Electronic Journal of Qualitative Theory of Differential Equations 2021, No. 69, 1–38; https://doi.org/10.14232/ejqtde.2021.1.69 www.math.u-szeged.hu/ejqtde/ Limit cycles of planar discontinuous piecewise linear Hamiltonian systems without equilibria separated by reducible cubics Rebiha Benterki1 , Johana Jimenez B 2 and Jaume Llibre3 1 Département de Mathématiques, Université Mohamed El Bachir El Ibrahimi, Bordj Bou Arréridj 34000, El Anasser, Algeria 2 Universidade Federal do Oeste da Bahia, 46470000 Bom Jesus da Lapa, Bahia, Brazil 3 Departament de Matematiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain Received 16 January 2021, appeared 10 September 2021 Communicated by Gabriele Villari Abstract. Due to their applications to many physical phenomena during these last decades the interest for studying the discontinuous piecewise differential systems has increased strongly. The limit cycles play a main role in the study of any planar dif- ferential system, but to determine the maximum number of limits cycles that a class of planar differential systems can have is one of the main problems in the qualitative theory of the planar differential systems. Thus in general to provide a sharp upper bound for the number of crossing limit cycles that a given class of piecewise linear dif- ferential system can have is a very difficult problem. In this paper we characterize the existence and the number of limit cycles for the piecewise linear differential systems formed by linear Hamiltonian systems without equilibria and separated by a reducible cubic curve, formed either by an ellipse and a straight line, or by a parabola and a straight line parallel to the tangent at the vertex of the parabola. Hence we have solved the extended 16th Hilbert problem to this class of piecewise differential systems. Keywords: limit cycles, discontinuous piecewise linear Hamiltonian systems, reducible cubic curves. 2020 Mathematics Subject Classification: 34C29, 34C25, 47H11. 1 Introduction and statement of the main results Andronov, Vitt and Khaikin [1] started around 1920’s the study of the piecewise differential systems mainly motivated for their applications to some mechanical systems, and nowadays these systems still continue to receive the attention of many researchers. Thus these differen- tial systems are widely used to model processes appearing in mechanics, electronics, economy, etc., see for instance the books [8] and [28], and the survey [25], as well as the hundreds of references cited there. B Corresponding author. Email:jjohanajimenez@gmail.com 2 R. Benterki, J. Jimenez and J. Llibre A limit cycle is a periodic orbit of the differential system isolated in the set of all periodic orbits of the system. Limit cycles are important in the study of the differential systems. Thus limit cycles have played and are playing a main role for explaining physical phenomena, see for instance the limit cycle of van der Pol equation [26, 27], or the one of the Belousov– Zhavotinskii model [3, 29], etc. The extended 16th Hilbert problem, that is, to find an upper bound for the maximum number of limit cycles that a given class of differential systems can exhibit, is in general an unsolved problem. Only for very few classes of differential system this problem has been solved. For the class of discontinuous piecewise differential systems here studied, we can obtain its solution by using the first integrals provided by the Hamiltonians of the systems which form the discontinuous piecewise differential systems. For the statement of the classical 16th Hilbert problem see [16, 18, 21]. Of course in order that a discontinuous piecewise differential system be defined on the dis- continuous line, which separates the different differential systems forming the discontinuous piecewise differential system, we follow the rules of Filippov, see [11]. The discontinuous piecewise differential systems formed by linear differential systems can exhibit two kinds of limit cycles, the crossing and the sliding limit cycles, the first are the ones which only contain isolated points of the line of discontinuity, and the second the ones which contains arcs of the line of discontinuity. Here we only study the crossing limit cycles. The simplest class of discontinuous piecewise differential systems are the planar ones formed by two pieces separated by a straight line having a linear differential system in each piece. Several authors have tried to determine the maximum number of crossing limit cycles for this class of discontinuous piecewise differential systems. Thus, in one of the first papers dedicated to this problem, Giannakopoulos and Pliete [14] in 2001, showed the existence of discontinuous piecewise linear differential systems with two crossing limit cycles. Then, in 2010 Han and Zhang [15] found other discontinuous piecewise linear differential systems with two crossing limit cycles and they conjectured that the maximum number of crossing limit cycles for discontinuous piecewise linear differential systems with two pieces separated by a straight line is two. But in 2012 Huan and Yang [17] provided numerical evidence of the existence of three crossing limit cycles in this class of discontinuous piecewise linear differential systems. In 2012, Llibre and Ponce [24] inspired by the numerical example of Huan and Yang, proved for the first time that there are discontinuous piecewise linear differential systems with two pieces separated by a straight line having three crossing limit cycles. Later on, other authors obtained also three crossing limit cycles for discontinuous piecewise linear differential systems with two pieces separated by a straight line, see Braga and Mello [9] in 2013, Buzzi, Pessoa and Torregrosa [10] in 2013, Liping Li [22] in 2014, Freire, Ponce and Torres [13] in 2014, and Llibre, Novaes and Teixeira [23] in 2015. But proving