curve Jeidy J. Jimenez1 , Jaume Llibre B 2 and João C. Medrado3 1 Universidade Federal do Oeste da Bahia, Bom Jesus da Lapa, 46470000, Bahia, Brasil 2 Departament de Matemàtiques, Universitat Autònoma de Barcelona, Bellaterra, Barcelona, 08193, Catalonia, Spain 3 Instituto de Matemática e Estatística, Universidade Federal de Goiás, Goiânia, 74001-970, Goiás, Brazil Received 27 August 2020, appeared 5 April 2020 Communicated by Gabriele Villari Abstract. As for the general planar differential systems one of the main problems for the piecewise linear differential systems is to determine the existence and the maxi- mum number of crossing limits cycles that these systems can exhibit. But in general to provide a sharp upper bound on the number of crossing limit cycles is a very difficult problem. In this work we study the existence of crossing limit cycles and their distribu- tion for piecewise linear differential systems formed by linear differential centers and separated by a reducible cubic curve, formed either by a circle and a straight line, or by a parabola and a straight line. Keywords: discontinuous piecewise linear differential centers, limit cycles, conics. 2010 Mathematics Subject Classification: 34C05, 34C07, 37G15. 1 Introduction and statement of the main results The discontinuous piecewise differential systems arose from the study of nonlinear oscilla- tions by Andronov, Vitt and Khaikin in [1]. And nowadays the qualitative theory of the discontinuous piecewise differential systems is a matter of great interest in the mathematical community because these systems arise naturally in the modeling of several real phenomena and processes for instance in electronics, mechanics, economy, biology, neuroscience etc., see [3, 4, 10, 13, 21, 23] and references quoted therein. One of the main problems in the qualitative theory of the discontinuous piecewise dif- ferential systems is to determine the maximum number of crossing limits cycles that these systems can have and their distribution. In this work we study the crossing limit cycles which are periodic orbits isolated in the set of all periodic orbits of the piecewise linear differential system, which only have isolated points of intersection with the discontinuity curve. We recall that the 16th Hilbert’s problem requests for the maximum number of limit cycles that can have a polynomial differential system in R2 in function of the degree of the system, B Corresponding author. Email: jllibre@mat.uab.cat 2 J. Jimenez, J. Llibre and J. C. Medrado see [11, 12]. Then the problem of establishing a sharp upper bound on the number of crossing limits cycles for the class of planar piecewise linear differential systems can be considered as an extension of the 16th Hilbert’s problem to this class and is in general a very difficult problem, because there are few developed techniques. In the plane the class of piecewise linear differential systems separated by a straight line is apparently the simplest class to study, and has been studied in several papers, see [2, 5–9, 16, 19, 22] but it is still an open problem to know if three is the maximum number of crossing limit cycles that this class can have. In particular when the class of piecewise linear differential systems separated by a straight line is formed by linear differential centers we know that these systems have no crossing limit cycles, see [15]. However, there are more recent works which study planar discontinuous piecewise linear differential centers where the curve of discontinuity is not a straight line, see [18, 20], there it was proved that there are crossing limit cycles in those systems. Moreover in the paper [14] it was provided the maximum number of crossing limit cycles for piecewise linear differential centers separated by any conic, then the objective of this work is to study the existence of crossing limit cycles of the discontinuous piecewise linear differential centers in R2 separated a reducible cubic curve, formed either by a circle and a straight line, or by a parabola and a straight line. In this paper we study the crossing limit cycles of the discontinuous piecewise linear differential centers separated by such reducible cubic curves which intersect either in two, or in four, or in six points the discontinuity curve. First we have the crossing limit cycles which intersect in two points the discontinuity curve. In [15] was proved that the class of linear differential centers separated by a straight line have no crossing limit cycles, then we can consider that those two intersection points on the discontinuity curve are on the circle or on the parabola and these two options were considered in the paper [14]. Second the crossing limit cycles intersect the discontinuity curve in exactly four points, here we consider that at least one of the four points is on the straight line, because the case which the four points are only on the circle or on the parabola was studied in [14]. Finally we have the crossing limit cycles such that intersect the discontinuity curve in six points. In this paper we study the crossing limit cycles with four points on discontinuity curve. In subsection 1.1 we consider the piecewise linear differential systems formed by linear differen- tial centers separated by the cubic Σk = ( x, y) ∈ R2 : ( x − k )( x2 + y2 − 1) = 0, k ∈ R, k ≥ 0 .  And in subsection 1.2 we consider the piecewise linear differential systems formed by linear differential centers separated by the cubic Σ̃k = ( x, y) ∈ R2 : (y − k )(y − x2 ) = 0, k ∈ R .  1.1 Crossing limit cycles intersecting the discontinuity curve Σk Let F1 be the family of piecewise linear differential centers separated by Σk with k > 1. Let F2 be the family of piecewise linear differential centers separated by Σk with k = 1. In these two cases we have the following regions in the plane R1 ={( x, y) ∈ R2 : x2 + y2 < 1}, R2 ={( x, y) ∈ R2 : x2 + y2 > 1 and x < k }, R3 ={( x, y) ∈ R2 : x2 + y2 > 1 and x > k }. Crossing limit cycles for piecewise linear differential centers 3 Figure 1.1: The regions for the familes F1 and F2 . Figure 1.2: The regions for the familes F3 . And finally let F3 be the family of piecewise linear differential centers separated by Σk with 0 ≤ k < 1. Here we have the following regions in the plane R1 ={( x, y) ∈ R2 : x2 + y2 < 1, and x > k }, R2 ={( x, y) ∈ R2 : x2 + y2 > 1, and x > k}, R3 ={( x, y) ∈ R2 : x2 + y2 > 1 and x < k }, R4 ={( x, y) ∈ R2 : x2 + y2 < 1 and x < k }. In the family F3 we have three types of crossing limit cycles. First crossing limit cy- cles such that are formed by parts of orbits of the four linear differential centers considered, namely crossing limit cycles of type 1, see Figure 2.3, second we have crossing limit cycles which intersect the regions R1 , R2 and R4 or crossing limit cycles that intersect the regions R1 , R4 and R3 , namely crossing limit cycles of type 2+ or crossing limit cycles of type 2− , respectively, see Figure 2.4. Without loss of generality we only study the crossing limit cycles of type 2+ because the analysis for the crossing limit cycles of type 2− is the same, moreover we observe that these two cases can not occur simultaneously, because the orbits of linear differential system in the region R4 which are pieces of ellipses would have these ellipses not nested in contradiction with the fact that the ellipses of a linear center are nested. And finally we have the crossing limit cycles such that are formed by parts of orbits of the three linear differential centers in the regions R1 , R2 and R3 , or crossing limit cycles formed by parts of orbits of the three linear differential centers in the regions R2 , R3 and R4 , namely crossing limit cycles of type 3+ and crossing limit cycles of type 3− , respectively, see Figure 2.5. Without loss of generality in Theorem 1.1 we study the crossing limit cycles of type 3+ because the study by the crossing limit cycles of type 3− is the same. We observe that these types of crossing limit cycles can not appear simultaneously, because the orbits of linear differential system in the region R3 which are pieces of ellipses would have these ellipses not nested in contradiction with the fact that the ellipses of a linear center are nested. If we study the piecewise linear dif- ferential centers in the family F3 which have simultaneously two types of crossing limit cycles 4 J. Jimenez, J. Llibre and J. C. Medrado we observe we would have three possible combinations between the three different crossing limit cycles types, 1, 2+ and 3+ , but we observe that the crossing limit cycles of types 2+ and 3+ can not appear simultaneously, because the orbits of linear differential system in the region R1 which are pieces of ellipses would have these ellipses not nested in contradiction with the fact that the ellipses of a linear center are nested. For this same reason there are no piecewise linear differential centers in F3 with three types of crossing limit cycles simultaneously. Then in the following theorem we provide examples of piecewise linear differential centers in F3 with crossing limit cycles of types 1, 2+ and 3+ separately and piecewise linear differential centers in F3 such that have simultaneously crossing limit cycles of types 1 and 2+ or of types 1 and 3+ . Theorem 1.1. The following statements hold. (a) There are piecewise linear differential systems in F1 and in F2 formed by three linear differential centers that have four crossing limit cycles, see Figures 2.1 and 2.2. (b) There are piecewise linear differential systems in F3 that have five crossing limit cycles of type 1, see Figure 2.3. (c) There are piecewise linear differential systems in F3 that have four crossing limit cycles of type 2+ , see Figure 2.4. (d) There are piecewise linear differential systems in F3 that have three crossing limit cycles of type 3+ , see Figure 2.5. (e) There are piecewise linear differential systems in F3 that have four crossing limit cycles of type 1 and two crossing limit cycles of type 2+ , see Figure 2.6. (f) There are piecewise linear differential systems in F3 that have four crossing limit cycles of type 1 and one crossing limit cycle of type 3+ , see Figure 2.7. Theorem 1.1 is proved in Section 2. By the numerical computations made for the families F1 , F2 and F3 and the illustrated examples of Theorem 1.1 we propose the following problem. Open problem 1. The numbers of crossing limit cycles determined in Theorem 1.1 for the families F1 , F2 and F3 are the maximum numbers of crossing limit cycles in each family. 1.2 Crossing limit cycles intersecting the discontinuity curve Σ̃k Let F4 be the family of piecewise linear differential centers separated by Σ̃k with k < 0. In this case, we have following three regions in the plane R1 ={( x, y) ∈ R2 : y > x2 }, R2 ={( x, y) ∈ R2 : y < x2 and y > k}, R3 ={( x, y) ∈ R2 : y < x2 and y < k}. For this family we have the following Theorem. Theorem 1.2. There are piecewise linear differential systems in F4 that have four crossing limit cycles with four points on Σ̃k , see Figure 3.1. Crossing limit cycles for piecewise linear differential centers 5 Figure 1.3: The regions for the familes F4 . Figure 1.4: The regions for the familes F5 . Theorem 1.2 is proved in Section 3. Let F5 be the family of piecewise linear differential centers separated by Σ̃k with k = 0. When the discontinuity curve is Σ̃0 we have following four regions in the plane R1 ={( x, y) ∈ R2 : y > x2 }, R2 ={( x, y) ∈ R2 : y < x2 and y > 0, x < 0}, R3 ={( x, y) ∈ R2 : y < x2 and y < 0}, R4 ={( x, y) ∈ R2 : y < x2 and y > 0, x > 0}. Here we have two types of crossing limit cycles, first crossing limit cycles formed by parts of orbits of the four linear differential centers considered, namely crossing limits cycles of type 4, see Figure 4.1. Second crossing limit cycles of type 5, see Figure 4.2, which intersect only three regions, in this case we have two options, first we have the case where the crossing limit cycles are formed by parts of orbits of the linear differential centers in the regions R1 , R3 and R4 and second the crossing limit cycles that intersect only the three regions R1 , R2 and R3 , without loss of generality we can consider the first case because the study for the second type of crossing limit cycles is the same. Here we observe that it is not possible to have crossing limit cycles of type 5 that satisfy those two cases simultaneously, because the orbits of linear differential system in the region R3 which are pieces of ellipses would have these ellipses not nested in contradiction with the fact that the ellipses of a linear center are nested. Therefore in the following Theorem we study the piecewise linear differential centers in F5 which have crossing limit cycles of types 4 and 5 separately, and crossing limit cycles of types 4 and 5 simultaneously. 