that discontinuous piecewise linear differential systems separated by a straight line have at most three crossing limit cycles is an open problem. Recently, in [4, 6, 7, 19, 20] the authors have studied the extended 16th Hilbert problem to discontinuous piecewise linear differential centers separated by either conics, or cubics. However for the discontinuous piecewise linear Hamiltonian systems without equilibrium points, it was proven in [12] that such systems separated by two parallel straight lines can have at most one crossing limit cycle. In [5] it was proven that there is an example of two crossing limit cycles when these systems are separated by three parallel straight lines, and they can also have two crossing limit cycles if the curve of separation is a parabola, and three crossing limit cycles if the curve of separation is either an ellipse or a hyperbola. In [2] the Limit cycles of DPWLHS without equilibria separated by reducible cubics 3 authors provided the maximum number of crossing limit cycles when the curve of separation of these systems is an irreducible cubic. In this paper we give the solution of the extended 16th Hilbert problem for discontinuous piecewise linear differential Hamiltonian systems without equilibrium points separated by two different reducible cubic curves, formed either by an ellipse and a straight line, or by a parabola and a straight line parallel to the tangent at the vertex of the parabola. More precisely, we provide the maximum number of crossing limit cycles for these systems, when these limit cycles intersected with the cubic of separation in four points. Note that if a crossing limit cycle of a discontinuous piecewise linear differential Hamilto- nian systems without equilibrium points intersects in two points the discontinuity line formed either by an ellipse and a straight line, or by a parabola and a straight line parallel to the tan- gent at the vertex of the parabola, this crossing limit cycle must intersect in two points either the straight line, or the ellipse or the parabola, and these types of crossing limit cycles already have been studied in [5, 12], as we have mention previously. For this reason in this paper we study the crossing limit cycles with intersect in four points the reducible cubic formed by either by an ellipse and a straight line, or by a parabola and a straight line parallel to the tangent at the vertex of the parabola. Doing an affine change if the reducible cubic is formed by an ellipse and a straight line we can transform it into the reducible cubic Γk = {( x, y) ∈ R2 : ( x − k )( x2 + y2 − 1) = 0, k ≥ 0}, formed by the circle x2 + y2 = 1 and the straight line x = k with k ≥ 0. In a similar way if the reducible cubic is formed by a parabola and a straight line parallel to the tangent at the vertex of the parabola we can transform it into the reducible cubic Σk = ( x, y) ∈ R2 : (y − k )(y − x2 ) = 0, k ∈ R ,  formed by the parabola y = x2 and the straight line y = k with k ∈ R First in Subsection 1.1 we shall consider the piecewise linear Hamiltonian systems without equilibrium points separated by the reducible cubic Γk , and after in Subsection 1.2 we shall consider the piecewise linear Hamiltonian systems without equilibrium points separated by the reducible cubic Σk . The next result is proved in [12]. Lemma 1.1. An arbitrary linear differential Hamiltonian system in R2 without equilibrium points can be written as ẋ = −λbx + by + µ, ẏ = −λ2 bx + λby + σ, where σ 6= λµ and b 6= 0. The Hamiltonian function of this Hamiltonian system is 1 b H ( x, y) = − λ2 bx2 + λbxy − y2 + σx − µy. (1.1) 2 2 Of course H ( x, y) is a first integral of the Hamiltonian system. 1.1 The line of discontinuity is a circle and a straight line We denote by C1 the class of planar discontinuous piecewise linear Hamiltonian systems with- out equilibrium points separated by Γk with k > 1. Let C2 be the class of planar discontinuous 4 R. Benterki, J. Jimenez and J. Llibre piecewise linear Hamiltonian systems without equilibrium points separated by Γk with k = 1. For these two classes we get the following three zones Z1 = {( x, y) ∈ R2 : x2 + y2 < 1}, Z2 = {( x, y) ∈ R2 : x2 + y2 > 1 and x < k }, (1.2) Z3 = {( x, y) ∈ R2 : x2 + y2 > 1 and x > k }. Now we denote by C3 the class of piecewise linear Hamiltonian systems without equilibrium points separated by Γk with 0 ≤ k < 1. In this case Γk separate the plane into four zones Z1 = {( x, y) ∈ R2 : x2 + y2 > 1 and x > k }, Z2 = {( x, y) ∈ R2 : x2 + y2 > 1 and x < k }, (1.3) Z3 = {( x, y) ∈ R2 : x2 + y2 < 1 and x < k }, Z4 = {( x, y) ∈ R2 : x2 + y2 < 1 and x > k }. 3 2 2 1 2 Z Z 1 1 1 2 3 3 4 0 Z Z Z 0 Z Z -1 -1 -2 -3 -2 -2 -1 0 1 2 3 -2 -1 0 1 2 (a) (b) Figure 1.1: (a) The three zones for the class C1 . (b) The four zones for the class C3 . We have three different configurations of crossing limit cycles for the class C3 . The first one which will be denoted by Conf 1, here we have the limit cycles formed by four pieces of orbits, such that in each zone of (1.3) we have one piece of orbit of each of the four Hamiltonian systems considered, see Figure 1.4a. The second configuration of limit cycles denoted by Conf 2, where we have the limit cycles formed by pieces of orbits belonging to the three zones either Z1 , Z2 and Z4 , or Z1 , Z2 and Z3 . We are going to consider only the three zones Z1 , Z2 and Z4 , because by a similar analysis we obtain the crossing limit cycles intersecting the three zones Z1 , Z2 and Z3 , for this configuration, see Figure 1.4b. Finally the third configuration namely Conf 3 where we have limit cycles formed by pieces of orbits belonging to the three zones either Z1 , Z3 and Z4 , or Z2 , Z3 and Z4 . For the same reason as in the second configuration, we are going to consider only the three zones Z1 , Z3 and Z4 , see Figure 1.4c. We notice that we can obtain two new configurations by combining the three previous ones, such as Conf 1 and Conf 2, Conf 1 and Conf 3. Note that we cannot have the configu- ration Conf 2 and Conf 3, and Conf 1, Conf 2 and Conf 3. Our main result on the crossing limit cycles of the discontinuous piecewise linear Hamil- tonian systems without equilibria when the discontinuity line is formed by a circle and a straight line is the following one. Limit cycles of DPWLHS without equilibria separated by reducible cubics 5 Theorem 1.2. The following statements hold for the discontinuous piecewise linear Hamiltonian sys- tems without equilibria when the discontinuity line is formed by a circle and a straight line. The maximum number of crossing limit cycles intersecting the cubic of separation in four points for the class (i) C1 or C2 is three and this maximum is reached in Example 1 for the class C1 and in Example 2 for the class C2 , see Figures 1.3a and 1.3b, respectively; (ii) C3 with Conf 1 is three and this maximum is reached in Example 3, see Figure 1.4a; (iii) C3 with Conf 2 is three and this maximum is reached in Example 4, see Figure 1.4b; (iv) C3 with Conf 3 is three and this maximum is reached in Example 5, see Figure 1.4c; (v) C3 with Conf 1 and Conf 2 simultaneously is six and this maximum is reached in Example 6, see Figure 1.5; (vi) C3 with Conf 1 and Conf 3 simultaneously is six and this maximum is reached in Example 7, see Figure 1.6. Theorem 1.2 is proved in Section 2. 1.2 The line of discontinuity is a parabola and a straight line parallel to the tan- gent at the vertex of the parabola 1.0 1.0 2.0 0.8 0.8 1.5 1 0.6 ZSk - 1 1 ZSk + ZS0 0.6 2 4 0.4 1.0 ZSk + ZSk + 2 4 0.2 0.5 ZS0 ZS0 0.4 5 0.0 2 ZSk- 0.2 ZSk + - 0.2 0.0 3 3- 3 0.0 ZSk + - 0.4 ZSk ZS0 - 0.5 -2 -1 0 1 2 -2 -1 0 1 2 - 1.5 - 1.0 - 0.5 0.0 0.5 1.0 1.5 (a) (b) (c) Figure 1.2: (a) Three zones for the class Ck− . (b) Four zones for the class C0 . (c) Five zones for the class Ck+ . Let CΣk− be the class of discontinuous piecewise linear Hamiltonian systems without equi- libria separated by Σk− with k < 0. In this case we have following three zones in the plane ZΣ1 k− = {( x, y) ∈ R2 : y > x2 }, ZΣ2 k− = {( x, y) ∈ R2 : y < x2 and y > k }, ZΣ3 − = {( x, y) ∈ R2 : y < x2 and y < k }, k see Figure 1.2a. Let CΣ0 be the class of discontinuous piecewise linear Hamiltonian systems without equilibria separated by Σk with k = 0. When the discontinuity curve is Σ0 we have 6 R. Benterki, J. Jimenez and J. Llibre following four zones in the plane ZΣ1 0 = {( x, y) ∈ R2 : y > x2 }, ZΣ2 0 = {( x, y) ∈ R2 : y < x2 and y > 0, x < 0}, ZΣ3 0 = {( x, y) ∈ R2 : y < x2 and y < 0}, ZΣ4 0 = {( x, y) ∈ R2 : y < x2 and y > 0, x > 0}, see Figure 1.2b. In this class we have two configurations of crossing limit cycles, first crossing limit cycles with Conf 4 which are constituted by pieces of orbits of the four Hamiltonian systems considered, see Figure 3.2a. Second crossing limit cycles with Conf 5 which intersect only three zones, 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 Hamiltonian systems in the zones ZΣ1 0 , ZΣ3 0 and ZΣ4 0 and second the crossing limit cycles that intersect only the three zones ZΣ1 0 , ZΣ2 0 and ZΣ3 0 , without loss of generality we can consider the first case because the study of the second is the same, see Figure 3.2b. Here we observe that it is not possible to have crossing limit cycles with Conf 5 that satisfy those two cases simultaneously, because the orbits of the Hamiltonian system in the zone ZΣ3 0 would not be nested. In statement (ii) of Theorem 1.3 we study the discontinuous piecewise linear Hamiltonian systems without equilibria in CΣ0 which have crossing limit cycles with Conf 4 and Conf 5 separately, and in statement (iii) of Theorem 1.3 we study the case when the crossing limit cycles with Conf 4 and Conf 5 appear simultaneously. Let CΣk+ be the class of discontinuous piecewise linear Hamiltonian systems without equi- libria separated by Σk with k > 0, in this case we have the following five zones in the plane ZΣ1 k+ = {( x, y) ∈ R2 : y > x2 and y > k }, √ ZΣ2 k+ = {( x, y) ∈ R2 : y < x2 and y > k, x < − k }, ZΣ3 + = {( x, y) ∈ R2 : y > x2 and y < k }, k √ ZΣ4 k+ = {( x, y) ∈ R2 : y < x2 and y > k, x > k }, ZΣ5 + k = {( x, y) ∈ R : x < y < k}, 2 2 see Figure 1.2c. In this class we have six different configurations of crossing limit cycles. First we have crossing limit cycles such that are formed by pieces of orbits of the four Hamiltonian systems in the zones ZΣ1 + , ZΣ5 + , ZΣ3 + and ZΣ4 + , or crossing limit cycles formed k k k k by pieces of orbits of the four Hamiltonian systems in the zones ZΣ1 + , ZΣ2 + , ZΣ3 + and ZΣ5 + , k k k k namely crossing limit cycles with Conf 6+ and crossing limit cycles with Conf 6− , respectively, see Figure 3.5. In statement (ii) of Theorem 1.3 we study the crossing limit cycles with Conf 6+ because the study for the case of crossing limit cycles with Conf 6− is the same. Second we have crossing limit cycles with Conf 7, which intersect the three zones ZΣ1 + , ZΣ5 + and k k ZΣ3 + , see Figure 3.3b. Third we have the crossing limit cycles with Conf 8, which intersect k the zones ZΣ1 + , ZΣ2 + , ZΣ3 + and ZΣ4 + , see Figure 3.3c. And finally we have the crossing limit k k k k cycles formed by pieces of orbits of the three Hamiltonian systems in the zones ZΣ1 + , ZΣ3 + and k k ZΣ4 + , or crossing limit cycles formed by pieces of orbits of the three Hamiltonian systems in k the zones ZΣ1 + , ZΣ2 + and ZΣ3 + , namely crossing limit cycles with Conf 9+ and crossing limit k k k cycles with Conf 9− , respectively, see Figure 3.3d. Without loss of generality in statement (ii) of Theorem 1.3 we study the crossing limit cycles with Conf 9+ because the study by Limit cycles of DPWLHS without equilibria separated by reducible cubics 7 the crossing limit cycles with Conf 9− is the same. We observe that there are no crossing limit cycles