6 J. Jimenez, J. Llibre and J. C. Medrado Theorem 1.3. The following statements hold. (a) There are piecewise linear differential systems in F5 that have four crossing limit cycles of type 4, see Figure 4.1. (b) There are piecewise linear differential systems in F5 that have three crossing limit cycles of type 5, see Figure 4.2. (c) There are piecewise linear differential systems in F5 that have simultaneously four crossing limit cycles of type 4 and two crossing limit cycles of type 5, see Figure 4.3. Theorem 1.3 is proved in Section 4. Figure 1.5: The regions for the familes F6 . Let F6 be the family of piecewise linear differential centers separated by Σ̃k with k > 0, in this case we have the following five regions in the plane √ R1 ={( x, y) ∈ R2 : y < x2 and y > k, x > k }, R2 ={( x, y) ∈ R2 : y > x2 and y > k }, √ R3 ={( x, y) ∈ R2 : y < x2 and y > k, x < − k }, R4 ={( x, y) ∈ R2 : y < x2 and y < k }, R5 ={( x, y) ∈ R2 : x2 < y < k }. Here we have six types of crossing limit cycles. First we have crossing limit cycles such that are formed by parts of orbits of the four linear differential centers in the regions R1 , R2 , R5 and R4 , or crossing limit cycles formed by parts of orbits of the four linear differential centers in the regions R2 , R3 , R4 and R5 , namely crossing limit cycles of type 6+ and crossing limit cycles of type 6− , respectively, see Figure 6.1. In Theorem 1.4 we study the crossing limit cycles of type 6+ because the study for the case of crossing limit cycles of type 6− is the same. Second we have crossing limit cycles type 7, see Figure 5.2, which intersect the three regions R2 , R5 and R4 . Third we have the crossing limit cycles of type 8, see Figure 5.3, which intersect the regions R1 , R2 , R3 and R4 . And finally we have the crossing limit cycles such that are formed by parts of orbits of the three linear differential centers in the regions R1 , R2 and R4 , or crossing limit cycles formed by parts of orbits of the three linear differential centers in the regions R2 , R3 and R4 , namely crossing limit cycles of type 9+ and crossing limit cycles of type 9− , respectively, see Figure 5.4. Without loss of generality in Theorem 1.4 we study the crossing limit cycles of type 9+ because the study by the crossing limit cycles of type 9− is the same. Then in Theorem 1.4 we study the crossing limit cycles of types 6+ , 7, 8 and 9+ . In Theorem 1.5 we study the piecewise linear differential centers in the family F6 which Crossing limit cycles for piecewise linear differential centers 7 have two types of crossing limit cycles simultaneously. And in Theorem 1.6 we study the piecewise linear differential centers in the family F6 which have three types of crossing limit cycles simultaneously. Theorem 1.4. The following statements hold. (a) There are piecewise linear differential systems in F6 that have five crossing limit cycles of type 6+ , see Figure 5.1. (b) There are piecewise linear differential systems in F6 that have three crossing limit cycles of type 7, see Figure 5.2. (c) There are piecewise linear differential systems in F6 that have four crossing limit cycles of type 8, see Figure 5.3. (d) There are piecewise linear differential systems in F6 that have three crossing limit cycles of type 9+ , see Figure 5.4. Theorem 1.4 is proved in Section 5. In Theorem 1.5 we would have fifteen possible combinations of pairs between the six different crossing limit cycles types, namely types 6+ , 6− , 7, 8, 9+ and 9− , we will analyze each one. Piecewise linear differential centers with crossing limit cycles of types 6+ and 6− are analyzed in statement ( a) of Theorem. The study for piecewise linear differential centers with crossing limit cycles of types 6+ and 7, or 6+ and 8, or 6+ and 9+ is the same for piecewise linear differential centers with crossing limit cycles of types 6− and 7, or 6− and 8, or 6− and 9− , respectively, and they are in statements (b), (c) and (d) of Theorem 1.5, respectively. The crossing limit cycles of types 6− and 9+ can not appear simultaneously because the orientation of these crossing limit cycles in region R4 would not be well defined, similarly happens with the crossing limit cycles of types 6+ and 9− . Piecewise linear differential centers with crossing limit cycles of types 7 and 8 are analyzed in statement (e) of Theorem 1.5. It is not possible to have crossing limit cycles of type 7 and 9+ , or 7 and 9− simultaneously, because the orbits of linear differential system in the region R2 would not be nested. Piecewise linear differential centers with crossing limit cycles of types 8 and 9+ are analyzed in statement ( f ) of Theorem 1.5, the case where appear crossing limit cycles of types 8 and 9− , simultaneously is the same. Finally we observe that it is not possible to have simultaneously crossing limit cycles of types 9+ and 9− , because the orbits of linear differential system in the region R4 would not be nested. Then we only have six cases analyzed in the following Theorem. Theorem 1.5. The following statements hold. (a) There are piecewise linear differential systems in F6 that have simultaneously four crossing limit cycles of type 6+ and four crossing limit cycles of type 6− , see Figure 6.1. (b) There are piecewise linear differential systems in F6 that have simultaneously four crossing limit cycles of type 6+ and two crossing limit cycles of type 7, see Figure 6.2. (c) There are piecewise linear differential systems in F6 that have simultaneously three crossing limit cycles of type 6+ and four crossing limit cycle of type 8, see Figure 6.3. (d) There are piecewise linear differential systems in F6 that have simultaneously four crossing limit cycles of type 6+ and two crossing limit cycles of type 9+ , see Figure 6.4. 8 J. Jimenez, J. Llibre and J. C. Medrado (e) There are piecewise linear differential systems in F6 that have simultaneously three crossing limit cycles of type 7 and four crossing limit cycle of type 8, see Figure 6.5. (f) There are piecewise linear differential systems in F6 that have simultaneously four crossing limit cycles of type 8 and two crossing limit cycles of type 9+ , see Figure 6.6. Theorem 1.5 is proved in Section 6. In Theorem 1.6 we would have twenty possible combinations of triplets between the six different crossing limit cycles types above, but we have fourteen combinations that include couples 6+ and 9− , 6− and 9+ , 7 and 9± , or 9+ and 9− and as it was said before these combinations are not possible. Therefore we have six options, first we observed that crossing limit cycles of types 6+ , 6− and 7, or 6+ , 6− and 8 can not appear simultaneously because the orientation of these crossing limit cycles in region R4 would not be well defined. Second we have that there are piecewise linear differential centers with crossing limit cycles of types 6+ , 7 and 8, this case is in statement ( a) of Theorem 1.6, the case where appear crossing limit cycles of types 6− , 7 and 8 is the same. Finally we have the piecewise linear differential centers with crossing limit cycles of types 6+ , 8 and 9+ , this case is in statement (b) of Theorem 1.6 and the case where appear crossing limit cycles of types 6− , 7 and 9− is the same. Theorem 1.6. The following statements hold. (a) There are piecewise linear differential systems in F6 that have simultaneously two crossing limit cycles of type 6+ , two crossing limit cycles of type 7 and four crossing limit cycles of type 8, see Figure 7.1. (b) There are piecewise linear differential systems in F6 that have simultaneously four crossing limit cycles of type 6+ , three crossing limit cycles of type 8 and two crossing limit cycles of type 9+ , see Figure 7.2. Theorem 1.6 is proved in Section 7. Similar to the previous case and by the illustrated examples in previous theorems we propose the following problem. Open problem 2. The numbers of crossing limit cycles determined in Theorems 1.2, 1.3, 1.4, 1.5 and 1.6 for the families F4 , F5 and F6 are the maximum numbers of crossing limit cycles in each family. By the previous analysis we observed that it is not possible to have piecewise linear differ- ential centers in F6 with four, five or six types of crossing limit cycles simultaneously. 2 Proof of Theorem 1.1 In the proof of the theorems will use the following lemma which provides a normal form for an arbitrary linear differential linear differential center, see a proof in [17]. Lemma 2.1. Through a linear change of variables and a rescaling of the independent variable every center in R2 can be written 4b2 + ω 2 ẋ = −bx − y + d, ẏ = ax + by + c, (2.1) 4a with a 6= 0 and ω > 0. This linear differential system has the first integral H1 ( x, y) = 4( ax + by)2 + 8a(cx − dy) + y2 ω 2 . (2.2) Crossing limit cycles for piecewise linear differential centers 9 Remark. As we shall see in the proofs of our results in order to find the crossing limit cycles we must solve equations of the form H1 ( x1 , y1 ) − H1 ( x2 , y2 ) = 0, (2.3) where H1 is defined in (2.2). Since a 6= 0 the solutions of equation (2.3) do not change if in it we change the function H1 by the function  2     b c d ω 2 2 H̄ ( x, y) = 4 x + y +8 x− y + y , a a a a because this is equivalent to divide equation (2.3) by the positive constant a2 > 0. Definining b c d ω = b̄, = c̄, = d,¯ = ω̄, a a a a the function H̄ ( x, y) is a first integral of the differential system 4b̄2 + ω̄ 2 ẋ = −b̄x − y + d,¯ ẏ = x + b̄y + c̄. (2.4) 4 Note that system (2.4) is essentially system (2.1) with a = 1. So in what follows we always will work with systems (2.1) with a = 1. In this way we shall work with systems having one parameter less and this will simplify a little the computations that will come. Proof of statement (a) for the family F1 of Theorem 1.1. By Lemma 2.1 we can consider the follow- ing piecewise linear differential center 4b12 + ω12 ẋ = −b1 x − y + d1 , ẏ = x + b1 y + c1 , in R1 , 4 4b2 + ω22 ẋ = −b2 x − 2 y + d2 , ẏ = x + b2 y + c2 , in R2 , (2.5) 4 4b2 + ω32 ẋ = −b3 x − 3 y + d3 , ẏ = x + b3 y + c3 , in R3 . 4 And the linear differential centers in (2.5) have the first integrals Hi ( x, y) = 4( x + bi y)2 + 8(ci x − di y) + y2 ωi2 , with i = 1, 2, 3, respectively. In order to have a crossing limit cycle, which intersects Σk in four different points p1 = (k, y1 ), p2 = ( x2 , y2 ), p3 = ( x3 , y3 ) and p4 = (k, y4 ), with p2 , p3 ∈ S1 , where S1 = ( x, y) : x2 + y2 = 1 . These points must satisfy the closing equations e1 = H2 (k, y1 ) − H2 ( x2 , y2 ) = 0, e2 = H1 ( x2 , y2 ) − H1 ( x3 , y3 ) = 0, e3 = H2 ( x3 , y3 ) − H2 ( k, y4 ) = 0, (2.6) e4 = H3 (k, y4 ) − H3 ( k, y1 ) = 0, x22 + y22 = 1, x32 + y23 = 1. For to build the example, we will impose the existence of periodic solutions and we will de- termine the parameters in (2.5) with the established conditions. First we fix the constant 10 J. Jimenez, J. Llibre and J. C. Medrado k = 2 and we assume that there is a real solution, namely q1 = (y11 , x21 , y12 , x31 , y13 , y14 ) = (3, cos (π/2), sin (π/2), cos (−π/3), sin (−π/3), −5/2), then by equations ei with i = 1, 2, 3, 4 in (2.6) we have the parameters ω22 d2 = 1 + b2 (3 + 2b2 ) + c2 + ; 2 1 √ √ d1 = − (−2 + 3)(−4 + 4b1 (2 3 + b1 ) − 16c1 + ω12 ); 16 √ √ √ √ 70 − 8 3 + 4b2 (10 − 5 3 + 31b2 − 4 3b2 ) + (31 − 4 3)ω22 c2 = √ ; 8(−8 + 3) 1 d3 = (4b3 (8 + b3 ) + ω32 ), 16 respectively. Now by the equation e4 we have 1 y4 = (1 − 2y1 ), 2 then we suppose that the point q2 = (y21 , x22 , y22 , x32 , y23 , y24 ) = (3, cos (π/2), sin (π/2), cos (−π/3), sin (−π/3), −5/2) is also a real solution of system (2.6), then by the equations e1 , e2 and e3 in (2.6) we obtain the following parameters 2 ω2 = − r √ √ q √ 3894 − 523 3 + 225 15 + 50 2(5 + 5)  √ √ √ q √ × −635 + 25 3 + 675 5 − 75 15 + 75 2(5 + 5)  √ √ √ q √ √ √ √  + 5 1468 + 34 3 + 100 5 − 50 15 + 5 2(5 + 5)(−68 + 3 − 8 5 + 15) b2  21 √ √ √ q √ + (−3894 + 523 3 + 25 5(70 − 9 3) − 50 2(5 + ; 5))b22 √ √  √ r  1 (−2 + 3) 5 + 5 (−4 + 8 3b1 + 4b12 + ω12 ) q √ 2 c1 = √ q √ + 6 ( 5 + 5))); 8(−1 + 5 − 2 2(5 + 5) b2 = 3.119845.., respectively. Now we fix the points x2 = cos (4π/7), y2 = sin (4π/7) and by equation e6 we have q y3 = − 1 − x32 , then by the equations e1 , e2 and e3qwe have y1 = 3.144465..; ω1 = −9.702226.. 0.042492.. + 0.031501..b1 − 0.042492..b12 ; x3 = 0.365470.., re- spectively. These conditions generate the real solution q3 = (3.144465.., cos (4π/7), sin (4π/7), 0.365470.., −0.930823.., −2.644465..). We build a fourth solution fixing the points x2 = −0.018219..; y2 = 0.999834..; therefore by the equations e1 , e2 and e3 we obtain y1 = 3.012016..; x3 = 0.489429..; b1 = 0.608380.., respectively. With these conditions we have the real solution q4 = (3.012016.., −0.018219.., 0.999834.., 0.489429.., −0.872042.., −2.512016..). With these four Crossing limit cycles for piecewise linear differential centers 11 real solutions we determined all the parameters that appear in system (2.6), even more in this particular case the parameters b3 , c3 , ω3 ∈ R, then we fix them, b3 = 1; c3 = 1/4; ω3 = 1. Therefore we obtain the following piecewise linear differential center ẋ = 0.977474.. − 0.608380..x − 1.451017..y, ẏ = −3.008357.. + x + 0.608380..y, in R1 , ẋ = 9.710162.. − 3.119845..x − 10.075224..y, ẏ = −20.799821.. + x + 3.119845..y, in R2 , (2.7) 37 5 1 ẋ = − x − y, ẏ = + x + y, in R3 . 