that intersect the five zones ZΣi + for i = 1, 2, 3, 4, 5. Then in statement (ii) of k Theorem 1.3 we study the crossing limit cycles with Conf 6+ , Conf 7, Conf 8 and Conf 9+ separately. In statements (iii)-(ix) of Theorem 1.3 we study the discontinuous piecewise linear Hamiltonian systems without equilibria in the class CΣk+ which have crossing limit cycles with two configurations simultaneously. Finally in statements (x)–(xii) we study the discontinuous piecewise linear Hamiltonian systems without equilibria in the class CΣk+ which have crossing limit cycles with three different configurations simultaneously. Our main result on the crossing limit cycles of the discontinuous piecewise linear Hamil- tonian systems without equilibria when the discontinuity curve is formed by a parabola and a straight line parallel to the tangent at the vertex of the parabola is the following one. Theorem 1.3. The following statements hold for the discontinuous piecewise linear Hamiltonian sys- tems without equilibria when the discontinuity line is formed by a parabola and a straight line parallel to the tangent at the vertex of the parabola. The maximum number of crossing limit cycles intersecting the cubic of separation in four points for the class (i) CΣk− is three and this maximum is reached, see Figure 3.1; (ii) CΣ0 or CΣk+ with either Conf 4, or Conf 5, or Conf 6+ , or Conf 7, or Conf 8, or Conf 9+ is three, respectively, see Figures 3.2a–3.3d; (iii) CΣk+ with Conf 4 and Conf 5 simultaneously is six, see Figure 3.4; (iv) CΣk+ with Conf 6+ and Conf 6− simultaneously is six, see Figure 3.5; (v) CΣk+ with Conf 6− and Conf 7 simultaneously is six, see Figure 3.6a; (vi) CΣk+ with Conf 6+ and Conf 8 simultaneously is six, see Figure 3.6b; (vii) CΣk+ with Conf 6+ and Conf 9+ simultaneously is six, see Figure 3.7; (viii) CΣk+ with Conf 7 and Conf 8 simultaneously is six, see Figure 3.8; (ix) CΣk+ with Conf 8 and Conf 9+ simultaneously is six, see Figure 3.9; (x) CΣk+ with Conf 6− , Conf 7 and Conf 8 simultaneously is nine, see Figure 3.10; (xi) CΣk+ with Conf 6+ , Conf 8 and Conf 9+ simultaneously is nine, see Figure 3.11; (xii) CΣk+ with Conf 6− , Conf 6+ and Conf 8 simultaneously is six with 2 (resp. 3) limit cycles with Conf 6− , 3 (resp. 2) limit cycles with Conf 6+ and 1 limit cycle with Conf 8, Figure 3.12 (resp. 3.13). Theorem 1.3 is proved in Section 3. 2 Proof of Theorem 1.2 Proof of statement (i ) of Theorem 1.2. We have to prove that the maximum number of crossing limit cycles of the class C1 intersecting the curve Γk in four points is three. In a similar way we should prove the statement for the classes C2 and C3 . 8 R. Benterki, J. Jimenez and J. Llibre 2 1 1 0 0 -1 -1 -2 -2 -1 0 1 2 3 4 -1 0 1 2 3 4 (a) (b) Figure 1.3: (a) The three limit cycles of the discontinuous piecewise differential system (2.3). (b) The three limit cycles of the discontinuous piecewise differen- tial system (2.4). By Lemma 1.1 we can consider the discontinuous piecewise linear Hamiltonian systems ẋ = −λi bi x + bi y + µi , ẏ = −λ2i bi x + λi bi y + σi , in the zone Zi , with i = 1, 2, 3. (2.1) with bi 6= 0 and σi 6= λi µi , and the three zones Zi are defined in (1.2). Their corresponding Hamiltonian first integrals are as (1.1) Hi ( x, y) = −(λ2i bi /2) x2 + λi bi xy − (bi /2)y2 + σi x − µi y, with i = 1, 2, 3. In order to have a crossing limit cycle which intersects Γk in the points Ai = ( xi , yi ), Bi = (zi , wi ), Ci = (k, f i ) and Di = (k, hi ), where k > 1, Ai and Bi are points on the circle x2 + y2 − 1 = 0, these points must satisfy the following system e1 = H1 ( xi , yi ) − H1 (zi , wi ) = 0, e2 = H2 ( xi , yi ) − H2 (k, f i ) = 0, e3 = H2 (zi , wi ) − H2 (k, hi ) = 0, (2.2) e4 = H3 (k, f i ) − H3 (k, hi ) = 0, xi2 + y2i − 1 = 0, z2i + wi2 − 1 = 0. We suppose that the discontinuous piecewise linear differential system (2.1) has four limit cy- cles. For this we must suppose that system (2.2) has four real solutions, namely ( Ai , Bi , Ci , Di ), i = 1, 2, 3, 4. The points Ai and Bi can take the form Ai = (cos ri , sin ri ), Bi = (cos si , sin si ). Then by solving e1 = 0 for the parameter σ1 and e4 = 0 for µ3 , we get 1     σ1 = b1 sin(r1 − s1 ) − λ21 − 1 sin(r1 + s1 ) − 2λ1 2(cos r1 − cos  s1 )  cos(r1 + s1 ) + 2µ1 (sin r1 − sin s1 ) , Limit cycles of DPWLHS without equilibria separated by reducible cubics 9 4 5 4 3 3 2 2 1 1 0 0 -1 -1 -1 0 1 2 3 -1.0 -0.5 0.0 0.5 1.0 (a) (b) 2.0 1.5 1.0 0.5 0.0 -0.5 -1.0 -1.5 -1 0 1 2 3 4 (c) Figure 1.4: (a) The three limit cycles of Conf 1 of the discontinuous piecewise differential system (2.5). (b) The three limit cycles of Conf 2 of the discontinuous piecewise differential system (2.7). (c) The three limit cycles of Conf 3 of the discontinuous piecewise differential system (2.9). b3 and µ3 = ( f 1 + h1 − 2kλ3 ), respectively. 2 Now we consider the second real solution of (2.2) for i = 2, and we fix the three points A2 = (cos r2 , sin r2 ), B2 = (cos s2 , sin s2 ) and (k, f 2 ), so by solving e1 = 0 for µ1 and e4 = 0 for h2 , we obtain 1    µ1 =      b1 csc r1 −2 s1 4 cos 12 (r1 − 2r2 + s1 ) − cos 12 (r1 + s1 − 2s2 )    − λ1 cos r1 sin(2r2 ) + cos r2 sin(r1 − s1 ) λ21 − 1 sin(r1 + s1 ) + 2λ1    cos(r1 + s1 ) − λ21 − 1 cos r1 sin(r2 − s2 ) sin(r2 + s2 ) + λ21 (− cos s2 ) sin(r1 − s1 ) sin(r1 + s1 ) + cos s2 sin(r1 − s1 ) sin(r1 + s1 ) − λ1 sin(2r1 ) cos s2 + λ1 cos r1 sin(2s2 ) − λ21 cos2 r2 cos s1 + cos s1 λ1 sin(2r2 )   − sin2 r2 + (sin s2 − λ1 cos s2 )2 + λ1 sin(2s1 ) cos s2 , 10 R. Benterki, J. Jimenez and J. Llibre 10 5 5 0 0 -5 -5 -10 -10 -15 -15 -20 -15 -10 -5 0 5 -1 0 1 2 3 4 5 6 Figure 1.5: Three limit cycles of Conf 1 and three limit cycles of Conf 2 for the class of the discontinuous piecewise differential system (2.10). 