16 4 4 The linear differential centers in (2.7) have the first integrals H1 ( x, y) = x2 + x (−6.016714.. + 1.216760..y) + y(−1.954949.. + 1.451017..y), H2 ( x, y) = x2 + x (−41.599643.. + 6.239690..y) + y(−19.420324.. + 10.075224..y), 37 H3 ( x, y) = 2x + 4x2 − y + 8xy + 5y2 , 2 respectively. Figure 2.1: Four crossing limit cycles of the discontinuous piecewise linear dif- ferential (2.7). These limit cycles are traveled in counterclockwise. In this case system (2.6) is equivalent to system 79.199286.. + x22 + 6.940944..y1 − 10.075224..y21 − 19.420324..y2 + 10.075224..y22 + x2 (−41.599643.. + 6.239690..y2 ) = 0, x22 − x32 − 1.954949..y2 + 1.451017..y22 + x2 (−6.016714.. + 1.216760..y2 ) + x3 (6.016714.. − 1.216760..y3 ) + 1.954949..y3 − 1.451017..y23 ) = 0, 79.199286.. + x32 − 19.420324..y3 + 10.075224..y23 (2.8) + x3 (−41.599643.. + 6.239690..y3 ) + 6.940944..y4 − 10.075224..y24 = 0,   5 (y1 − y4 ) − + 5y1 + 5y4 = 0, 2 2 2 x2 + y2 = 1, x32 + y23 = 1. Taking into account that the solutions qi = (y1i , x2i , y2i , x3i , y3i , y4i ) of system (2.8) must satisfy y4i < y1i we have that the unique reals solutions are the points q1 , q2 , q3 and q4 which provide four crossing limit cycles of the piecewise linear differential center (2.7). See these crossing limit cycles in Figure 2.1. 12 J. Jimenez, J. Llibre and J. C. Medrado Proof of statement (a) for the family F2 of Theorem 1.1. Following the steps illustrated in the pre- vious case we obtain a discontinuous piecewise linear differential system which is formed by the following linear differential centers in each region. First in the region R1 we have 3 3 ẋ = 2.185588.. − x − 6.201094..y, ẏ = −6.726549.. + x + y. (2.9) 20 20 This linear differential center has the first integral H1 ( x, y) = x2 + x (−13.453098.. + 3y/10) + y(−4.371176.. + 6.201094..y). In the region R2 we consider the linear differential center ẋ = −0.263120.. − 0.874044..x − 4.914345..y, ẏ = −23.305757.. + x + 0.874044..y, (2.10) which has the first integral H2 ( x, y) = x2 + x (−46.611514.. + 1.748088..y) + y(0.526241.. + 4.914345..y). And in the region R3 we have the linear differential center 21 5 1 ẋ = − x − y, ẏ = + x + y, (2.11) 16 4 4 which has the first integral H3 ( x, y) = 2x + 4x2 − 21y/2 + 8xy + 5y2 . In order to have a Figure 2.2: Four crossing limit cycles of the discontinuous piecewise linear dif- ferential formed by (2.9), (2.10) and (2.11) and separated by Σ1 . These limit cycles are traveled in counterclockwise. crossing limit cycle, which intersects Σ1 in four different points p1 = (1, y1 ), p2 = ( x2 , y2 ), p3 = ( x3 , y3 ) and p4 = (1, y4 ), with p2 , p3 ∈ S1 , these points must satisfy the closing equations given in (2.6). Then for the piecewise linear differential system formed by the centers (2.9), (2.10) and (2.11) we have that system (2.6) is equivalent to system 45.611514.. + x22 − 2.274330..y1 − 4.914345..y21 + 0.526241..y2 +4.914345..y22 + x2 (−46.611514.. + 1.748088..y2 ) = 0,   2 2 3 x2 − x3 + x2 −13.453098.. + y2 − 4.371176..y2 10   3 +6.201094..y22 + x3 13.453098.. − y3 + 4.371176..y3 − 6.201094..y23 = 0, 10 (2.12) 45.611514.. + x32 + 0.526241..y3 + 4.914345..y23 + x3 (−46.611514.. + 1.748088..y3 ) − 2.274330..y4 − 4.914345..y24 = 0,   5 (y1 − y4 ) − + 5y1 + 5y4 = 0, 2 2 2 x2 + y2 = 1, x32 + y23 = 1, Crossing limit cycles for piecewise linear differential centers 13 Therefore the unique real solutions qi = (y1i , x2i , y2i , x3i , y3i , y4i ) for system (2.12) that satisfy the condition y4i < y1i , are the points q1 = (3, cos (π/2), sin (π/2), cos (−π/3), sin (−π/3), −5/2); q2 = ( 17/5, cos (3π/5), sin (3π/5), cos (−2π/5), sin (−2π/5), −29/10); q3 = (3.294676.., cos (4π/7), sin (4π/7), 0.362651.., −0.931924.., −2.794676..) and q4 = (1.287554.., 0.814865.., 0.579649.., 0.966364.., −0.257177.., −0.787554), which generated four crossing limit cycles. See these crossing limit cycles of the piecewise linear differential center formed by (2.9), (2.10) and (2.11) in Figure 2.2. Proof of statement (b) of Theorem 1.1. We consider the piecewise linear differential center such that in the region R1 it has the linear differential center ẋ = 0.309248.. − 0.237408..x − 0.439335..y, ẏ = −0.478770.. + x + 0.237408..y, (2.13) this system has the first integral H1 ( x, y) = x2 + x (−0.957540.. + 0.474817..y) + (−0.618496.. + 0.439335..y)y. In the region R2 we have the linear differential center ẋ = 0.396090.. − 0.335276..x − 0.180370..y, ẏ = −0.861570.. + x + 0.335276..y, (2.14) which has the first integral H2 ( x, y) = x2 + x (−1.723140.. + 0.670553..y) + (−0.792181.. + 0.180370..y)y. In the region R3 we have the linear differential center ẋ = 0.242967.. + 0.112091..x − 0.194871..y, ẏ = 0.375114.. + x − 0.112091..y, (2.15) this system has the first integral H3 ( x, y) = x2 + x (0.750229.. − 0.224182..y) + (−0.485935.. + 0.194871..y)y. And in the region R4 we have the linear differential center ẋ = 0.394133.. + 0.278957..x − 0.25146..y, ẏ = 0.516804.. + x − 0.278957..y, (2.16) which has the first integral H4 ( x, y) = x2 + x (1.033609.. − 0.557914..y) − (0.788267.. −0.251469..y)y. In order to have a crossing limit cycle of type 1, which intersects the dis- Figure 2.3: Five crossing limit cycles of type 1 of the discontinuous piecewise linear differential system formed by the centers (2.13), (2.14), (2.15) and (2.16). These limit cycles are traveled in counterclockwise. 14 J. Jimenez, J. Llibre and J. C. Medrado continuity curve Σk in four different points p1 = (k, y1 ), p2 = ( x2 , y2 ), p3 = (k, y3 ) and p4 = ( x4 , y4 ), with p2 , p4 ∈ S1 , then these points must satisfy the system H1 (k, y1 ) = H1 ( x2 , y2 ), H2 ( x2 , y2 ) = H2 (k, y3 ), H3 (k, y3 ) = H3 ( x4 , y4 ), (2.17) H4 ( x4 , y4 ) = H4 (k, y1 ), x22 + y22 = 1, x42 + y24 = 1, Considering k = 0 and the previous piecewise linear differential center, system (2.17) is equiv- alent to system x22 + 0.618497..y1 − 0.439336..y21 + x2 (−0.957541.. + 0.474817..y2 ) − 0.618497..y2 +0.439336..y22 = 0, 4x22 − 3.168726..y2 + 0.721481..y22 + x2 (−6.892562.. + 2.682214..y2 ) + 3.168726..y3 −0.721481..y23 = 0, x42 + 0.485936..y3 − 0.194871..y23 + x4 (0.750229.. − 0.224183..y4 ) − 0.485936..y4 (2.18) +0.194871..y24 = 0, 4x42 + 3.153071..y1 − 1.005879..y21 + x4 (4.134439.. − 2.231658..y4 ) − 3.153071..y4 +1.005879..y24 = 0, x22 + y22 = 1, x42 + y24 = 1. Therefore discontinuous piecewise differential system formed by the linear differential centers (2.13), (2.14), (2.15) and (2.16) has five crossing limit cycles of type 1, because system (2.18) has five real solutions qi = (y1i , x2i , y2i , y3i , x4i , y4i ), for i = 1, 2, 3, 4, 5 that satisfy the conditions −1 < y1i < 1 < y3i ; x2i > 0 and x4i < 0. Where q1 = (1/3, cos (π/4), sin (π/4), 5/2, cos (5π/6), sin (5π/6)); q2 = ( 2/5, cos (27π/10), sin (27π/10), 12/5, cos (81π/100), cos (81π/100)); q3 = (1/5, cos (π/5), sin (π/5), 27/10, cos (89π/100), sin (89π/100)); q4 = (1/10, cos (3π/20), sin (3π/20), 57/20, cos (19π/20), sin (19π/20)) and q5 = (0.157052.., 0.843891.., 0.536513.., 2.764619.., −0.962848.., 0.270041..). See these five crossing limit cycles of type 1 in Figure 2.3. Proof of statement (c) of Theorem 1.1. We consider the following discontinuous piecewise linear differential system ẋ = −0.045605.. + 0.048166..x − 0.671455..y, ẏ = −0.418364.. + x − 0.048166..y, in R1 , x y ẋ = 0.058276.. + − 0.178664..y, ẏ = −0.763833.. + x − , in R2 , 100 100 901 x 901 1 y ẋ = − − y, ẏ = + x + , in R4 . 50000 50 2500 10 50 (2.19) The linear differential centers in (2.19) have the first integrals H1 ( x, y) = x2 + x (−0.836729.. − 0.096332..y) + (0.091210.. + 0.671455..y)y,  y H2 ( x, y) = x2 + x −1.527667.. − + (−0.116553.. + 0.178664..y)y, 50 4 901y(−1 + 10y) H4 ( x, y) = 4x2 + x (5 + y) + , 25 6250 Crossing limit cycles for piecewise linear differential centers 15 Figure 2.4: Four crossing limit cycles of type 2+ of the discontinuous piecewise linear differential center (2.19). These limit cycles are traveled in counterclock- wise. respectively. In order to have a crossing limit cycle of type 2+ , which intersects Σk in four different points p1 = ( x1 , y1 ), p2 = (k, y2 ), p3 = (k, y3 ) and p4 = ( x4 , y4 ), with p1 , p4 ∈ S1 , these points must satisfy the system H1 ( x1 , y1 ) = H1 ( k, y2 ), H4 ( k, y2 ) = H4 ( k, y3 ), H1 ( k, y3 ) = H1 ( x4 , y4 ), (2.20) H2 ( x4 , y4 ) = H2 ( x1 , y1 ), x12 + y21 = 1, x42 + y24 = 1. Then for the piecewise linear differential system (2.19) we have that system (2.20) becomes 4x12 + x1 (−3.346917.. − 0.385331..y1 ) + y1 (0.364840.. + 2.685822..y1 ) +(−0.364840..  − 2.685822..y2 )y2 = 0, 901 901 ( y2 − y3 ) − + (y2 + y3 ) = 0, 6250 625 −4x42 + y3 (0.364840.. + 2.685822..y3 ) + x4 (3.346917.. + 0.385331..y4 ) (2.21)   0.364840.. − 2.685822..y4 )y4 = 0, +(− 2 −4x12 + 4x42 + x1 6.110671.. + y1 + (0.466214.. − 0.714659..y1 )y1  25 2 + x4 −6.110671.. − y4 + (−0.466214.. + 0.714659..y4 )y4 = 0, 25 x12 + y21 = 1, x42 + y24 = 1, where k = 0. Therefore the unique real solutions qi = ( x1i , y1i , y2i , y3i , x4i , y4i ) for system (2.21) that satisfy the conditions −1 < y3i < y2i < 1; x1i > 0 and x4i > 0 are the points q1 = (cos (2π/5), sin (2π/5), 8/10, −7/10, cos (−3π/10), sin (−3π/10)); q2 = (cos (π/3), sin (π/3), 17/25, −29/50, cos (−π/10), sin (−π/10)); q3 = (cos (41π/100), sin (41π/100), 0.819235.., −0.719235.., 0.541860.., −0.840468..) and q4 = (0.256532.., 0.966535.., 0.833667.., −0.733667.., 0.508672.., −0.860960..). These four real solutions generated four crossing limit cycles of type 2+ . See these crossing limit cycles of the piecewise linear differential center (2.19) in Figure 2.4. 16 J. Jimenez, J. Llibre and J. C. Medrado Proof of statement (d) of Theorem 1.1. We consider the following discontinuous piecewise linear differential system 51 51 ẋ = 1.018312.. + x + 9.463668..y, ẏ = −5.008011.. − x − y, in R1 , 40 40 ẋ = 0.712799.. − 0.278320..x − 0.250791..y, ẏ = −1.026464.. + x + 0.278320..y, in R2 , 969 x 17 1 y ẋ = + − y, ẏ = + x − , in R3 . 1280 8 64 8 8 (2.22) The linear differential centers in (2.22) have the first integrals   2 51 H1 ( x, y) = 4x + x 40.064090.. + y + y(8.146500.. + 37.854675..y), 5 H2 ( x, y) = x2 + x (−2.052928.. + 0.556641..y) + (−1.425599.. + 0.250791..y)y, 17 H3 ( x, y) = x + 4x2 − xy + y(−57 + 10y). 160 respectively. In order to have a crossing limit cycle of type 3+ , which intersects the discontinu- ity curve Σk in four different points p1 = (k, y1 ), p2 = ( x2 , y2 ), p3 = (k, y3 ) and p4 = ( x4 , y4 ), with p2 , p4 ∈ S1 , these points must satisfy the system H2 ( x1 , y1 ) = H2 (k, y2 ), H3 (k, y2 ) = H3 (k, y3 ), H2 (k, y3 ) = H2 ( x4 , y4 ), (2.23) H1 ( x4 , y4 ) = H1 ( x1 , y1 ), x12 + y21 = 1, x42 + y24 = 1, Considering k = 0, system (2.23) is equivalent to system 4x12 + y1 (−5.702397.. + 1.003165..y1 ) + x1 (−8.211712.. + 2.226564..y1 ) +5.702397..y2 − 1.003165..y22 = 0, (y2 − y3 )(−57 + 10y2 + 10y3 ) = 0, 2 x4 + (1.425599.. − 0.250791..y3 )y3 + x4 (−2.052928.. + 0.556641..y4 ) 2   −1.425599..y4 + 0.250791..y4 = 0, (2.24) 51 x12 − x42 + x1 10.016022.. + y1 + y1 (2.036625.. + 9.463668..y1 )  20 51 + x4 −10.016022.. − y4 + (−2.036625.. − 9.463668..y4 )y4 = 0, 20 x12 + y21 = 1, x42 + y24 = 1. Therefore discontinuous piecewise differential (2.22) has three crossing limit cycles of type 3+ , because system (2.24) has three real solutions qi = ( x1i , y1i , y2i , y3i , x4i , y4i ), for i = 1, 2, 3 that satisfy the conditions 0 < x4i < x1i and 1 < y3i < y2i . Where q1 = (cos (π/5), sin (π/5), 43/10, 7/5 cos(2π/5), sin(2π/5)); q2 = (cos(16π/125), sin(16π/125), 447/100, 123/100 cos (9π/50), cos (9π/50)) and q3 = (cos (17π/100), sin (17π/100), 4.366812.., 1.333187.., 0.242211.., 0.970223..). See these three crossing limit cycles of type 3+ in Figure 2.5. Crossing limit cycles for piecewise linear differential centers 17 Figure 2.5: Three crossing limit cycles of type 3+ of the discontinuous piecewise linear differential center (2.22). These limit cycles are traveled in counterclock- wise. Proof of statement (e) of Theorem 1.1. We consider the following discontinuous piecewise linear differential system ẋ = 0.244909.. − 0.132672..x − 0.724279..y, ẏ = −0.471887.. + x + 0.132672..y, in R1 , ẋ = 0.668802.. − 0.514522..x − 0.636209..y, ẏ = −0.985653.. + x + 0.514522..y, in R2 , ẋ = −0.081198.. − 0.207828..x − 0.061343..y, ẏ = −0.124956.. + x + 0.207828..y, in R3 , ẋ = 0.211524.. − 0.634777..x − 0.705080..y, ẏ = −0.356652.. + x + 0.634777..y, in R4 . (2.25) The linear differential centers in (2.25) have the first integrals H1 ( x, y) = x2 + x (−0.943775.. + 0.265344..y) + (−0.489818.. + 0.724279..y)y, H2 ( x, y) = x2 + (−1.337605.. + 0.636209..y)y + x (−1.971307.. + 1.029044..y), H3 ( x, y) = x2 + x (−0.249913.. + 0.415657..y) + (0.162397.. + 0.061343..y)y, H4 ( x, y) = x2 + (−0.423048.. + 0.705080..y)y + x (−0.713304.. + 1.269555..y), respectively. In order to have crossing limit cycles of types 1 and 2+ , simultaneously, such that the crossing limit cycles of type 1 intersect the discontinuity curve Σ0 in four different points p1 = (0, y1 ), p2 = ( x2 , y2 ), p3 = (0, y3 ) and p4 = ( x4 , y4 ), with −1 < y1 < 1 < y3 and x4 < 0 < x2 and p2 , p4 ∈ S1 ; and the crossing limit cycles of type 2+ intersect the discontinuity curve Σ0 in four different points p5 = ( x5 , y5 ), p6 = (0, y6 ), p7 = (0, y7 ) and p8 = ( x8 , y8 ), with −1 < y7 < y6 < 1 and x5 , x8 > 0, with p5 , p8 ∈ S1 . These points must satisfy systems (2.17) and (2.20), respectively. Considering the piecewise linear differential center (2.25) systems 18 J. Jimenez, J. Llibre and J. C. Medrado Figure 2.6: Four crossing limit cycles of type 1 and two crossing limit cycles of type 2+ (black and magenta) of the discontinuous piecewise linear differential center (2.25). These limit cycles are traveled in counterclockwise. (2.17) and (2.20) become x22 + x2 (−0.943775.. + 0.265344..y2 ) − 0.489818..y2 + 0.724279..y22 +0.489818..y1 − 0.724279..y21 = 0, 4x22 − 5.350421..y2 + 2.544838..y22 + x2 (−7.885229.. + 4.116178..y2 ) +5.350421..y3 − 2.544838..y23 = 0, x42 − 0.162397..y3 − 0.061343..y23 + x4 (−0.249913.. + 0.415657..y4 ) +0.162397..y4 + 0.061343..y24 = 0, 4x42 − 1.692192..y4 + 2.820321..y24 + x4 (−2.853217.. + 5.078222..y4 ) +1.692192..y1 − 2.820321..y21 = 0, (2.26) 4x52 − 1.959275..y5 + 2.897117..y25 + x5 (−3.775101.. + 1.061377..y5 ) +1.959275..y6 − 2.897117..y26 = 0, (y6 − y7 )(−1.692192.. + 2.820321..