2.0 1.5 1.0 0.5 0.0 -0.5 -1.0 -1 0 1 2 3 4 Figure 1.6: Three limit cycles of Conf 1 and three limit cycles of Conf 3 for the class of the discontinuous piecewise differential system (2.11). and h2 = f 1 − f 2 + h1 . Likewise, the points A3 = (cos r3 , sin r3 ), B3 = (cos s3 , sin s3 ), (k, f 3 ) and (k, h3 ) are solution of (2.2), we fix A3 , B3 and (k, f 3 ), then by solving equation e4 = 0 for h√3 and e1 = 0 for λ1 we have h3 = f 1 − f 3 + h1 and we get the two values λ1,2 1 1 = ( A ± 2 2 sin 2 (r1 − r2 + √ s1 − s2 ) B)/C given in the appendix. Finally, if we fix the three points A4 = (cos r4 , sin r4 ), B4 = (cos s4 , sin s4 ), and (k, f 4 ), then from the equation e4 = 0 and e1 = 0 we have that h4 = f 1 − f 4 + h1 and b1 = 0 which is a contradiction to the assumptions. Therefore we have proved that the maximum number of crossing limit cycles for the class C1 intersecting the curve Γk in four points is three. Now we shall provide differential systems of class C1 , C2 and C3 separated by Γk with three limit cycles. We will explain the method for constructing an example of three crossing limit cycles intersecting Γk in four points, and by a similar way we build the remaining examples. Example 1: Three crossing limit cycles for the class C 1 . Here we consider the three zones Limit cycles of DPWLHS without equilibria separated by reducible cubics 11 defined in (1.2) for k = 2.5. We consider the Hamiltonian systems ẋ = − 0.02..x + 0.2..y + 0.316667.., ẏ = 0.02..y − 0.002..x in Z1 , ẋ = 10.8..x + 18y − 3, ẏ = −6.48..x − 10.8..y in Z2 , (2.3) 3 ẋ = 5.x − 3y − 1.38889.., ẏ = 8.33333..x − 5y in Z . The first integrals of the linear Hamiltonian systems (2.3) are H1 ( x, y) = − 0.001..x2 + 0.02..xy − 0.1..y2 − 0.316667..y, H2 ( x, y) = − 3.24..x2 − 10.8..xy − 9y2 + 3y, H3 ( x, y) = 4.16667..x2 − 5xy + 1.5y2 + 1.38889..y, respectively. The discontinuous piecewise linear differential system formed by the linear Hamiltonian systems (2.3) has exactly three crossing limit cycles, because the system of equations (2.2) has the three real solutions Si = ( xi , yi , zi , wi , f i , hi ) for i = 1, 2, 3, where S1 = (0.244811.., −0.969571.., 0.767202.., 0.641406.., −2.05982.., −0.60685..), S2 = (0.390566.., −0.920575.., 0.912879.., 0.40823.., −1.8861.., −0.780563..), S3 = (0.535321.., −0.844649.., 0.979509.., 0.201401.., −1.62201.., 1.04466..). Then these three limit cycles are drawn in 1.3a. Example 2: Three crossing limit cycles for the class C 2 . We consider the three zones defined in (1.2) with k = 1. We consider the Hamiltonian systems ẋ = 15x − 3y − 11.25.., ẏ = 75.x − 15.y + 22.5 in Z1 , ẋ = 4x + 20y − 3, ẏ = −0.8x − 4y + 6 in Z2 , (2.4) 3 ẋ = − 0.4x + 4y + 0.6, ẏ = −0.04x + 0.4y − 1 in Z . The first integrals of the Hamiltonian systems (2.4) are 3y2 H1 ( x, y) = 37.5..x2 − 15xy + 22.5..x + + 11.25..y, 2 H2 ( x, y) = − 0.4x2 − 4xy + 6x − 10y2 + 3y, H3 ( x, y) = − 0.02..x2 + 0.4..xy − x − 2y2 − 0.6..y, respectively. The discontinuous piecewise linear Hamiltonian system (2.4) has exactly three crossing limit cycles, because the system of equations (2.2) has the three real solutions Si = ( xi , yi , zi , wi , f i , hi ) for i = 1, 2, 3, where S1 = (0.559983.., 0.828504.., 0.619895.., −0.784685.., 0.878709.., −0.978709..), S2 = (0.755607.., 0.655025.., 0.754335.., −0.65649.., 0.7, −0.8), S3 = (0.903742.., 0.881627.., −0.471947.., 0.428077.., 0.462348.., −0.562348..). These solutions provide three crossing limit cycles of the piecewise linear differential Hamil- tonian system (2.2), which are illustrate in Figure 1.3b. This completes the proof of state- ment (i). 12 R. Benterki, J. Jimenez and J. Llibre To complete the proof of statements (ii)–(iv) of Theorem 1.2 we shall provide discontinuous piecewise linear Hamiltonian systems without equilibrium points separated by the cubic curve Γk with three limit cycles for the class C3 of Conf 1; Conf 2; Conf 3. Example 3: Three crossing limit cycles of Conf 1 for the class C 3 . For this class we consider the four zones defined in (1.3). We consider the Hamiltonian systems ẋ = − 6.8..x + 4y − 2, ẏ = −11.56x + 6.8y − 2 in Z1 , ẋ = 1.06216..x + 2y − 1.28925.., ẏ = −0.564089..x − 1.06216..y + 3.92358.. in Z2 , (2.5) ẋ = − 4x + 2y − 2.8.., ẏ = −8x + 4y − 1 in Z3 , ẋ = 121.33..x + 3y + 508.239.., ẏ = −4907.01..x − 121.33..y + 611.017.. in Z4 . The linear Hamiltonian systems in (2.5) have the first integrals H1 ( x, y) = − 5.78..x2 + 6.8..xy − 2x − 2y2 + 2y, H2 ( x, y) = − 0.282045..x2 − 1.06216..xy + 3.92358..x − y2 + 1.28925..y, H3 ( x, y) = − 4x2 + 4xy − x − y2 + 2.8..y, H4 ( x, y) = − 2453.5..x2 − 121.33..xy + 611.017..x − 1.5y2 − 508.239..y, respectively. The discontinuous piecewise linear Hamiltonian system (2.5) has exactly three crossing limit cycles intersecting Γk in the points Ai = ( xi , yi ), Bi = (zi , wi ), Ci = (k, f i ) and D4 = (k, hi ) for i = 1, 2, 3, where Ai and Bi are points on the circle x2 + y2 − 1 = 0, because the system of equations H1 ( xi , yi ) − H1 (k, f i ) = 0, H2 (zi , wi ) − H2 (k, f i ) = 0, H3 (zi , wi ) − H3 (k, hi ) = 0, (2.6) H4 ( xi , yi ) − H4 (k, hi ) = 0, xi2 + y2i − 1 = 0, z2i + wi2 − 1 = 0, with k = 0, has only three real solutions Si = ( xi , y, i, zi , wi , f i , hi ) for i = 1, 2, 3, where S1 = (0.859402.., 0.5113.., −0.573716.., 0.819054.., 3.12047.., −0.724745..), S2 = (0.795991.., 0.605309.., −0.403541.., 0.914962.., 2.8.., −0.5..), S3 = (0.708174.., 0.706038.., −0.208691.., 0.977982.., 2.3798.., −0.207107..). These three limit cycles are drawn in Figure 1.4a. This completes the proof of statement (ii). Example 4: Three crossing limit cycles of Conf 2 for the class C 3 . In (1.3), we work only with the three zones Z1 , Z2 and Z4 , with k = 0, and we consider the Hamiltonian systems ẋ = 19 − 18x − 3y, ẏ = −68 + 108x + 18y in Z1 , ẋ = −3.88389x − 2y + 5.99641.., ẏ = 7.54231..x + 3.88389..y − 7.99048.. in Z2 , (2.7) 4 ẋ = 6 + 2x − 2y, ẏ = −2 + 2x − 2y in Z . Limit cycles of DPWLHS without equilibria separated by reducible cubics 13 The first integrals of the Hamiltonian systems (2.7) are 3y2 H1 ( x, y) = 54x2 + 18xy − 68x + − 19y, 2 H2 ( x, y) = 3.77115..x2 + 3.88389..xy − 7.99048..x + y2 − 5.99641..y, H4 ( x, y) = x2 − 2xy − 2x + y2 − 6y, respectively The discontinuous piecewise linear differential system formed by the linear Hamiltonian systems (2.7) has exactly three crossing limit cycles, because the system of equations H1 ( xi , yi ) − H1 (k, f i ) = 0, H1 (zi , wi ) − H1 (k, hi ) = 0, H2 (k, hi ) − H2 (k, f i ) = 0, (2.8) H4 ( xi , yi ) − H4 (zi , wi ) = 0, xi2 + y2i − 1 = 0, z2i + wi2 − 1 = 0, for k = 0 has the three real solutions Si = ( xi , yi , zi , wi , f i , hi ) for i = 1, 2, 3, where S1 = (0.597407.., 0.801938.., 0.29046.., 0.956887.., 4.80282.., 1.19718..), S2 = (0.736107.., 0.676866.., 0.161682.., 0.986843.., 4.86511.., 1.13489..), S3 = (0.831057.., 0.556188.., 0.0773343.., 0.997005.., 4.92764.., 1.07236..). These three limit cycles are drawn in Figure 1.4b. This completes the proof of statement (iii). Example 5: Three crossing limit cycles of Conf 3 for the class C 3 . Here we consider the three zones Z1 , Z3 and Z4 defined in (1.3) with k = 0. 43x 43x 43y ẋ = − + 43y + 6, ẏ = − + − 2, in Z1 , 2 4 2 ẋ = − 5.01788..x + 10y + 1.37209.., ẏ = −2.51792..x + 5.01788..y (2.9) − 0.356396.., in Z4 , ẋ = − 5.2..x + 13y + 1.78427.., ẏ = −2.08..x + 5.2..y + 7, in Z3 . The first integrals of the Hamiltonian systems (2.9) are 43x2 43xy 43y2 H1 ( x, y) = − + − 2x − − 6y, 8 2 2 H2 ( x, y) = − 1.25896..x2 + 5.01788..xy − 0.356396..x − 5y2 − 1.37209..y, 13y2 H3 ( x, y) = − 1.04..x2 + 5.2xy + 7x − − 1.78427..y, 2 respectively. The discontinuous piecewise linear differential system formed by the linear Hamiltonian systems (2.9) has exactly three crossing limit cycles, because the system of equations (2.2) has the solutions Si = ( xi , yi , zi , wi , f i , hi ) for i = 1, 2, 3, where S1 = (0.92178.., −0.387712.., 0.478499.., 0.878088.., −0.974503.., 0.7), S2 = (0.988715.., −0.149808.., 0.647429.., 0.762126.., −0.819428.., 0.544924..), S3 = (0.980618.., 0.195928.., 0.855019.., 0.518597.., −0.616986.., 0.342483..). These three limit cycles are drawn in Figure 1.4c. This completes the proof of statement (iv). 14 R. Benterki, J. Jimenez and J. Llibre Proof of statement (v) of Theorem 1.2. In order to have limit cycles with Conf 1 and Conf 2 si- multaneously, the intersection points of the limit cycles of Conf 1 with Γk must satisfy system (2.6) with k = 0, and the points of intersection of the limit cycles with Conf 2 with Γk must satisfy system (2.8). In statement (ii ) and (iii ) of Theorem 1.2 we proved that the maximum number of limit cycles with Conf 1 and Conf 2 is three , then we know that the upper bound of maximum number of limit cycles with both configurations is six. Example 6: Six crossing limit cycles for the class C 3 , with three limit cycles of Conf 1 and three limit cycles of Conf 2. Here we consider the four zones defined in (1.3). ẋ = − 3.37125..x − y + 3.95604.., ẏ = 11.3653..x + 3.37125..y − 11.7972.. in Z1 , 1 ẋ = − 0.121473..x − y + 2.02017.., ẏ = 0.0295115..x + 0.121473..y − 0.684232.. in Z2 , 2 (2.10) 3 ẋ = 0.328515..x + y + 3, ẏ = −0.107922..x − 0.328515..y − 1.29868.. in Z , ẋ = − 9.2x − 2.3y + 17, ẏ = 36.8x + 9.2y − 56 in Z4 . The first integrals of the Hamiltonian systems (2.10) are y2 H1 ( x, y) = 5.68265..x2 + 3.37125..xy − 11.7972..x + − 3.95604..y, 2 1 H2 ( x, y) = 0.0147557..x2 + 0.121473..xy − 0.684232..x + y2 − 2.02017..y, 4 y 2 H3 ( x, y) = − 0.0539609x2 − 0.328515xy − 1.29868x − − 3y, 2 H4 ( x, y) = 18.4x2 + 9.2xy − 56x + 1.15y2 − 17y, respectively. For the discontinuous piecewise differential system (2.11), system (2.6) with k = 0, has the three real solutions S1 = (0.224513.., −0.974471.., −0.98, 0.198997, 8.21167.., −0.231664..), S2 = (0.359928.., −0.93298.., −0.812094.., 0.583526.., 7.77944.., 0.239163..), S2 = (0.503738.., −0.863856.., −0.41, 0.912086.., 7.31697.., 0.743252..). and system (2.8), has the three real solutions S1 = (0.65827.., −0.752782.., 0.093398.., 0.995629.., 6.82398.., 1.25669..), S2 = (0.825187.., −0.56486.., 0.309897.., 0.95077.., 6.31504.., 1.76563..), S2 = (0.986374.., −0.164516.., 0.630863.., 0.775894.., 5.87164.., 2.20904..). These six limit cycles are presented in Figure 1.5. This completes the proof of statement (v). Proof of statement (vi ) of Theorem 1.2. To get limit cycles with Conf 1 and Conf 3 simultane- ously, the points of intersection of the limit cycles with Conf 1 and Conf 3 with Γk must satisfy system (2.6) and (2.2), respectively, with k = 0. In statement (ii ) and (iv) of Theorem 1.2 we showed that the maximum number of limit cycles with Conf 1 and Conf 3 is three , then we know that the upper bound of maximum number of limit cycles with both configurations is six. Limit cycles of DPWLHS without equilibria separated by reducible cubics 15 Example 7: Six crossing limit cycles for the class C 3 , with three limit cycles of Conf 1 and three others of Conf 3. Here we consider the four zones defined in (1.3) with k = 0 with the following Hamiltonian systems ẋ = −8.8x + 22y − 3, ẏ = −3.52x + 8.8y − 4 in Z1 , ẋ = 30.9637..x + 30y + 0.9.., ẏ = −31.9584..x − 30.9637..y + 24.1071.. in Z2 , (2.11) ẋ = 0.713131..x + 0.9y − 0.162525.., ẏ = −0.565063..x − 0.713131..y + 0.620587.. in Z3 , ẋ = −8.37872..x + 22y − 3.97708.., ẏ = −3.19104..x + 8.37872..y − 3.05205.. in Z4 . The first integrals of the Hamiltonian systems (2.11) are H1 ( x, y) = −1.76x2 + 8.8xy − 4x − 11y2 + 3y, H2 ( x, y) = −13.0028..x2 − 27.9315..xy + 24.0252..x − 15y2 − 0.9y, H3 ( x, y) = −0.282531..x2 − 0.713131..xy + 0.620587..x − 0.45..y2 + 0.162525..y, H4 ( x, y) = −1.59552..x2 + 8.37872..xy − 3.05205..x − 11y2 + 3.97708..y, respectively. For the discontinuous piecewise differential system (2.11), system (2.6) with k = 0, has the three real solutions S1 = (0.859956.., −0.510369.., 0.89, 0.45596.., 1.232.., −0.895261..), S2 = (0.925727.., −0.378193.., −0.818732.., 0.574176.., 1.14562.., −0.79916..), S3 = (0.969836.., −0.243758.., −0.7, 0.714143.., 1.05112.., −0.694334..), and system (2.2), has the three real solutions S1 = (0.995048.., −0.0993944.., 0.167496.., 0.985873.., 0.937707.., −0.576541..), S2 = (0.997733.., 0.0672986.., 0.41691.., 0.908948.., 0.799221.., −0.438055..), S3 = (0.954489.., 0.298247.., 0.659704.., 0.751525..0.621163.., −0.259997..). These six limit cycles are drawn in 1.6. This completes the proof of statement (vi). 