(y6 + y7 )) = 0, x82 + 0.489818..y7 − 0.724279..y27 + x8 (−0.943775.. + 0.265344..y8 ) −0.489818..y8 + 0.724279..y28 ) = 0, x52 − x82 − 1.337605..y5 + 0.636209..y25 + x5 (−1.971307.. + 1.029044..y5 ) + x8 (1.971307.. − 1.029044..y8 ) + 1.337605..y8 − 0.636209..y28 = 0, x22 + y22 = 1, x42 + y24 = 1, x52 + y25 = 1, x82 + y28 = 1. We have four real solutions qi = (y1i , x2i , y2i , y3i , x4i , y4i , x5i , y5i , y6i , y7i , x8i , y8i ) with i = 1, 2, 3, 4, for system (2.26) that satisfy the above conditions, namely q1 = ( −1/3, cos (−π/6), sin (−π/6), 3/2, cos (2π/3), sin (2π/3), cos (π/3), sin (π/3), 7/10, −1/10, 1, 0); q2 = (−0.654342.., cos (−π/3), sin (−π/3), 12/5, cos (79π/100), sin (79π/100), cos (11π/50), sin (11π/50), 63/100, −3/100, 0.975733.., 0.216981..); q3 = (−0.447098.., cos (−23π/100), sin (−23π/100), 1.882264.., cos (18π/25), sin (18π/25), −0.654342.., cos (11π/50), sin (11π/50), 63/100, −3/100, 0.975733.., 0.216981..); q4 = ( −0.305568.., cos (−3π/20), sin (−3π/20), 1.365012.., −0.441883.., 0.897073.., cos (11π/50), sin (11π/50), 63/100, −3/100, 0.975733.., 0.216981..), these four solutions generated four crossing limit cycles of type 1 and two crossing limit cycles of type 2+ . See these crossing limit cycles of the piecewise linear differential center (2.25) in Figure 2.6. Crossing limit cycles for piecewise linear differential centers 19 Here we observed that we obtain a total of six crossing limit cycles between limit cycles of type 1 and of type 2+ , moreover these six crossing limit cycles have the configuration (4, 2), this is, 4-crossing limit cycle of type 1 and 2-crossing limit cycles of type 2+ . Clearly this lower bound for the maximum number of crossing limit cycles of types 1 and 2+ simultaneously, could be also obtained with the configurations (3, 3) or (2, 4). But after several numeric com- putations we could not build a third limit cycle of type 2+ , previously fixing two limit cycles of type 1, so we only get those lower bound with the configuration (4, 2). Proof of statement (f) of Theorem 1.1. We consider the following discontinuous piecewise linear differential system ẋ = 0.078341.. + 0.855624..x + 1.571418..y, ẏ = −0.065526.. − x − 0.855624..y, in R1 , ẋ = 0.496667.. + 0.078616..x − 0.193136..y, ẏ = −0.471461.. + x − 0.078616..y, in R2 , ẋ = 5.276135.. + 0.212817..x − 1.851275..y, ẏ = −5.383865.. + x − 0.212817..y, in R3 , ẋ = 0.484115.. + 0.548314..x − 0.303113..y, ẏ = 0.569064.. + x − 0.548314..y, in R4 . (2.27) The linear differential centers in (2.27) have the first integrals H1 ( x, y) = x2 + y(0.156682.. + 1.571418..y) + x (0.131053.. + 1.711249..y), H2 ( x, y) = x2 + x (−0.942922.. − 0.157232..y) + (−0.993334.. + 0.193136..y)y, H3 ( x, y) = x2 + x (−10.767731.. − 0.425635..y) + y(−10.552270.. + 1.851275..y), H4 ( x, y) = x2 + x (1.138128.. − 1.096628..y) + (−0.968231.. + 0.303113..y)y, respectively. In order to have crossing limit cycles of types 1 and 3+ , simultaneously, such that the crossing limit cycles of type 1 intersect the discontinuity curve Σ0 in four different points p1 = (0, y1 ), p2 = ( x2 , y2 ), p3 = (0, y3 ) and p4 = ( x4 , y4 ),with −1 < y1 < 1 < y3 and x4 < 0 < x2 and p2 , p4 ∈ S1 ; and the crossing limit cycles of type 3+ intersect the discontinuity curve Σ0 in four different points p5 = ( x5 , y5 ), p6 = (0, y6 ), p7 = (0, y7 ) and p8 = ( x8 , y8 ), with 1 < y7 < y6 , x5 , x8 > 0 and p5 , p8 ∈ S1 , these points must satisfy systems (2.17) and (2.23), respectively. Considering the piecewise linear differential center (2.27) systems (2.17) and (2.23) become −0.524214..x2 − 4x22 − 6.844999..x2 y2 + (y1 − y2 )(0.626729.. + 6.285673..(y1 + y2 )) = 0, −3.771690..x2 + 4x22 − 0.628929..x2 y2 + (y2 − y3 )(−3.973339.. + 0.772547..(y2 + y3 )) = 0, 43.070926..x4 − 4x42 + 1.702543..x4 y4 + (y3 − y4 )(−42.209082.. + 7.405102..(y3 + y4 )) = 0, 4.552513..x4 + 4x42 − 4.386514..x4 y4 − (y1 − y4 )(−3.872927.. + 1.212454..(y1 + y4 )) = 0, −3.771690..x5 + 4x52 − 0.628929..x5 y5 + (y5 − y6 )(−3.973339.. + 0.772547..(y5 + y6 )) = 0, (y6 − y7 )(−42.209082.. + 7.405102..(y6 + y7 )) = 0, 3.771690..x8 − 4x82 + 0.628929..x8 y8 + (y7 − y8 )(−3.973339.. + 0.772547..(y7 + y8 )) = 0, −4x52 + 4x82 + x5 (−0.524214.. − 6.844999..y5 ) + (−0.626729.. −6.285673..y5 )y5 + y8 (0.626729.. + 6.285673..y8 ) + x8 (0.524214.. + 6.844999..y8 ) = 0, x22 + y22 = 1, x42 + y24 = 1, x52 + y25 = 1, x82 + y28 = 1. (2.28) i i i i i i i We have four real solutions q = (y1 , x2 , y2 , y3 , x4 , y4 , x5 , y5 , y6 , y7 , x8 , y8 ) with i = 1, 2, 3, 4, for system (2.28) that satisfy the above conditions, namely q1 = (4/5, 1, 0, 26/5, cos (3π/5), sin (3π/5), cos (π/5), sin (π/5), 43/10, 7/5, cos (2π/5), sin (2π/5)); q2 = (53/100, 20 J. Jimenez, J. Llibre and J. C. Medrado Figure 2.7: Four crossing limit cycles of type 1 and one crossing limit cycle of type 3+ (black) of the discontinuous piecewise linear differential center (2.27). These limit cycles are traveled in counterclockwise. cos (−13π/100), sin (−13π/100), 557/100, cos (17π/25), sin (17π/25), cos (π/5), sin (π/5), 43/10, 7/5, cos (2π/5), sin (2π/5)); q3 = (1/2, cos (−3π/20), sin (−3π/20), 5.611962.., cos (17239π/25000), sin (17239π/25000), cos (π/5), sin (π/5), 43/10, 7/5, cos (2π/5), sin (2π/5)); q4 = (0.993727.., cos (12π/125), sin (12π/125), 4.808026.., −0.066301.., 0.997799.., cos (π/5), sin (π/5), 43/10, 7/5, cos (2π/5), sin (2π/5)), these four solutions generated four crossing limit cycles of type 1 and one crossing limit cycle of type 3+ . See these crossing limit cycles of the piecewise linear differential center (2.27) in Figure 2.7. Here we observed that we obtain a total of five crossing limit cycles between limit cycles of type 1 and of type 3+ , moreover these five crossing limit cycles have the configuration (4, 1), this is, 4-crossing limit cycle of type 1 and 1-crossing limit cycles of type 3+ . In order to obtain a result similar to the previous statement, this is, an example with a configuration (4, 2), we tried to build a second cycle of type 3+ but when building this second cycle we lost a cycle of type 1, so we only got a configuration (3, 2). If we consider the piecewise linear system Figure 2.8: Three crossing limit cycles of type 1 and two crossing limit cycle of type 3+ (black and orange) of the discontinuous piecewise linear differential center (2.29). These limit cycles are traveled in counterclockwise. Crossing limit cycles for piecewise linear differential centers 21 ẋ = −0.128852.. − 0.332114..x − 0.791281..y, ẏ = −0.143708.. + x + 0.332114..y, in R1 , ẋ = 0.597908.. + 0.108856..x − 0.227688..y, ẏ = −0.530777.. + x − 0.108856..y, in R2 , ẋ = 0.716356.. + 0.457342..x − 0.251353..y, ẏ = −0.189975.. + x − 0.457342..y, in R3 , 4 4 ẋ = 1.857676.. − x − 0.688147..y, ẏ = −1.219907.. + x + y, in R4 . 5 5 (2.29) It is possible verify that we obtain the configuration (3, 2), see Figure 2.8. But after several numeric computations we could not build a third limit cycle of type 3+ , previously fixing two limit cycles of type 1, so we only get those lower bound by the maximum number of types 1 and 3+ , simultaneously, with the configurations (4, 1) and (3, 2). 3 Proof of Theorem 1.2 We consider the following piecewise linear differential center 111 111 ẋ = −124.644504.. + x − 6.045715..y, ẏ = −148.901657.. + x − y, in R1 , 50 50 ẋ = 0.236087.. + 0.003662..x − 0.009243..y, ẏ = −0.402647.. + x − 0.003662..y, in R2 , x 9 y ẋ = 1 + − 0.102500..y, ẏ = − + x − , in R3 . 5 20 5 (3.1) The linear differential centers in (3.1) have the first integrals   2 111 H1 ( x, y) = x + x −297.803314.. − y + y(249.289008.. + 6.045715..y), 25 H2 ( x, y) = x2 + x (−0.805295.. − 0.007324..y) + (−0.472175.. + 0.009243..y)y,   2 9 2 H3 ( x, y) = x + x − − y + (−2 + 0.102500..y)y, 10 5 respectively. Figure 3.1: Four crossing limit cycles of the discontinuous piecewise linear dif- ferential system (3.1). These limit cycles are traveled in counterclockwise. For piecewise linear differential systems in the family F4 we have crossing limit cycles which intersect the discontinuity curve Σ̃k in four different points p1 = ( x1 , x12 ), p2 = ( x2 , x22 ), 22 J. Jimenez, J. Llibre and J. C. Medrado p3 = ( x3 , k ) and p4 = ( x4 , k ), if these points satisfy the system H1 ( x1 , x12 ) = H1 ( x2 , x22 ), H2 ( x2 , x22 ) = H2 ( x3 , k ), (3.2) H3 ( x3 , k ) = H3 ( x4 , k ), H2 ( x4 , k ) = H2 ( x1 , x12 ). Then for the piecewise linear differential centers (3.1) and Σ̃k considering k = −1, system (3.2) becomes x1 (−1191.213259.. + x1 (1001.156032.. + x1 (−44425 + 24.182863..x1 ))) + x2 (1191.213259.. + x2 (−1001.156032.. + (−44425 − 24.182863..x2 ) x2 )) = 0, −1.925675.. + x2 (−3.221182.. + x2 (2.111297.. + (−0.029297.. + 0.036973..x2 ) x2 )) +(3.191885.. − 4x3 ) x3 = 0,  (3.3) 1 ( x3 − x4 ) − + x3 + x4 = 0, 2 1.925675.. + x1 (3.221182.. + x1 (−2.111297.. + (0.029297.. − 0.036973..x1 ) x1 )) + x4 (−3.191885.. + 4x4 ) = 0. Taking into account that the solutions ( x1 , x2 , x3 , x4 ) must satisfy x2 < x1 and x3 < x4 , system (3.3) has four real solutions qi = ( x1i , x2i , x3i , x4i ), with i = 1, 2, 3, 4. Namely, q1 = (3, −2, −3/2, 2); q2 = (457/100, −3.753677.., −3.116713.., 3.616713..); q3 = (5.820000.., −5.115260.., −4.592690.., 5.092690..) and q4 = (41.045251.., −40.667957.., −162.945374.., 163.445374..). Which provide four crossing limit cycles of the piecewise linear differential center (3.1). See these four crossing limit cycles in Figure 3.1. Here we observe that there is a duality between the crossing limit cycles that intersect the discontinuity curve Σ̃−1 and the crossing limit cycles that intersect the discontinuity curve Σ2 for the family F1 studied in statement (a) of Theorem 1.1, where we also got four crossing limit cycles, see Figures 2.1 and 3.1. 4 Proof of Theorem 1.3 Proof of statement (a) of Theorem 1.3. We consider the following piecewise linear differential center 11 4 4 4 ẋ = + x − y, ẏ = 1 + x − y, in R1 , 10 5 5 5 17 3 17 61 3 ẋ = − x− y, ẏ = − + x + y, in R2 , 75 10 150 20 10 (4.1) 1 25 1 ẋ = + x − y, ẏ = − + x − y, in R3 , 6 16 4 133 x 7 543 y ẋ = + − y, ẏ = + x − , in R4 . 36 10 45 20 10 The linear differential centers in (4.1) have the first integrals H1 ( x, y) = 5x (2 + x ) − (11 + 8x )y + 4y2 , H2 ( x, y) = 150x2 + 17(−4 + y)y + 15x (−61 + 6y), y H3 ( x, y) = 4x2 − 2x (1 + 4y) + (−16 + 75y), 12 2 H4 ( x, y) = 90x + 9x (543 − 2y) + 7y(−95 + 2y), Crossing limit cycles for piecewise linear differential centers 23 Figure 4.1: Four crossing limit cycles of type 4 of the discontinuous piecewise linear differential system (4.1). These limit cycles are traveled in counterclock- wise. respectively. In order to have a crossing limit cycle of type 4, which intersects the discontinuity curve Σ̃0 in four different points p1 = ( x1 , x12 ), p2 = ( x2 , x22 ), p3 = ( x3 , 0) and p4 = ( x4 , 0), these points must satisfy system H1 ( x1 , x12 ) = H1 ( x2 , x22 ), H2 ( x2 , x22 ) = H2 ( x3 , 0), (4.2) H3 ( x3 , 0) = H3 ( x4 , 0), H4 ( x4 , 0) = H4 ( x1 , x12 ). Considering the piecewise linear differential center (4.1) system (4.2) becomes ( x1 − x2 )(−1 + x1 + x2 )(−5 + 2(−1 + x1 ) x1 + 2(−1 + x2 ) x2 ) = 0, 2x2 (−915 + x2 (82 + x2 (90 + 17x2 ))) +30(61 − 10x3 ) x3 = 0, 1 (4.3) 4( x3 − x4 ) − + x3 + x4 = 0, 2 2x1 (−4887 + x1 (575 + 2(9 − 7x1 ) x1 )) + 18x4 (543 + 10x4 ) = 0. In this case we have that the solutions qi = ( x1i , x2i , x3i , x4i ) must satisfy x2i < 0 < x1i and x3i < 0 < x4i then we have four real solutions q1 = (3, −2, −3/2, 2); q2 = (4, −3, −2, 5/2); q3 = (5, −4, 27/10, 16/5) and q4 = (10.440607.., −9.440607.., −19.555603.., 20.055606..) of sys- tem (4.3), which provide four crossing limit cycles of type 4 of the piecewise linear differential center (4.1). See these four crossing limit cycles in Figure 4.1. Here we observe that there is a duality between the crossing limit cycles of type 4 that in- tersect the discontinuity curve Σ̃0 and the crossing limit cycles that intersect the discontinuity curve Σ1 for the family F2 studied in statement (a) of Theorem 1.1, where we also got four crossing limit cycles, see Figures 2.2 and 4.1. Proof of statement (b) of Theorem 1.3. In this case we consider the following piecewise linear dif- ferential center 24 J. Jimenez, J. Llibre and J. C. Medrado Figure 4.2: Three crossing limit cycles of type 5 of the discontinuous piecewise linear differential system (4.4). These limit cycles are traveled in counterclock- wise. 2 2 ẋ = 0.100318.. − x + 0.161744..y ẏ = 0.260062.. − x + y, in R1 , 5 5 13 31 ẋ = 1 − x − y, ẏ = − + x + y, in R3 , 4 30 ẋ = −0.399222.. + 0.378090..x − 0.144616..y, ẏ = −1.020635.. + x − 0.378090..y, in R4 . (4.4) The linear differential centers in (4.4) have the first integrals   2 4 H1 ( x, y) = x + x −0.520124.. − y + (0.200636.. + 0.161744..y)y, 5 124 H3 ( x, y) = − x − 8y + 9y2 + 4( x + y)2 , 15 H4 ( x, y) = 4( x − 0.378090..y)2 + 8(−1.020635..x + 0.399222..y) + 0.006657..y2 , respectively. In order to have a crossing limit cycle of type 5, which intersects the discontinuity curve Σ̃0 in four different points p1 = ( x1 , x12 ), p2 = ( x2 , x22 ), p3 = ( x3 , 0) and p4 = ( x4 , 0), with 0 < x2 < x1 and 0 < x3 < x4 , these points must satisfy system H1 ( x1 , x12 ) = H1 ( x2 , x22 ), H4 ( x2 , x22 ) = H4 ( x3 , 0), (4.5) H3 ( x3 , 0) = H3 ( x4 , 0), H4 ( x4 , 0) = H4 ( x1 , x12 ). Considering the piecewise linear differential center (4.4) system (4.5) becomes −2.080498..x1 + 4.802546..x12 − 3.199999..x13 + 0.646977..x14 + x2 (2.080498.. − 4.802546..x2 + 3.199999..x22 − 0.646977..x23 ) = 0, x2 (−2591625737556 + x2 (2283329836763 + 50x2 (−19201143493 + 3672147700x2 ))) −324x3 (−7998844869+ 3918560960x3) = 0, 31 4( x3 − x4 ) − + x3 + x4 = 0, 15 x1 (2591625737556 + x1 (−2283329836763 + 50(19201143493 − 3672147700x1 ) x1 )) +324x4 (−7998844869 + 3918560960x4 ) = 0. (4.6) Crossing limit cycles for piecewise linear differential centers 25 In this case system (4.6) has three real solutions qi =( x1i , x2i , x3i , x4i ), where q1 =(2, 1/2, 2/5, 5/3); q2 = (93/50, 63/100, 47/100, 479/300) and q3 = (17/10, 0.785691.., 0.534387.., 1.532279..) which provide three crossing limit cycles of type 5 of the piecewise linear differential center (4.4). See these three crossing limit cycles in Figure 4.2. Proof of statement (c) of Theorem 1.3. We consider the following piecewise linear differential center x y ẋ = 45.736851.. − − 7.515818..y, ẏ = −1146.321640.. + x + , in R1 , 2 2 ẋ = −0.320594.. − 0.199436..x − 0.051960..y, ẏ = 0.460058.. + x + 0.199436..y, in R2 , x 13 23 y ẋ = 2 + − y, ẏ = − + x − , in R3 , 20 200 4 20 ẋ = −0.457007.. + 0.276952..x − 0.076768..y, ẏ = −4.377702.. + x − 0.276952..y, in R4 . (4.7) The linear differential centers in (4.7) have the first integrals H1 ( x, y) = x2 + x (−2292.643280.. + y) + y(−91.473702.. + 7.515818..y), H2 ( x, y) = x2 + x (0.920117.. + 0.398872..y) + (0.641188.. + 0.051960..y)yx2 + x (0.920117.. + 0.398872..y) + (0.641188.. + 0.051960..y)y, 2 13 H3 ( x, y) = 2x (−23 + 2x ) − (40 + x )y + y2 , 5 50 2 H4 ( x, y) = x + x (−8.755405.. − 0.553904..y) + (0.914014.. + 0.076768..y)y, respectively In order to have crossing limit cycles of type 4 and 5, simultaneously, such that the crossing limit cycles of type 4 intersect the discontinuity curve Σ̃0 in four different points p1 = ( x1 , x12 ), p2 = ( x2 , x22 ), p3 = ( x3 , 0) and p4 = ( x4 , 0), with x2 < 0 < x1 and x3 < 0 < x4 , and the crossing limit cycles of type 5 intersect the discontinuity curve Σ̃0 in four different points p5 = ( x5 , x52 ), p6 = ( x6 , x62 ), p7 = ( x7 , 0) and p8 = ( x8 , 0), with 0 < x6 < x5 and 0 < x7 < x8 , these points must satisfy systems (4.2) and (4.5), respectively. Considering the piecewise linear differential center (4.7) systems (4.2) and (4.5) become x1 (−9170.573120.. + x1 (−361.894811.. + x1 (3.999999.. + 30.063275..x1 ))) + x2 (9170.573120.. + x2 (361.894811.. + (−3.999999.. − 30.063275..x2 ) x2 )) = 0, x2 (3.680468.. + x2 (6.564754.. + (1.595489.. + 0.207843..x2 ) x2 )) − 3.680468..x3 − 4x32 = 0, ( x3 − x4 ) (−23 + 2x3 + 2x4 ) = 0, x1 (35.021620.. + x1 (−7.656056.. + (2.215618.. − 0.307072..x1 ) x1 )) − 35.021620..x4 + 4x42 = 0, x5 (−9170.573120.. + x5 (−361.894811.. + x5 (3.999999.. + 30.063275..x5 ))) + x6 (9170.573120.. + x6 (361.894811.. + (−3.999999.. − 30.063275..x6 ) x6 )) = 0, x6 (−35.021620.. + x6 (7.656056.. + (−2.215618.. + 0.307072..x6 ) x6 )) + 35.021620..x7 − 4x72 = 0, ( x7 − x8 ) (−23 + 2x7 + 2x8 ) = 0, x5 (35.021620.. + x5 (−7.656056.. + (2.215618.. − 0.307072..x5 ) x5 )) − 35.021620..x8 + 4x82 = 0. (4.8) i i i i i In this case system (4.8) has four real solutions q = ( x1 , x2 , x3 , x4 , x5 , x6 , x7 , x8 ), that sat- isfy the necessary conditions to have crossing limit cycles of types 4 and 5. Namely, q1 = (8, −16/5, −3, 29/2, 6, 3, 16/5, 83/10); q2 = (823/100, −413/100, −96/25, 767/50, 6, 3, 16/5, 26 J. Jimenez, J. Llibre and J. C. Medrado 83/10); q3 = (841/100, −4.737905.., −4.516438.., 16.016438.., 6.040228.., 2.934482.., 3.093430.., 8.406569..) and q4 = (429/50, −5.236369.., −5.170738.., 16.670738.., 6.040228.., 2.934482.., 3.093430.., 8.406569..). These solutions provide four crossing limit cycles of type 4 and two crossing limit cycles of type 5 of the piecewise linear differential center (4.7). See these cross- ing limit cycles in Figure 4.3. Here we observed that we obtain a total of six crossing limit cycles between limit cycles of type 4 and of type 5, moreover these six crossing limit cycles have the configuration (4, 2), this is, 4-crossing limit cycle of type 4 and 2-crossing limit cycles of type 5. We know that this lower bound for the maximum number of crossing limit cycles of types 4 and 5 simultaneously, could be also obtained with the configuration (3, 3). But if we previously fixing two limit cycles of each type after several numeric computations we could not build a third limit cycle of type 5, then we only get those lower bound with the configuration (4, 2). Figure 4.3: Four crossing limit cycles of type 4 and two crossing limit cycles of type 5 (black and orange) of the discontinuous piecewise linear differential system (4.7). These limit cycles are traveled in counterclockwise. 5 Proof of Theorem 1.4 Proof of statement (a) of Theorem 1.4. We consider the following piecewise linear differential center ẋ = −0.678037.. + 0.111302..x − 0.025436..y, ẏ = −3.106005.. + x − 0.111302..y, in R1 , ẋ = −0.133244.. + 0.232759..x − 0.058573..y, ẏ = −0.290609.. + x − 0.232759..y, in R2 , ẋ = 3.074032.. + 0.434135..x − 2.713559..y, ẏ = −3.035258.. + x − 0.434135..y, in R4 , ẋ = 1.427543.. + 0.059092..x − 0.651180..y, ẏ = −1.450367.. + x − 0.059092..y, in R5 . (5.1) The linear differential centers in (5.1) have the first integrals H1 ( x, y) = x2 + x (−6.212010.. − 0.222604..y) + (1.356074.. + 0.025436..y)y, H2 ( x, y) = x2 + x (−0.581218.. − 0.465518..y) + (0.266488.. + 0.058573..y)y, H4 ( x, y) = x2 + x (−6.070516.. − 0.868271..y) + y(−6.148064.. + 2.713559..y), H5 ( x, y) = x2 + x (−2.900734.. − 0.118185..y) + (−2.855087.. + 0.651180..y)y, respectively. In order to have a crossing limit cycle of type 6+ , which intersects the discontinu- Crossing limit cycles for piecewise linear differential centers 27 Figure 5.1: Five crossing limit cycles of type 6+ of the discontinuous piecewise linear differential system (5.1). These limit cycles are traveled in counterclock- wise. ity curve Σ̃k in four different points p1 = ( x1 , x12 ), p2 = ( x2 , k ), p3 = ( x3 , x32 ) and p4 = ( x4 , k ), these points must satisfy system H1 ( x4 , k ) = H1 ( x1 , x12 ), H2 ( x1 , x12 ) = H2 ( x2 , k), (5.2) H5 ( x2 , k ) = H5 ( x3 , x32 ), H4 ( x3 , x32 ) = H4 ( x4 , k ). Considering piecewise linear differential center (5.1) and k = 4, system (5.2) becomes −8.012495.. + x1 (−2.324875.. + x1 (5.065954.. + (−1.862075.. + 0.234292..x1 ) x1 )) +(9.773178.. − 3.999999..x2 ) x2 = 0, −1.001459.. + (−3.373476.. + x2 ) x2 + x3 (2.900734.. + x3 (1.855087.. + (0.118185.. −0.651180..x3 ) x3 )) = 0, (5.3) −75.298768.. + x3 (−24.282066.. + x3 (−20.592258.. + x3 (−3.473086.. +10.854237..x3 ))) + (38.174413.. − 4x4 ) x4 = 0, 23.325149.. + x1 (24.848040.. + x1 (−9.424297.. + (0.890418.. − 0.101747..x1 ) x1 )) + x4 (−28.409714.. + 4x4 ) = 0, In this case system (5.3) has five real solutions qi = ( x1i , x2i , x3i , x4i ) that satisfy the condi- tions −2 < x2i < 2 < x1i and −2 < x3i < 2 < x4i . We have q1 = (4, −2/5, −1/5, 7); q2 = (193/50, −31/100, −1/20, 683/100); q3 = (7/2, −3/25, 9/50, 641/100); q4 = (159/50, 1/100, 3/10, 303/50) and q5 = (4.149236.., −0.507154.., −0.449658.., 7.185104..), which provide five crossing limit cycles of type 6+ of the piecewise linear differential center (5.1). See these crossing limit cycles in Figure 5.1. Here we observe that there is a duality between the crossing limit cycles of type 6+ that intersect the discontinuity curve Σ̃4 and the crossing limit cycles of type 1 for the family F3 that intersect the discontinuity curve Σ0 studied in statement (b) of Theorem 1.1, where we also got five crossing limit cycles, see Figures 2.3 and 5.1. 28 J. Jimenez, J. Llibre and J. C. Medrado Proof of statement (b) of Theorem 1.4. We consider the following piecewise linear differential center x 17 21 y ẋ = 3 + − y, ẏ = + x − , in R2 , 4 16 20 4 36 ẋ = 3.601959.. − x − 5.323060..y, ẏ = − + x + y, in R4 , (5.4) 25 11827667 91445 8433175 26369 91445 ẋ = − x− y, ẏ = +x+ y, in R5 . 24434928 6205696 97739712 1108160 6205696 The linear differential centers in (5.4) have the first integrals 2 17 H2 ( x, y) = x (21 + 10x ) − 2(12 + x )y + y2 , 5   4 72 H4 ( x, y) = x2 + x − + 2y + y(−7.203918.. + 5.323060..y), 25 H5 ( x, y) = 977397120x2 + 63x (738332 + 457225y) + 10y(−94621336 + 8433175y), respectively. In order to have a crossing limit cycle of type 7, which intersects the discontinuity curve Σ̃k in four different points p1 = ( x1 , k ), p2 = ( x2 , k ), p3 = ( x3 , x32 ) and p4 = ( x4 , x42 ), these points must satisfy system H2 ( x1 , k ) = H2 ( x2 , k ), H5 ( x2 , k ) = H5 ( x3 , x32 ), (5.5) H4 ( x3 , x32 ) = H4 ( x4 , x42 ), H5 ( x4 , x42 ) = H5 ( x1 , k ). In this case considering k = 4, system (5.5) becomes Figure 5.2: Three crossing limit cycles of type 7 of the discontinuous piecewise linear differential system (5.4). These limit cycles are traveled in counterclock- wise. Crossing limit cycles for piecewise linear differential centers 29   1 4( x1 − x2 ) + x1 + x2 = 0, 10 −2435545440 + 4032x2 (40113 + 242410x2 ) − x3 (46514916 + 5x3 (6236752 +5x3 (1152207 + 3373270x3 ))) = 0,   288 (5.6) x3 − + x3 (−24.815674.. + x3 (8 + 21.292240..x3 ))  25  288 + x4 − + x4 (24.815674.. + (−8 − 21.292240..x4 ) x4 ) = 0, 25 2435545440 − 4032x1 (40113 + 242410x1 ) + x4 (46514916 + 5x4 (6236752 +5x4 (1152207 + 3373270x4 ))) = 0. System (5.6) has three real solutions qi = ( x1i , x2i , x3i , x4i ) that satisfy the conditions −2 < x2i < x1i < 2 and −2 < x3i < x4i < 2. They are q1 = (17/10, −9/5, −8/5, 3/2); q2 = (8/5, −17/10, −6/5, 6/5) and q3 = (89/50, −47/25, −1.788665.., 1.667136..), which provide three crossing limit cycles of type 7 of the piecewise linear differential center (5.4). See these three crossing limit cycles in Figure 5.2. Proof of statement (c) of Theorem 1.4. We consider the following piecewise linear differential center ẋ = −0.228658.. + 0.153388..x − 0.043263..y, ẏ = −1.233713.. + x − 0.153388..y, in R1 , 52 ẋ = + x − 5y, ẏ = 2 + x − y, in R2 , 5 ẋ = −0.208786.. − 0.135584..x − 0.040106..y, ẏ = 1.549735 + x + 0.135584..y, in R3 , x 5 41 y ẋ = 2 − − y, ẏ = − + x + , in R4 . 2 4 20 2 (5.7) The linear differential centers in (5.7) have the first integrals H1 ( x, y) = 15298879995x2 + 5y(1399284923 + 132375500y) − 6x (6291478429 + 782226050y), 8 H2 ( x, y) = 4x (4 + x ) − (52 + 5x )y + 20y2 , 5 H3 ( x, y) = 57070082030x2 + 15y(1588730299 + 152593500y) + x (176887019081 + 15475638300y),   2 82 H4 ( x, y) = 4x + x − + 4y + y(−16 + 5y), 5 respectively. In order to have a crossing limit cycle of type 8, which intersects the discontinuity curve Σ̃k in four different points p1 = ( x1 , k ), p2 = ( x2 , x22 ), p3 = ( x3 , x32 ) and p4 = ( x4 , k ), these points must satisfy system H2 ( x1 , x12 ) = H2 ( x2 , x22 ), H3 ( x2 , x22 ) = H3 ( x3 , k ), (5.8) H4 ( x3 , k ) = H4 ( x4 , k ), H1 ( x4 , k ) = H1 ( x1 , x12 ). In this case considering k = 4, system (5.8) becomes 30 J. Jimenez, J. Llibre and J. C. Medrado ( x1 − x2 )(−1 + 5x1 + 5x2 )(−20 + x1 (−1 + 5x1 ) + x2 (−1 + 5x2 )) = 0, 2 (−131946257940 + x2 (176887019081 + 5x2 (16180207303 28535041015 854345518x3 +60x2 (51585461 + 7629675x2 )))) − − 4x32 = 0, 51046585  1  (5.9) 4( x3 − x4 ) − + x3 + x4 = 0, 10 8 (19287869230 + x1 (18874435287 − 5x1 (2229530461 15298879995 1196239064x4 +10x1 (−46933563 + 6618775x1 )))) − + 4x42 = 0. 80946455 System (5.9) has four real solutions qi = ( x1i , x2i , x3i , x4i ) that satisfy the conditions x3i < −2 < 2 < x4i and x2i < −2 < 2 < x1i . They are q1 = (5/2, −23/10, −13/5, 27/10); q2 = (29/10, −27/10, −3, 31/10); q3 = (17/5, −16/5, −7/2, 18/5) and q4 = (98/25, −93/25, −203/50, 104/25) which provide four crossing limit cycles of type 8 of the piecewise linear differential center (5.7). See these four crossing limit cycles in Figure 5.3. Here we observe that there is a duality between the crossing limit cycles for family F4 stud- ied in Theorem 1.2, the crossing limit cycles of type 4 for the family F5 studied in statement (a) of Theorem 1.3 and crossing limit cycles of type 8 for the family F6 studied in statement (c) of Theorem 1.4. In these three cases we got four crossing limit cycles. See Figures 3.1, 4.1 and 5.3. Figure 5.3: Four crossing limit cycles of type 8 of the discontinuous piecewise linear differential system (5.7). These limit cycles are traveled in counterclock- wise. Proof of statement (d) of Theorem 1.4. We consider the following piecewise linear differential center 243469 1826 9088 614289 1826 ẋ = + x− y, ẏ = − +x− y, in R1 , 1620885 77185 324177 154370 77185 7 7 ẋ = −0.229652.. + x − 0.020472..y, ẏ = −1.718896.. + x − y, in R2 , (5.10) 5 5 9 53 1 9 ẋ = 1 + x − y, ẏ = − + x − y, in R4 . 