3 Proof of Theorem 1.3 We will prove the statement (i). For the other statements the proof is completely analogous. Proof of statement (i) of Theorem 1.3. From Lemma 1.1 we can consider an arbitrary piecewise linear differential Hamiltonian system in CΣk− formed by the following three linear Hamilto- nian systems without equilibrium points ẋ = −λi bi x + bi y + µi , ẏ = −λ2i bi x + λi bi y + σi in ZΣi k− , (3.1) for i = 1, 2, 3, where σi 6= λi µi and bi 6= 0. The Hamiltonian functions associated to these systems are 1 bi Hi ( x, y) = − λ2i bi x2 + λi bi xy − y2 + σi x − µi y, in ZΣi − for i = 1, 2, 3. 2 2 k 16 R. Benterki, J. Jimenez and J. Llibre 8 6 4 2 0 -2 -4 -2 0 2 4 Figure 3.1: Three limit cycles of system (3.3) intersecting Σ−1 . In order to have a limit cycle which intersects Σk− in four different points ( x1 , x12 ), ( x2 , x22 ), ( x3 , k) and ( x4 , k) with k < 0, these points must satisfy the system H1 ( x1 , x12 ) − H1 ( x2 , x22 ) = 0, H2 ( x2 , x22 ) − H2 ( x3 , k ) = 0, (3.2) H3 ( x3 , k ) − H3 ( x4 , k ) = 0, H2 ( x4 , k ) − H2 ( x1 , x12 ) = 0, k < 0. Assume that the discontinuous piecewise linear differential system (3.1) has four limit cycles. (i ) (i ) (i ) (i ) For this we must suppose that system (3.2) has four real solutions, namely ( x1 , x2 , x3 , x4 ), (1) (1) (1) (1) with i = 1, 2, 3, 4. Firstly we consider that ( x1 , x2 , x3 , x4 ) satisfies system (3.2). From the (1) (1) first equation, and by assuming that x1 + x2 6= 0, we obtain the expression (1) (1) (1) (1) (1) (1) (1) (1) µ1 = (2σ1 − b1 ( x1 + x2 − λ1 )(( x1 )2 + ( x2 )2 − ( x1 + x2 )λ1 ))/(2( x1 + x2 )). By the second equation we get µ2 , (1) (1) (1) (1) (1) µ2 = (−b2 ( x2 )2 + b2 ( x2 )4 − 2b2 ( x2 )3 λ2 + 2b2 kx3 λ2 + b2 ( x2 )2 λ22 (1) (1) (1) (1) − b2 ( x3 )2 λ22 − 2x2 σ2 + 2x3 σ2 )/2(k − ( x2 )2 ). (1) We observed that k − ( x2 )2 < 0, since k < 0. Solving the third equation we have the parameter σ3 , (1) (1) σ3 = b3 λ3 (−2k + ( x3 + x4 )λ3 )/2. By the fourth equation we obtain (1) (1) (1) (1) (1) (1) (1) σ2 = (−b2 k3 + b2 k2 ( x1 )2 + b2 k ( x2 )4 − b2 ( x1 )2 ( x2 )4 − 2b2 k ( x2 )3 λ2 + 2b2 ( x1 )2 ( x2 )3 λ2 (1) (1) (1) (1) (1) (1) (1) + 2b2 k2 x3 λ2 − 2b2 k( x1 )2 x3 λ2 + b2 k( x2 )2 λ22 − b2 ( x1 )2 ( x2 )2 λ22 − b2 k( x3 )2 λ22 (1) (1) (1) (1) (1) (1) (1) + b2 ( x1 )2 ( x3 )2 λ22 + (k − ( x2 )2 )b2 (k − ( x1 )2 + x1 λ2 − x4 λ2 )(k + ( x1 )2 (1) (1) (1) (1) (1) (1) (1) (1) − ( x1 + x4 )λ2 ))/2(( x2 − x3 )(k − ( x1 )2 ) + ( x4 − x1 )(k − ( x2 )2 )), Limit cycles of DPWLHS without equilibria separated by reducible cubics 17 (1) (1) (1) (1) (1) (1) considering ( x2 − x3 )(k − ( x1 )2 ) + ( x4 − x1 )(k − ( x2 )2 ) 6= 0. (2) (2) Now we suppose the second solution of system (3.2), we fixed the points ( x2 , x3 ), then by the second equation we obtain the parameter λ2 , then we have determined the values of the parameters µ2 , σ2 and λ2 of the Hamiltonian function H2 in the zone ZΣ2 − . By solving the (2) (1) (1) (2) third equation we get that x4 = x3 + x4 − x3 , solving the fourth equation we get the (2) point x1 which depends of parameters µ2 , σ2 , λ2 and b2 , moreover the parameter λ2 depends (2) (2) (2) of parameters µ2 , σ2 and b2 , therefore we write x1 depends of λ2 , this is x1 = x1 (λ2 ). With (2) (2) (2) (2) these points ( x1 , x2 , x3 , x4 ) and solving first equation we obtain the parameter σ1 (1) (1) (2) (2) (1) (1) (2) (2) (2) (1) σ1 = − b1 ( x1 + x2 )( x2 + x1 )(−( x1 )2 − ( x2 )2 + ( x2 )2 + ( x1 )2 ) + 2( x2 (( x1 )2 (1) (1) (1) (1) (1) (2) (1) (1) (1) (1) (1) + x1 x2 + ( x2 )2 − ( x1 + x2 ) x2 ) + (( x1 )2 + x1 x2 + ( x2 )2 − ( x1 (1) (2) (2) (1) (1) (2) (1) (1) (2) (2) + ( x2 )) x2 ) x1 − ( x1 + x2 )( x1 )2 )λ1 /(2(− x1 − x2 + x2 + x1 )),  (1) (1) (2) (2) considering (− x1 − x2 + x2 + x1 ) 6= 0. (3) Likewise, we consider the third solution, and we fixed the point x2 . Then by the second (3) equation we obtain the point x3 which depends of parameter λ2 , solving the third equation (3) (1) (1) (3) (3) we get that x4 = x3 + x4 − x3 and by fourth equation we obtain the point x1 which (3) (3) (3) (1) (1) (3) depends of parameter λ2 , finally with these points ( x1 , x2 , x3 , x3 + x4 − x3 ) and by the first equation we obtain λ1 = A/B with B 6= 0, where (1) (2) (2) (3) (3) (1) (1) (2) (2) (3) (3) (1) (2) A = ( x1 )3 ( x1 + x2 − x1 − x2 ) + ( x1 )2 x2 ( x1 + x2 − x1 − x2 ) + ( x2 )3 ( x1 (2) (3) (3) (2) (2) (3) (3) (2) (2) (3) (3) + x2 − x1 − x2 ) + ( x1 + x2 )( x1 + x2 )(( x1 )2 + ( x2 )2 − ( x1 )2 − ( x2 )2 ) (1) (2) (2) (2) (2) (2) (2) (3) (3) (3) + x2 (−( x1 )3 − ( x1 )2 x2 − x1 ( x2 )2 − ( x2 )3 + ( x1 )3 + ( x1 )2 x2 (3) (3) (3) (1) (2) (2) (2) (2) (2) (2) (3) + x1 ( x2 )2 + ( x2 )3 ) + x1 (−( x1 )3 − ( x1 )2 x2 − x1 ( x2 )2 − ( x2 )3 + ( x1 )3 (1) (2) (2) (3) (3) (3) (3) (3) (3) (3) + ( x2 )2 ( x1 + x2 − x1 − x2 ) + ( x1 )2 x2 + x1 ( x2 )2 + ( x2 )3 ) ,  (2) (3) (2) (2) (3) (2) (3) (2) (3) (2) (3) (1) (2) B = 2 ( x1 )2 x1 + x1 x2 x1 + ( x2 )2 x1 − x1 ( x1 )2 − x2 ( x1 )2 + ( x1 )2 ( x1 (2) (3) (3) (1) (2) (2) (3) (3) (2) (3) (2) (2) (3) + x2 − x1 − x2 ) + ( x2 )2 ( x1 + x2 − x1 − x2 ) + ( x1 )2 x2 + x1 x2 x2 (2) (3) (2) (3) (3) (2) (3) (3) (2) (3) (2) (3) (1) (2) + ( x2 )2 x2 − x1 x1 x2 − x2 x1 x2 − x1 ( x2 )2 − x2 ( x2 )2 + x2 (−( x1 )2 (2) (2) (2) (3) (3) (3) (3) (1) (2) (2) (2) (2) − x1 x2 − ( x2 )2 + ( x1 )2 + x1 x2 + ( x2 )2 ) + x1 (−( x1 )2 − x1 x2 − ( x2 )2 (3) (1) (2) (2) (3) (3) (3) (3) (3) + ( x1 )2 + x2 ( x1 + x2 − x1 − x2 ) + x1 x2 + ( x2 )2 ) .  