10 50 2 10 Crossing limit cycles for piecewise linear differential centers 31 Figure 5.4: Three crossing limit cycles of type 9+ of the discontinuous piecewise linear differential system (5.10). These limit cycles are traveled in counterclock- wise. The linear differential centers in (5.10) have the first integrals H1 ( x, y) = 21x (−614289 + 77185x ) − 2(243469 + 38346x )y + 45440y2 ,   14 H2 ( x, y) = x2 + x −3.437793.. − y + (0.459305.. + 0.020472..y)y, 5  2 9 H4 ( x, y) = 4 x − y + y2 − 4( x + 2y), 10 respectively. In order to have a crossing limit cycle of type 9+ , which intersects the discontinu- ity curve Σ̃k in four different points p1 = ( x1 , x12 ), p2 = ( x2 , x22 ), p3 = ( x3 , k ) and p4 = ( x4 , k ), these points must satisfy system H2 ( x1 , x12 ) = H2 ( x2 , x22 ), H1 ( x2 , x22 ) = H1 ( x3 , k ), (5.11) H4 ( x3 , k ) = H4 ( x4 , k ), H1 ( x4 , k ) = H1 ( x1 , x12 ). Considering k = 4, system (5.11) becomes      23 x1 −13.751172.. + x1 5.837222.. + − + 0.081889..x1 x1 + x2 (13.751172.. 25     23 + x2 −5.837222.. + − − 0.081889..x2 x2 = 0, 25 x2 (12900069 − x2 (1133947 − 76692x2 + 45440x22 )) − 3(406904 + 7(628897 − 77185x3 ) x3 ) = 0,   41 4( x3 − x4 ) − + x3 + x4 = 0, 5 x1 (12900069 − x1 (1133947 − 76692x1 + 45440x12 )) − 3(406904 + 7(628897 − 77185x4 ) x4 ) = 0, (5.12) And we have that system (5.12) has three real solutions qi = ( x1i , x2i , x3i , x4i ) that satisfy the con- ditions 2 < x2i < x1i and 2 < x3i < x4i . They are q1 = (4, 3, 16/5, 5); q2 = (15/4, 33/10, 7/2, 47/10) and q3 = (41/10, 2.879320.., 3.058075.., 5.141924..) which provide three crossing limit cycles of type 9+ of the piecewise linear differential center (5.10). See these three crossing limit cycles in Figure 5.4. 32 J. Jimenez, J. Llibre and J. C. Medrado Here we observe that there is a duality between the crossing limit cycles of type 3+ for family F3 studied in statement (d) of Theorem 1.1, the crossing limit cycles of type 5 for the family F5 studied in statement (b) of Theorem 1.3 and crossing limit cycles of type 9+ for the family F6 studied in statement (d) of Theorem 1.4. In these three cases we got three crossing limit cycles. See Figures 2.5, 4.2 and 5.4. 6 Proof of Theorem 1.5 Proof of statement (a) of Theorem 1.5. We consider the following discontinuous piecewise linear differential system ẋ = 0.751960.. − 0.008805..x − 0.043938..y, ẏ = −1.117055.. + x + 0.008805..y, in R1 , 4701043 122761 91946 42715283 122761 ẋ = − − x+ y, ẏ = − −x+ y, in R2 , 7161144 156650025 31330005 313300050 156650025 ẋ = 0.041424.. − 0.228644..x − 0.115044..y, ẏ = 2.030027.. + x + 0.228644..y, in R3 , ẋ = 6.094659.. − 0.970562..x − 1.475325..y, ẏ = −4.066695 + x + 0.970562..y, in R4 , ẋ = −0.014046.. − 0.011408..x + 0.000796..y, ẏ = −0.900270.. − x + 0.011408..y, in R5 . (6.1) Figure 6.1: Four crossing limit cycles of type 6+ in the right hand side and four crossing limit cycles of type 6− in the left hand side, of the discontinu- ous piecewise linear differential system (6.1). These limit cycles are traveled in counterclockwise. The linear differential centers in (6.1) have the first integrals H1 ( x, y) = x2 + x (−2.234111.. + 0.017610..y) + (−1.503920.. + 0.043938..y)y, H2 ( x, y) = 626600100x2 + x (170861132 − 982088y) + 5y(−164536505 + 367784y), H3 ( x, y) = x2 + x (4.060055.. + 0.457288..y) + (−0.082848.. + 0.115044..y)y, H4 ( x, y) = x (−5448004792428006890183 + 669831938277330213420x ) − 160y (51029434834312436627 − 8126422570764957500x ) + 988220002292252000000y2 , H5 ( x, y) = 17172023317192110696x2 + x (30918934250652233287 − 391817091205831000y) + 6y(−80400672913407451 + 2279188834700000y), Crossing limit cycles for piecewise linear differential centers 33 respectively. In order to have simultaneously crossing limit cycles of types 6+ and 6− , such that the crossing limit cycles of type 6+ intersect the discontinuity curve Σ̃k in four different points p1 = ( x1 , x12 ), p2 = ( x2 , k ), p3 = ( x3 , x32 ) and p4 = ( x4 , k ), with −2 < x2 < 2 < x1 and −2 < x3 < 2 < x4 , and the crossing limit cycles of type 6− intersect the discontinuity curve Σ̃k in four different points p5 = ( x5 , x52 ), p6 = ( x6 , k ), p7 = ( x7 , x72 ) and p8 = ( x8 , k ), with x5 < −2 < x7 < 2 and x6 < −2 < x8 < 2, these points must satisfy systems (5.2) and H3 ( x5 , x52 ) = H3 ( x6 , k ), H4 ( x6 , k ) = H4 ( x7 , x72 ), (6.2) H5 ( x7 , x72 ) = H5 ( x8 , k ), H2 ( x8 , k ) = H2 ( x5 , x52 ), respectively. Considering the piecewise linear differential center (6.1) and k = 4, systems (5.2) and (6.2) become 170861132x1 − 196082425x12 − 982088x13 + 1838920x14 − 60(−54355123 + 2782213x2 +10443335x22 ) = 0, −1710814021790578824 + 29351665885828909287x2 + 17172023317192110696x22 −30918934250652233287x3 − 16689619279711665990x32 + 391817091205831000x33 −13675133008200000x34 = 0, −5448004792428006890183x3 − 7494877635212659646900x32 + 1300227611322393200000x33 + 988220002292252000000x34 + 21(802253250346853687680 −11766397482782575723x4 + 31896758965587153020x42 ) = 0, −21.250638.. + 8.936444..x1 + 2.015680..x12 − 0.070440..x13 −0.175755..x14 − 8.654682..x4 + 4x42 = 0, −6.037269.. + 16.240221..x5 + 3.668606..x52 + 1.829154..x53 +0.460177..x54 − 23.556840..x6 − 4x62 = 0, 16847318257283927441280 + 247094347138434090183x6 − 669831938277330213420x62 −5448004792428006890183x7 − 7494877635212659646900x72 +1300227611322393200000x73 + 988220002292252000000x74 = 0, 30918934250652233287x7 + 16689619279711665990x72 − 391817091205831000x73 +13675133008200000x74 − 21(−81467334370979944 + 1397698375515662347x8 +817715396056767176x82 ) = 0, −170861132x5 + 196082425x52 + 982088x53 − 1838920x54 + 60(−54355123 + 2782213x8 +10443335x82 ) = 0. (6.3) We have four real solutions qi = ( x1i , x2i , x3i , x4i , x5i , x6i , x7i , x8i ) with i = 1, 2, 3, 4, for system (6.3) that satisfy the above conditions namely q1 = (5, 1/2, 9/50, 23/5, −18/5, −9/2, −49/50, −1); q2 = (9/2, 19/20, 91/100, 7/5, 3, −17/5, −303/200, −3/2); q3 = (41/10, 1.208958.., 1.176604.., 2.657283..,−2.816357..,−31/10, −1.626433..,−1.613770..), and q4 = (51/10, 0.368157.., 0.315951.., 4.829311.., −3.059352.., −7/2, −1.475955.., −1.460360..), these four solutions generated four crossing limit cycles of type 6+ and four crossing limit cycles of type 6− . See these cross- ing limit cycles of the piecewise linear differential center (6.1) in Figure 6.1. Here we obtain a total of eight crossing limit cycles of types 6+ and 6− simultaneously, with a configuration (4, 4). And observed that it is possible obtain this lower bound with the configurations (5, 3) or (3, 5), but here we only present the example with the configuration (4, 4). 34 J. Jimenez, J. Llibre and J. C. Medrado Proof of statement (b) of Theorem 1.5. We consider the following discontinuous piecewise linear differential system ẋ = 1.717686.. + 0.650612..x − 0.423688..y, ẏ = 0.850546.. + x − 0.650612..y, in R1 , ẋ = 0.516832.. + 0.082481..x − 0.038759..y, ẏ = 0.179926.. + x − 0.082481..y, in R2 , ẋ = 1.470269.. + 0.406982..x − 3.640154..y, ẏ = −0.122065.. + x − 0.406982..y, in R4 , ẋ = 0.685228.. + 0.043300..x − 0.293631..y, ẏ = 0.017396.. + x − 0.043300..y, in R5 . (6.4) The linear differential centers in (6.4) have the first integrals H1 ( x, y) = x2 + x (1.701093.. − 1.301224..y) + (−3.435373.. + 0.423688..y)y, H2 ( x, y) = x2 + x (0.359853.. − 0.164963..y) + (−1.033664.. + 0.038759..y)y, H4 ( x, y) = x2 + x (−0.244130.. − 0.813965..y) + y(−2.940538.. + 3.640154..y), H5 ( x, y) = x2 + x (0.034792.. − 0.086601..y) + (−1.370456.. + 0.293631..y)y, respectively. In order to have simultaneously crossing limit cycles of types 6+ and 7, such that the crossing limit cycles of type 6+ intersect the discontinuity curve Σ̃k in four different points p1 = ( x1 , x12 ), p2 = ( x2 , k ), p3 = ( x3 , x32 ) and p4 = ( x4 , k ), with −2 < x2 < 2 < x1 and −2 < x3 < 2 < x4 , and the crossing limit cycles of type 7 intersect the discontinuity curve Σ̃k in four different points p5 = ( x5 , k ), p6 = ( x6 , k), p7 = ( x7 , x72 ) and p8 = ( x8 , x82 ), with −2 < x6 < x5 < 2 and −2 < x7 < x8 < 2 these points must satisfy systems (5.2) and (5.5), respectively. Considering the piecewise linear differential center (6.4) and k = 4, systems (5.2) and (5.5) become 14.058034.. + 1.439414..x1 − 0.134656..x12 − 0.659853..x13 + 0.155036..x14 6 + x2 − 4x22 = 0, 5 −0.783728.. − 0.311613..x2 + x2 − 0.034792..x3 + 0.370456..x3 + 0.086601..x33 2 2 −0.293631..x34 = 0, −185.921253.. − 0.976522..x3 − 7.762153..x32 − 3.255860..x33 + 14.560616..x34 +13.999964..x4 − 4x42 = 0, −27.849933.. − 6.804375..x1 + 9.741494..x12 + 5.204898..x13 − 1.694752..x14 −14.015217..x4 + 4x42 = 0, (6.5)   3 4( x5 − x6 ) − + x5 + x6 = 0, 10 −0.783728.. − 0.311613..x6 + x6 − 0.034792..x7 + 0.370456..x72 2 +0.086601..x73 − 0.293631..x74 = 0, −0.976522..x7 − 7.762153..x72 − 3.255860..x73 + 14.560616..x74 + x8 (0.976522.. + 7.762153..x8 + 3.255860..x82 − 14.560616..x83 ) = 0, −0.783728.. − 0.311613..x5 + x52 − 0.034792..x8 + 0.370456..x82 +0.086601..x83 − 0.293631..x84 = 0. We have four real solutions qi = ( x1i , x2i , x3i , x4i , x5 , x6 , x7 , x8 ) with i = 1, 2, 3, 4, for system (6.5) that satisfy the above conditions. We have q1 = (4, −9/5, −19/10, 7/2, 1, −7/10, −9/10, 11/10); q2 = (106/25, −39/20, −1.975633.., 51/10, 1, −7/10, −9/10, 11/10); q3 = (413/100, −469/250, −1.938820.., 4.420122.., 101/100, −71/100, −941/1000, 1.132764..) and q4 = Crossing limit cycles for piecewise linear differential centers 35 Figure 6.2: Four crossing limit cycles of type 6+ and two crossing limit cycles of type 7 (black and orange) of the discontinuous piecewise linear differential system (6.4). These limit cycles are traveled in counterclockwise. (401/100, −1.805407.., −1.902798.., 3.579564.., 101/100, −71/100, −941/1000, 1.132764..). These four real solutions generated four crossing limit cycles of type 6+ and two crossing limit cy- cles of type 7. See these crossing limit cycles of the piecewise linear differential center (6.4) in Figure 6.2. Here we observed that we obtain a total of six crossing limit cycles between limit cycles of type 6+ and of type 7, moreover these six crossing limit cycles have the configuration (4, 2). We observe that this lower bound for the maximum number of crossing limit cycles of types 6+ and 7 simultaneously, could be also obtained with the configuration (3, 3). But if we previously fixing two limit cycles of type 6+ after several numeric computations we could not build a third limit cycle of type 7, then we only get those lower bound with the configuration (4, 2). We can also observe that there is a duality between the case studied in statement (e) of Theorem 1.1, where we have studied simultaneously crossing limit cycles of types 1 and 2+ and this case, where study the crossing limit cycles of types 6+ and 7, simultaneously. In these two cases we got the configuration (4, 2). See Figures 2.6 and 6.2. Figure 6.3: Three crossing limit cycles of type 6+ (purple, green and black) and four crossing limit cycles of type 8 (orange, blue, magenta and light blue) of the discontinuous piecewise linear differential system (6.6). These limit cycles are traveled in counterclockwise. Proof of statement (c) of Theorem 1.5. We consider the following discontinuous piecewise linear 36 J. Jimenez, J. Llibre and J. C. Medrado differential system ẋ = 0.212208.. − 0.051128..x − 0.004724..y, ẏ = −3.713538.. + x + 0.051128..y, in R1 , ẋ = 0.592855.. − 0.098217..x − 0.044462..y, ẏ = −1.739750.. + x + 0.098217..y, in R2 , ẋ = −0.324307.. − 0.152006..x − 0.023227..y, ẏ = 2.010345.. + x + 0.152006y, in R3 , ẋ = 5.173755.. − 0.530837..x − 1.789344..y, ẏ = −2.823348.. + x + 0.530837..y, in R4 , 9 9 ẋ = 0.905547.. + x + 0.037591..y, ẏ = −2.213772.. − x − y, in R5 . 50 50 (6.6) The linear differential centers in (6.6) have the first integrals H1 ( x, y) = 92350000x2 + 2y(−19597489 + 218145y) + x (−685890524 + 9443461y), H2 ( x, y) = x (−2350427721 + 675507095x ) + 2(−400478067 + 66346510x )y + 30034700y2 , H3 ( x, y) = x2 + x (4.020691.. + 0.304014..y) + (0.648615.. + 0.023227..y)y, H4 ( x, y) = 2.248715.. × 1016 x2 − 5x (2.539563.. × 1016 − 4.774807.. × 1015 y) + y(−2.326860.. × 1017 + 4.023727.. × 1016 y), H5 ( x, y) = − 5.437818.. × 1022 x2 + 6x (−4.012698.. × 1022 − 3.262691.. × 1021 y) + 5(−1.969681.. × 1022 − 4.088345.. × 1020 y)y, respectively. In order to have crossing limit cycles of types 6+ and 8, simultaneously, such that the crossing limit cycles of type 6+ intersect the discontinuity curve Σ̃k in four different points p1 = ( x1 , x12 ), p2 = ( x2 , k ), p3 = ( x3 , x32 ) and p4 = ( x4 , k ), with −2 < x2 < 2 < x4 and −2 < x3 < 2 < x1 , and the crossing limit cycles of type 8 intersect the discontinuity curve Σ̃k in four different points p5 = ( x5 , x52 ), p6 = ( x6 , x62 ), p7 = ( x7 , k ) and p8 = ( x8 , k ), with x7 < −2 < 2 < x8 and x6 < −2 < 2 < x5 , these points must satisfy systems (5.2) and (5.8), respectively. Considering the piecewise linear differential center (6.6) and k = 4, systems (5.2) and (5.8) become 16.125777.. − 13.918004..x1 − 0.742843..x12 + 0.785738..x13 + 0.177849..x14 +10.775049..x2 − 4x22 = 0, 36 31.383400.. + 23.470181..x2 + 4x22 − 17.710181..x3 − 11.244381..x32 − x33 25 −0.150367..x34 = 0, 51.042105.. − 22.586789..x3 − 37.390043..x32 + 4.246697..x33 + 7.157379..x34 +5.599999..x4 − 4x42 = 0, −6.488327.. + 29.708306..x1 − 2.302329..x12 − 0.409029..x13 − 0.018897..x14 −28.072189..x4 + 4x42 = 0, (6.7) −149799272 − 648116680x8 + 92350000x82 + 685890524x5 − 53155022x52 −9443461x53 − 436290x54 = 0, −2350427721x5 − 125449039x52 + 132693020x53 + 30034700x54 + x6 (2350427721 + 125449039x6 − 132693020x62 − 30034700x63 ) = 0, −11.864396.. + 16.082766..x6 + 6.594461..x62 + 1.216054..x63 + 0.092909..x64 −20.946982..x7 − 4x72 = 0, ( x8 − x7 )(−7 + 5x8 + 5x7 ) = 0. We have four real solutions qi = ( x1i , x2i , x3i , x4i , x5i , x6i , x7i , x8i ) with i = 1, 2, 3, 4, for system (6.7) that satisfy the above conditions. We have q1 = (7/2, −6/5, 2/5, 19/5, 4, −3, −16/5, 23/5); Crossing limit cycles for piecewise linear differential centers 37 q2 = (18/5, −7/5, 3/10, 199/50, 41/10, −37/10, −3351/1000, 4751/1000); q3 = (71/20, −1.299400.., 7/20, 3.893976.., 4.132430.., −3.871790..,−17/5, 24/5) and q4 = (71/20, −1.299400.., 7/20, 3.893976.., 178349/20000, 108083/10000, −119/10, 133/10). These four real solutions generated three crossing limit cycles of type 6+ and four crossing limit cycle of type 8. See these crossing limit cycles of the piecewise linear differential center (6.6) in Figure 6.3. Here we observed that we obtain a total of seven crossing limit cycles between limit cycles of type 6+ and of type 8, moreover in this example, the seven crossing limit cycles have the configuration (3, 4). We observe that this lower bound for the maximum number of crossing limit cycles of types 6+ and 8 simultaneously, could be also obtained with the configurations (4, 3). And we obtain a example with this configuration in the proof of statement (b) of Theorem 1.6 with piecewise linear differential center (7.3), see Figure 7.2. Figure 6.4: Four crossing limit cycles of type 6+ and two crossing limit cycles of type 9+ (black and orange) of the discontinuous piecewise linear differential system (6.8). These limit cycles are traveled in counterclockwise. Proof of statement (d) of Theorem 1.5. We consider the following discontinuous piecewise linear differential system ẋ = −0.478750.. + 0.183274..x − 0.037189..y, ẏ = −4.300673.. + x − 0.183274..y, in R1 , ẋ = 0.122511.. + 0.079715..x − 0.013506..y, ẏ = −1.007263.. + x − 0.079715..y, in R2 , ẋ = −1.261810.. + 0.053348..x − 0.212413..y, ẏ = −4.836606.. + x − 0.053348..y, in R4 , ẋ = 0.060157.. + 0.062627..x − 0.047729..y, ẏ = −0.739728.. + x − 0.062627..y, in R5 . (6.8) The linear differential centers in (6.8) have the first integrals H1 ( x, y) = x2 + x (−8.601346.. − 0.366548..y) + (0.957501401147845‘ + 0.037189..y)y, H2 ( x, y) = x2 + x (−2.014527.. − 0.159430..y) + (−0.245022.. + 0.013506..y)y, H4 ( x, y) = x2 + x (−9.673213.. − 0.106696..y) + (2.523620.. + 0.212413..y)y, H5 ( x, y) = x2 + x (−1.479456.. − 0.125255..y) + (−0.120314.. + 0.047729..y)y, respectively. In order to have simultaneously crossing limit cycles of types 6+ and 9+ , such that the crossing limit cycles of type 6+ intersect the discontinuity curve Σ̃k in four different points p1 = ( x1 , x12 ), p2 = ( x2 , k ), p3 = ( x3 , x32 ) and p4 = ( x4 , k ), with −2 < x2 < 2 < x4 and −2 < x3 < 2 < x1 , and the crossing limit cycles of type 9+ intersect the discontinuity curve Σ̃k in four different points p5 = ( x5 , x52 ), p6 = ( x6 , x62 ), p7 = ( x7 , k ) and p8 = ( x8 , k ), with 2 < x6 < x5 and 2 < x7 < x8 , these points must satisfy systems (5.2) and (5.11), respectively. 38 J. Jimenez, J. Llibre and J. C. Medrado Considering the piecewise linear differential center (6.8) and k = 4, systems (5.2) and (5.11) become 3.055923.. + x1 (−8.058108.. + x1 (3.019909.. + (−0.637722.. + 0.054027..x1 ) x1 )) +(10.608997.. − 4x2 ) x2 = 0, 0.282412.. + (−1.980480.. + x2 ) x2 + x3 (1.479456.. + x3 (−0.879685.. + (0.125255.. −0.047729..x3 ) x3 )) = 0, −53.972411.. + x3 (−38.692854.. + x3 (14.094480.. + (−0.426786.. + 0.849655..x3 ) x3 )) +(40.4000000.. − 3.999999..x4 ) x4 = 0, 17.700131.. + x1 (34.405384.. + x1 (−7.8300056.. + (1.466193.. − 0.148756..x1 ) x1 )) + x4 (−40.270159.. + 4x4 ) = 0, −8.058108..x5 + 3.019909..x52 − 0.637722..x53 + 0.054027..x54 + x6 (8.058108.. −3.019909..x6 + 0.637722..x62 − 0.054027..x63 ) = 0, −17.700131.. − 34.405384..x6 + 7.830005..x62 − 1.466193..x63 + 0.148756..x64 +40.270159..x7 − 4x72 = 0, 4( x7 − x8 )(−10.100000.. + x7 + x8 ) = 0, 17.700131.. + 34.405384..x5 − 7.830005..x52 + 1.466193..x53 − 0.148756..x54 −40.270159..x8 + 4x82 = 0. (6.9) i i i i i We have four real solutions q = ( x1 , x2 , x3 , x4 , x5 , x6 , x7 , x8 ) with i = 1, 2, 3, 4, for system (6.9) that satisfy the above conditions. We have q1 = (6, 1/2, 4/10, 8, 5, 14/5, 3, 71/10); q2 = (317/50, 19/100, 1/25, 423/50, 5, 14/5, 3, 71/10); q3 = (291/50, 0.664193.., 3/5, 7.554404.., 487/100, 3.986608.., 3.058022.., 7.041977..) and q4 = (61/10, 0.409425.., 0.293958.., 8.128324.., 487/100, 3.986608.., 3.058022.., 7.041977..) These four real solutions generated four crossing limit cycles of type 6+ and two crossing limit cycles of type 9+ . See these crossing limit cycles of the piecewise linear differential center (6.8) in Figure 6.4. Here we obtain a total of six crossing limit cycles between limit cycles of type 6+ and of type 9+ , moreover these six crossing limit cycles have the configuration (4, 2). We observed that this lower bound for the maximum number of crossing limit cycles of types 6+ and 9+ simultaneously, could be also obtained with the configuration (3, 3). But if we build two cross- ing limit cycles of type 6+ and two of type 9+ , simultaneously, we have that all the parameters that appear in system (5.11) are determined, where this system is such that generated limit cycles of type 9+ , then it is no possible to build a third crossing limit cycle of type 9+ and therefore we can not obtain the configuration (3, 3). Proof of statement (e) of Theorem 1.5. We consider the following discontinuous piecewise linear differential system ẋ = −0.147861.. + 0.083875..x − 0.018000..y, ẏ = −3.106437.. + x − 0.083875..y, in R1 , 7769951 176465 204250 6997939 176465 ẋ = + x− y, ẏ = +x− y, in R2 , 9492348 2373087 2373087 47461740 2373087 ẋ = −0.284659.. − 0.174915..x − 0.046689..y, ẏ = 1.660380.. + x + 0.174915..y, in R3 , 3871251 3 4335 19 3 ẋ = − + x− y, ẏ = − + x − y, in R4 , 31913000 10 31913 20 10 ẋ = 0.206531.. + 0.150466..x − 0.054352..y, ẏ = 0.451143.. + x − 0.150466..y, in R5 . (6.10) Crossing limit cycles for piecewise linear differential centers 39 The linear differential centers in (6.10) have the first integrals H1 ( x, y) = (58546435625x2 + 4y(4328392296 + 263466775y) − 15x (24249448597 + 654747306y), H2 ( x, y) = x (6997939 + 23730870x ) − 5(7769951 + 705860x )y + 2042500y972 , H3 ( x, y) = 1.054579.. × 10−58 (3.792980.. × 1058 x2 + y(2.15949717.. × 1058 + 1.770939.. × 1057 y) + x (1.259558..1059 + 1.326899.. × 1058 y)), 2 3y(129041797 + 722500y) H4 ( x, y) = 4x2 + x (19 − 6y) + , 5 3989125 H5 ( x, y) = 16x (472818597 + 524021995x ) − 75(46176919 + 33641680x )y + 45571250097y2 , respectively. In order to have crossing limit cycles of types 7 and 8, simultaneously, such that the crossing limit cycles of type 7 intersect the discontinuity curve Σ̃k in four different points p1 = ( x1 , k), p2 = ( x2 , k ), p3 = ( x3 , x32 ) and p4 = ( x4 , x42 ), with −2 < x2 < x1 < 2 and −2 < x3 < x4 < 2, and the crossing limit cycles of type 8 intersect the discontinuity curve Σ̃k in four different points p5 = ( x5 , x52 ), p6 = ( x6 , x62 ), p7 = ( x7 , k ) and p8 = ( x8 , k ), with x6 < −2 < 2 < x5 and x7 < −2 < 2 < x8 , these points must satisfy systems (5.5) and (5.8), respectively. Considering the piecewise linear differential center (6.10) and k = 4, systems (5.5) and (5.8) become   3 4( x1 − x2 ) − + x1 + x2 = 0, 10 −6561675700 − 2527406448x2 + 8384351920x22 − 7565097552x3 −4921082995x32 + 2523126000x33 − 455712500x34 = 0, 30317350x3 + 19827751x32 − 9573900x33 + 2167500x34 − x4 (30317350 + 19827751x4 − 9573900x42 + 2167500x43 ) = 0, 6561675700 + 2527406448x1 − 8384351920x12 + 7565097552x4 +4921082995x42 − 2523126000x43 + 455712500x44 = 0, (6.11) 86116150336 − 403026567315x8 + 58546435625x82 + 363741728955x5 −75860004809x52 + 9821209590x53 − 1053867100x54 = 0, 6997939x5 − 15118885x52 − 3529300x53 + 2042500x54 + x6 (−6997939 +15118885x6 + 3529300x62 − 2042500x63 ) = 0, −1.030050.. + 8(1.660379.. + 0.284660..x6 ) x6 + 4(1 + 0.174915..x6 )2 x62 +0.064378..x64 + 8(−1.138640.. − 1.660379..x7 ) − 4(0.699661.. + x7 )2 = 0,   1 −4( x8 − x7 ) − + x8 + x7 = 0. 2 We have four real solutions qi = ( x1i , x2i , x3i , x4i , x5i , x6i , x7i , x8i ) with i = 1, 2, 3, 4, for system (6.11) that satisfy the above conditions. We have q1 = (1, −7/10, −9/10, −1/10, 37/10, −5/2, −3, 7/2); q2 = (1, −7/10, −9/10, −1/10, 4, −29/10, −33/10, 19/5); q3 = (11/10, −8/10, −26/25, 1/10, 21/5, −157/100, −7/2, 4) and q4 = (1.194602.., −0.894602.., −1.147986.., 0.273096.., 87/20, −3.312719.., −3653/1000, 4153/1000). These four real solutions generated three crossing limit cycles of type 7 and four crossing limit cycle of type 8. See these crossing limit cycles of the piecewise linear differential center (6.10) in Figure 6.5. Here we obtain a total of seven crossing limit cycles between limit cycles of type 7 and of type 8, moreover these seven crossing limit cycles have the configuration (3, 4). By our numer- 40 J. Jimenez, J. Llibre and J. C. Medrado Figure 6.5: Three crossing limit cycles of type 7 (purple, green and black) and four crossing limit cycles of type 8 of the discontinuous piecewise linear differ- ential system (6.10). These limit cycles are traveled in counterclockwise. ical computations we observed that this lower bound for the maximum number of crossing limit cycles of types 7 and 8 simultaneously, could not be obtained with the configuration (4, 3), because in the statement (b) of Theorem 1.4 we only got three crossing limit cycle of type 7. Proof of statement (f) of Theorem 1.5. We consider the following discontinuous piecewise linear differential system ẋ = −0.224106.. + 0.256615..x − 0.075244..y, ẏ = −3.489877.. + x − 0.256615..y, in R1 , x y ẋ = 33.031408.. − − 5.321982..y, ẏ = −816.418879.. + x + , in R2 , 2 2 ẋ = −0.151463.. − 0.173662..x − 0.047290..y, ẏ = 0.297861.. + x + 0.173662..y, in R3 , x 13 111 y ẋ = 2 + − y, ẏ = − + x − , in R4 . 20 200 20 20 (6.12) The linear differential centers in (6.12) have the first integrals H1 ( x, y) = x2 + x (−6.979755.. − 0.513231..y) + (0.448213.. + 0.075244..y)y, H2 ( x, y) = x2 + x (−1632.837759.. + y) + y(−66.062816.. + 5.321982..y), H3 ( x, y) = x2 + x (0.595723.. + 0.347324..y) + (0.302926.. + 0.047290..y)y, 13 2 H4 ( x, y) = 4x2 − 16y + y2 − x (111 + y), 50 5 respectively. In order to have simultaneously crossing limit cycles of types 8 and 9+ , such that the crossing limit cycles of type 8 intersect the discontinuity curve Σ̃k in four different points p1 = ( x1 , x12 ), p2 = ( x2 , x22 ), p3 = ( x3 , k ) and p4 = ( x4 , k ), with x2 < −2 < 2 < x1 and x3 < −2 < 2 < x4 , and the crossing limit cycles of type 9+ intersect the discontinuity curve Σ̃k in four different points p5 = ( x5 , x52 ), p6 = ( x6 , x62 ), p7 = ( x7 , k) and p8 = ( x8 , k), with 2 < x6 < x5 and 2 < x7 < x8 , these points must satisfy systems (5.8) and (5.11), respectively. Considering the piecewise linear differential center (6.12) and k = 4, systems (5.8) and (5.11) Crossing limit cycles for piecewise linear differential centers 41 become −6531.351039..x1 − 260.251264..x12 + 4x13 + 21.287931..x14 + x2 (6531.351039.. +260.251264..x2 − 4x22 − 21.287931..x23 ) = 0, −7.873414.. + 2.382895..x2 + 5.211706..x22 + 1.389297..x23 + 0.189161..x24 −7.940084..x3 − 4x32 = 0,   23 4( x3 − x4 ) − + x3 + x4 = 0, 2 11.987037.. + 27.919023..x1 − 5.792854..x1 + 2.052924..x1 − 0.300976..x14 2 3 −36.130722..x4 + 4x42 = 0 x5 (−6531.351039.. + x5 (−260.251264.. + x5 (4 + 21.287931..x5 ))) + x6 (6531.351039.. + x6 (260.251264.. + (−4 − 21.287931..x6 ) x6 )) = 0, −11.987037.. + x6 (−27.919023.. + x6 (5.792854.. + (−2.052924.. + 0.300976..x6 ) x6 )) +(36.130722.. − 4x7 ) x7 = 0,   23 4( x7 − x8 ) − + x7 + x8 = 0, 2 11.987037.. + x5 (27.919023.. + x5 (−5.792854.. + (2.052924.. − 0.300976..x5 ) x5 )) + x8 (−36.130722.. + 4x8 ) = 0, (6.13) Figure 6.6: Four crossing limit cycles of type 8 and two crossing limit cycles of type 9+ (black and orange) of the discontinuous piecewise linear differential system (6.12). These limit cycles are traveled in counterclockwise. We have four real solutions qi = ( x1i , x2i , x3i , x4i , x5 , x6 , x7 , x8 ) with i = 1, 2, 3, 4, for system (6.13) that satisfy the above conditions. We have q1 = (8, −16/5, −3, 29/2, 6, 3, 16/5, 83/10); q2 = (823/100, −4.136449.., −3.840062.., 15.340062.., 6, 3, 16/5, 83/10); q3 = ( 841/100, − 4.748093.., −4.516514.., 16.016514..587/100, 3.203924.., 177/50, 199/25) and q4 = (429/50, − 5.249123.., −5.170790.., 16.670790.., 587/100, 3.203924.., 177/50, 199/25). These four real so- lutions generated four crossing limit cycles of type 8 and two crossing limit cycles of type 9+ . See these crossing limit cycles of the piecewise linear differential center (6.12) in Figure 6.6. Here we obtain a total of six crossing limit cycles between limit cycles of type 8 and of type + 9 , moreover these six crossing limit cycles have the configuration (4, 2). We observed that this lower bound for the maximum number of crossing limit cycles of types 8 and 9+ simul- taneously, could be also obtained with the configurations (3, 3). But if we build two crossing limit cycles of type 8 and two of type 9+ , simultaneously, we have that all the parameters that 42 J. Jimenez, J. Llibre and J. C. Medrado appear in system (5.11) are determined, where this system is such that generated limit cycles of type 9+ , then it is no possible to build a third crossing limit cycle of type 9+ and therefore we can not obtain the configurations (3, 3). We can also observe that there is a duality between the case studied in statement (c) of Theorem 1.3, where we have studied simultaneously crossing limit cycles of types 4 and 5 and this case, where study the crossing limit cycles of types 8 and 9+ , simultaneously. In these two cases we got the configuration (4, 2). See Figures 4.3 and 6.6. 7 Proof of Theorem 1.6 Proof of statement (a) of Theorem 1.6. We consider the following discontinuous piecewise linear differential system ẋ = −0.107128.. + 0.268308..x − 0.095415..y, ẏ = −2.390037.. + x − 0.268308..y, in R1 , ẋ = 0.492346.. + 0.144928..x − 0.061289..y, ẏ = 0.429713.. + x − 0.144928..y, in R2 , ẋ = 1.394400.. + 0.300769..x − 0.091362..y, ẏ = 2.707746.. + x − 0.300769..y, in R3 , ẋ = 0.976917.. + 0.400189..x − 4.241691..y, ẏ = −0.349243.. + x − 0.400189..y, in R4 , ẋ = 0.685228.. + 0.043300..x − 0.293631..y, ẏ = 0.017396.. + x − 0.043300..y, in R5 . (7.1) The linear differential centers in (7.1) have the first integrals Figure 7.1: Two crossing limit cycle of type 6+ (magenta and blue), two cross- ing limit cycles of type 7 (black and orange) and four crossing limit cycles of type 8 (green, purple, brown and cyan) of the discontinuous piecewise linear differential system (7.1). These limit cycles are traveled in counterclockwise. H1 ( x, y) = x2 + x (−4.780074.. − 0.536616..y) + (0.214257.. + 0.095415..y)y, H2 ( x, y) = x2 + x (0.859427.. − 0.289856..y) + (−0.984693.. + 0.061289..y)y, H3 ( x, y) = x2 + x (5.415492.. − 0.601538..y) + (−2.788801.. + 0.091362..y)y, H4 ( x, y) = x2 + x (−0.698486.. − 0.800378..y) + y(−1.953834.. + 4.241691..y), H5 ( x, y) = x2 + x (0.034792.. − 0.086601..y) + (−1.370456.. + 0.293631..y)y, respectively. In order to have crossing limit cycles of types 6+ , 7 and 8 simultaneously, such that the crossing limit cycles of type 6+ intersect the discontinuity curve Σ̃k in four different Crossing limit cycles for piecewise linear differential centers 43 points p1 = ( x1 , x12 ), p2 = ( x2 , k ), p3 = ( x3 , x32 ) and p4 = ( x4 , k ), with −2 < x2 < 2 < x1 and −2 < x3 < 2 < x4 , the crossing limit cycles of type 7 intersect the discontinuity curve Σ̃k in four different points p5 = ( x5 , k ), p6 = ( x6 , k ), p7 = ( x7 , x72 ) and p8 = ( x8 , x82 ), with x5 < −2 < x7 < 2 and x6 < −2 < x8 < 2 and the crossing limit cycles of type 8 intersect the discontinuity curve Σ̃k in four different points p9 = ( x9 , x92 ), p10 = ( x10 , x102 ), p = ( x , k ) 11 11 and p12 = ( x12 , k ), with x10 < −2 < 2 < x9 and x11 < −2 < 2 < x12 these points must satisfy systems (5.2), (5.5) and (5.8) respectively. Considering the piecewise linear differential center (7.1) and k = 4, systems (5.2), (5.5) and (5.8) become 11.832571.. + 3.437710..x1 + 0.061227..x12 − 1.159427..x13 + 0.245157..x14 +1.200000..x2 − 3.999999..x22 = 0, −0.783728.. − 0.311613..x2 + x22 − 0.034792..x3 + 0.370456..x32 + 0.086601..x33 −0.293631..x34 = 0, −240.206876.. − 2.793946..x3 − 3.815339..x32 − 3.201513..x33 + 16.966764..x34 +15.600000..x4 − 4x42 = 0, 9.534728.. + 19.120296..x1 − 4.857030..x12 + 2.146465..x13 − 0.381662..x14 −27.706159..x4 + 4x42 = 0, 4( x5 − x6 )(−0.300000.. + x5 + x6 ) = 0, −0.783728.. − 0.311613..x6 + x62 − 0.034792..x7 + 0.370456..x72 + 0.086601..x73 −0.293631..x74 = 0, (7.2) −2.793946..x7 − 3.815339..x72 − 3.201513..x73 + 16.966764..x74 + x8 (2.793946.. +3.815339..x8 + 3.201513..x82 − 16.966764..x83 ) = 0, −0.783728.. − 0.311613..x5 + x52 − 0.034792..x8 + 0.370456..x82 +0.086601..x83 − 0.293631..x84 = 0, −3.437710..x10 − 0.061227..x102 + 1.159427..x 3 − 0.245157..x 4 10 10 + x9 (3.437710.. + 0.061227..x9 − 1.159427..x92 + 0.245157..x93 ) = 0, 2 − 2.406152..x 3 + 0.365448..x 4 38.773655.. + 21.661968..x10 − 7.155207..x10 10 10 2 = 0, −12.037359..x11 − 4x11 4( x11 − x12 )(−3.900000.. + x11 + x12 ) = 0, 2 + 4.780074..x − 1.214257..x 2 2.383682.. − 6.926539..x12 + x12 9 9 +0.536616..x9 − 0.095415..x94 = 0. 3 We have four real solutions qi = ( x1i , x2i , x3i , x4i , x5i , x6i , x7i , x8i , x9i , x10 i , x i , x i ) with i = 1, 2, 3, 4, 11 12 for system (7.2) that satisfy the above conditions, namely q1 = (4, −9/5, −19/10, 7/2, 1, −7/10, −9/10, 11/10, 5, −27/10, −5/2, 32/5); q2 = (2007/500, −181/100, −1.905170.., 3.692535.., 101/100, −71/100, −941/1000, 1.132764.., 511/100, −2.805313.., −139/50, 167/25); q3 = (2007/500, −181/100, −1.905170.., 3.692535.., 101/100, −71/100, −941/1000, 1.132764.., 26/5, −2.891869.., −3.012824.., 6.912824..) and q4 = (2007/500, −181/100, −1.905170.., 3.692535.., 101/100, −71/100, −941/1000, 1.132764.., 549/10, −52.535582.., −883.528310.., 887.428310..). These four real solutions generated two crossing limit cycles of type 6+ , two crossing limit cycles of type 7 and four crossing limit cycles of type 8. See these crossing limit cycles of the piecewise linear differential center (7.1) in Figure 7.1. Here we obtain a total of eight crossing limit cycles between limit cycles of types 6+ , 7 and 8, moreover these eight crossing limit cycles have the configuration (2, 2, 4), this is 2-crossing limit cycles of type 6+ , 2-crossing limit cycles of type 7 and 4-crossing limit of type 8. We observed that this lower bound for the maximum number of crossing limit cycles of types 44 J. Jimenez, J. Llibre and J. C. Medrado 6+ , 7 and 8 simultaneously, could be also obtained with other configurations. But if we build two crossing limit cycles of each type we obtain that all parameters of systems (5.2) and (5.5) are determined, and these systems are such that generated the limit cycles of types 6+ and 7, then we can not build more than two crossing limit cycles of types 6+ or 7 when we have previously fixed two crossing limit cycles of each type. Then we only obtain the configuration obtained here, namely (2, 2, 4). Figure 7.2: Four crossing limit cycles of type 6+ (green, magenta, cyan and purple), three crossing limit cycles of type 8 (yellow, brown and blue) and two crossing limit cycles of type 9+ (black and orange) of the discontinuous piece- wise linear differential system (7.3). These limit cycles are traveled in counter- clockwise. Proof of statement (b) of Theorem 1.6. We consider the following discontinuous piecewise linear differential system ẋ = −0.312756.. + 0.105676..x − 0.022483..y, ẏ = −4.523476.. + x − 0.105676..y, in R1 , ẋ = −0.158662.. + 0.176712..x − 0.031977..y, ẏ = −1.018470.. + x − 0.176712..y, in R2 , x y ẋ = 0.893671.. + − 0.055338..y, ẏ = 1.647781.. + x − , in R3 , (7.3) 10 10 ẋ = −1.521810.. + 0.129660..x − 0.102089..y, ẏ = −4.531357.. + x − 0.129660..y, in R4 , ẋ = 2.392166.. + 0.863445..x − 1.210282..y, ẏ = 11.457801.. + x − 0.863445..y, in R5 . The linear differential centers in (7.3) have the first integrals H1 ( x, y) = x2 + x (−9.046952.. − 0.211353..y) + (0.625512.. + 0.022483..y)y, H2 ( x, y) = x2 + x (−2.03694.. − 0.353424..y) + (0.317325.. + 0.031977..y)y,  y H3 ( x, y) = x2 + x 3.295563.. − + (−1.787342.. + 0.055338..y)y, 5 H4 ( x, y) = x2 + x (−9.062715.. − 0.259321..y) + (3.043621.. + 0.102089..y)y, H5 ( x, y) = x2 + x (22.915603.. − 1.726890..y) + y(−4.784333.. + 1.210282..y), respectively. In order to have crossing limit cycles of types 6+ , 8 and 9+ simultaneously, such that the crossing limit cycles of type 6+ intersect the discontinuity curve Σ̃k in four different points p1 = ( x1 , x12 ), p2 = ( x2 , k ), p3 = ( x3 , x32 ) and p4 = ( x4 , k ), with −2 < x2 < 2 < x1 and −2 < x3 < 2 < x4 , the crossing limit cycles of type 8 intersect the discontinuity curve Crossing limit cycles for piecewise linear differential centers 45 Σ̃k in four different points p5 = ( x5 , x52 ), p6 = ( x6 , x62 ), p7 = ( x7 , k ) and p8 = ( x8 , k ), with x6 < −2 < 2 < x5 and x7 < −2 < 2 < x8 and the crossing limit cycles of type 9+ intersect the discontinuity curve Σ̃k in four different points p9 = ( x9 , x92 ), p10 = ( x10 , x10 2 ), p = ( x , k ) 11 11 and p12 = ( x12 , k ), with 2 < x10 < x9 and 2 < x11 < x12 these points must satisfy systems (5.2), (5.8) and (5.11) respectively. Considering the piecewise linear differential center (7.3) and k = 4, systems (5.2), (5.8) and (5.11) become −7.123782.. + x1 (−8.147767.. + x1 (5.269300.. + (−1.413698.. + 0.127911..x1 ) x1 )) +(13.802561.. − 4x2 ) x2 = 0, 0.227189.. + x2 (16.008041.. + x2 ) + x3 (−22.915603.. + x3 (3.784333.. + (1.726890.. −1.210282..x3 ) x3 )) = 0, −55.231640.. + x3 (−36.250863.. + x3 (16.174485.. + (−1.037284.. + 0.408356..x  3) x3 )) 202 + − 4x4 x4 = 0, 5 11.447141.. + x1 (36.187810.. + x1 (−6.502051.. + (0.845414.. − 0.089933..x1 ) x1 )) + x4 (−39.569467.. + 4x4 ) = 0, x5 (−8.147767.. + x5 (5.269300.. + (−1.413698.. + 0.127911..x5 ) x5 )) + x6 (8.147767.. + x6 (−5.269300.. + (1.413698.. − 0.127911..x6 ) x6 )) = 0,   4 25.055786.. + x6 (13.182255.. + x6 (−3.149369.. + − + 0.221355..x6 x6 )) 5 +(−9.982255.. − 4x7 ) x7 = 0,   101 4( x7 − x8 ) − + x7 + x8 = 0, 10 11.447141.. + x5 (36.187810.. + x5 (−6.502051.. + (0.845414.. − 0.089933..x5 ) x5 )) + x8 (−39.569467.. + 4x8 ) = 0, x10 (8.147767.. + x10 (−5.269300.. + (1.413698.. − 0.127911..x10 ) x10 )) + x9 (−8.147767.. + x9 (5.269300.. + (−1.413698.. + 0.127911..x9 ) x9 )) = 0, −11.447141.. + x10 (−36.187810.. + x10 (6.502051.. + (−0.845414.. +0.089933..x10 ) x10 )) + (39.569467.. − 4x11 ) x11 = 0,   101 4( x11 − x12 ) − + x11 + x12 = 0, 10 2.861785.. + (−9.892366.. + x12 ) x12 + x9 (9.046952.. + x9 (−1.625512.. + (0.211353.. −0.022483..x9 ) x9 )) = 0. (7.4) We have four real solutions qi = ( x1i , x2i , x3i , x4i , x5i , x6i , x7i , x8i , x9i , x10 i , x i , x i ) with i = 1, 2, 3, 4, 11 12 for system (7.4) that satisfy the above conditions, namely q1 = (6, 1/2, 2/5, 8, 87/10, −31/10, −23/10, 62/5, 5, 19/5, 3, 71/10); q2 = (317/50, 0.042569.., 1/25, 8.417274.., 861/100, −3.007479.., −2.117234.., 12.217234.., 5, 19/5, 3, 71/10); q3 = (1479/250, −0.610424.., −1/2, 7.904488.., 883/100, −3.233408.., −2.568105.., 12.668105.., 51/10, 3.582979.., 2.936322.., 7.163677..), and q4 = (15/2, −1.752776.., −1.049779.., 10.157706.., 883/100, −3.233408.., −2.568105.., 12.668105.., 51/10, 3.582979.., 2.936322.., 7.163677..) these four solutions generated four crossing limit cy- cles of type 6+ , three crossing limit cycles of type 8 and two crossing limit cycle of type 9+ . See these crossing limit cycles of the piecewise linear differential center (7.3) in Figure 7.2. Here we obtain a total of nine crossing limit cycles between limit cycles of types 6+ , 8 and + 9 , moreover these nine crossing limit cycles have the configuration (4, 3, 2). We observed that this lower bound for the maximum number of crossing limit cycles of types 6+ , 8 and 46 J. Jimenez, J. Llibre and J. C. Medrado 9+ simultaneously, could be also obtained with other configurations. When we build two crossing limit cycles of each type we obtain that system (5.11) has all parameters determined, and therefore we can not build a third crossing limit cycle of type 9+ . Systems (5.2), (5.8) which generated the limit cycles of types 8 and 9+ would still have free parameters and it is possible verify that we can have the configurations (4, 3, 2) or (3, 4, 2). Here we have illustrated the configuration (4, 3, 2). Acknowledgements We thank to the reviewer his/her comments that help us to improve the presentation of this paper. The first author is partially supported by CAPES grant number 88881.188516/2018-01. The second author is supported by the Ministerio de Ciencia, Innovación y Universidades, Agencia Estatal de Investigación grants MTM2016-77278-P (FEDER) and MDM-2014-0445, the Agència de Gestió d’Ajuts Universitaris i de Recerca grant 2017SGR1617, and the H2020 Eu- ropean Research Council grant MSCA-RISE-2017-777911. The third author is partially sup- ported by the Brazilian agencies FAPESP (Grants 2013/24541-0 and 2017/03352-6), CAPES (Grant PROCAD 88881.068462/2014-01), CNPq (Grant 308006/2015-1), FAPEG (29199/2018), and FAPEG/CNPq (Grant PRONEX 2017-10267000508). References [1] A. Andronov, C. E. Chaikin, Theory of oscillations. English language edition edited under the direction of Solomon Lefschetz, Princeton University Press, Princeton, N.J., 1949. MR0029027 [2] J. C. Artés, J. Llibre, J. C. Medrado, M. A. Teixeira, Piecewise linear differential systems with two real saddles, Math. Comput. 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