We observed that we have determined the values of the parameters µ1 , σ1 and λ1 of the Hamiltonian function H1 in the zone ZΣ1 − . (4) By a similar way, we consider the fourth solution, and we fixed the point x2 , then by the (4) (4) (1) second equation we obtain the point x3 , solving the third equation we get that x4 = x3 + (1) (4) (4) (4) (4) (4) (1) x4 − x3 , by the fourth equation we get the point x1 . With these points ( x1 , x2 , x3 , x3 + (1) (4) x4 − x3 ) from the first equation we have that b1 = 0 which is a contradiction, because from Lemma 1.1 bi 6= 0 for i = 1, 2, 3. Therefore the maximum number of limit cycles in this case is three. Now we prove that this upper bound is attached. We have that the unique restriction of value k is that the denominator in the expressions of σ2 is different from zero. We ob- (1) (1) (1) (1) served that it is possible to choose values to the points x1 , x2 , x3 and x4 such that 18 R. Benterki, J. Jimenez and J. Llibre (1) (2) (1) (1) (2) (1)  (1) (1) (1) (1)  k 6= ( x1 )2 ( x2 − x3 ) + ( x1 )2 ( x2 − x3 ) / x2 − x3 + x4 − x1 , for instance if we (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) consider x2 < 0, x1 > − x2 , x1 < x4 < x1 − x2 and x3 < ( x1 )2 x2 − x1 ( x2 )2 + (1) (1)  (1) (1) (2) (1) (1) (2) (1)  ( x2 )2 x4 /( x1 )2 we have that the expression ( x1 )2 ( x2 − x3 ) + ( x1 )2 ( x2 − x3 ) / (1) (1) (1) (1)  x2 − x3 + x4 − x1 is always positive therefore it is different of value of k, since that k < 0. Then we can consider without loss of generality that k = −1. We consider the discon- tinuous piecewise linear differential system defined by the following three linear Hamiltonian systems 3 ẋ = −24.293899.. − 0.692634..x + y, ẏ = −19.232427.. − 0.319828..x + 0.692634..y, 2 ẋ = −378.204351.. + 62.383901..x − 4y, ẏ = 916.621187.. + 972.937795..x − 62.383901..y, (3.3) 9 7 35 7 7 7 ẋ = − x − y, ẏ = + x + y, 10 2 4 2 5 2 in the zones ZΣ1 −1 , ZΣ2 −1 and ZΣ3 −1 , respectively. Then for system (3.3), we have that system (3.2) (i ) (i ) (i ) (i ) has three real solutions ( x1 , x2 , x3 , x4 ), i = 1, 2, 3, namely 3, −2, − 72 , 72 , 2.625658.., − 17 31 31 14    10 .., − 10 , 10 , 5 , −1.843412.., −3.287307.., 3.287307.. . These three real solutions provide the three limit cycles intersecting Σ−1 shown in Figure 3.1. This completes the proof of statement (i). 4 20 3 5 15 2 4 10 3 1 2 0 5 1 -1 0 0 -1 -2 -1 0 1 2 -4 -2 0 2 4 6 8 10 0.0 0.5 1.0 1.5 2.0 2.5 3.0 (a) (b) Figure 3.2: (a) Three limit cycles with Conf 4 of system (3.4). (b) Three limit cycles with Conf 5 of system (3.6). Proof of statement (ii) of Theorem 1.3. The proof in this statement is similar to the proof of state- ment (i). For each configuration of limit cycles that intersect Σk with k ≥ 0 we have that the upper bound of limit cycles is three. In what follows we show examples of piecewise linear differential system in CΣ0 with three limit cycles with Conf 4 and Conf 5, respectively. And piecewise linear differential system in CΣk+ with three limit cycles with Conf 6+ , Conf 7, Conf 8 and Conf 9+ , respectively. Crossing limit cycles with Conf 4: In order to have a limit cycle with Conf 4 which intersects Σ0 in four different points ( x1 , x12 ), ( x2 , x22 ), ( x3 , 0) and ( x4 , 0), these points must satisfy system Limit cycles of DPWLHS without equilibria separated by reducible cubics 19 (3.2) with k = 0. We consider the discontinuous piecewise linear differential system defined by the following four linear Hamiltonian systems 17 207 69 53 621 207 ẋ = − x − y, ẏ = − + x+ y, 2 50 10 10 250 50 31 ẋ = 48.069511.. + 11.825263..x − y, ẏ = −7.155434.. + 22.554330..x − 11.825263..y, 5 (3.4) 9 156 39 13 156 ẋ = + x − y, ẏ = − + 9.984000..x − y, 2 25 10 10 25 27 ẋ = 17.727172.. − 7.176019..x − y, ẏ = −1.428092.. + 9.536159..x + 7.176019..y, 5 in the zones ZΣ1 0 , ZΣ2 0 , ZΣ3 0 and ZΣ4 0 , respectively. For the discontinuous piecewise differential (i ) (i ) (i ) (i ) system (3.4), system (3.2) has three real solutions ( x1 , x2 , x3 , x4 ), i = 1, 2, 3 given by  1.517382.., −2.102549.., −1.142394.., 1.402811.. ,  1.474836.., −2.058730.., −0.973819.., 1.234236.. ,  1.427170.., −2.00939.., −0.774355.., 1.034772.. . These solutions provide the three limit cycles with Conf 4 shown in Figure 3.2a. Crossing limit cycles with Conf 5: In order to have a limit cycle with Conf 5 which intersects Σ0 in the four different points ( x1 , x12 ), ( x2 , x22 ), ( x3 , 0) and ( x4 , 0), they must satisfy H1 ( x1 , x12 ) − H1 ( x2 , x22 ) = 0, H4 ( x2 , x22 ) − H4 ( x3 , k ) = 0, (3.5) H3 ( x3 , k ) − H3 ( x4 , k ) = 0, H4 ( x4 , k) − H4 ( x1 , x12 ) = 0, with k = 0. We consider the discontinuous piecewise linear differential system defined by the following three linear Hamiltonian systems 3 ẋ = −4.711119.. + 3.915394..x − y, ẏ = −11.965988.. + 10.220210..x − 3.915394..y, 2 9 27 3 243 27 ẋ = + x − y, ẏ = −5.022000.. + x − y, (3.6) 10 10 2 50 10 11 ẋ = −3.005265.. + 2.848936..x − y, ẏ = −7.616106.. + 7.378583..x − 2.848936..y, 10 in the zones ZΣ1 0 , ZΣ3 0 and ZΣ4 0 , respectively. For the discontinuous piecewise differential system (i ) (i ) (i ) (i ) (3.6), system (3.5) has three real solutions ( x1 , x2 , x3 , x4 ), i = 1, 2, 3 given by 2, 12 , 25 , 35 , 93 47 479    50 , 0.628914.., , 100 300 , 1.696225.., 0.780317.., 0.534387.., 1.532279.. . These solutions provide the three limit cycles with Conf 5 shown in Figure 3.2b. Crossing limit cycles with Conf 6+ : In order to have a limit cycle with Conf 6+ which intersects Σ+ in four different points ( x1 , x12 ), ( x2 , k ), ( x3 , x32 ) and ( x4 , k ), these points must satisfy H1 ( x1 , x12 ) − H1 ( x2 , k ) = 0, H5 ( x2 , k ) − H5 ( x3 , x32 ) = 0, (3.7) H3 ( x3 , x32 ) − H3 ( x4 , k ) = 0, H4 ( x4 , k ) − H4 ( x1 , x12 ) = 0, for k > 0. 20 R. Benterki, J. Jimenez and J. Llibre 8 15 6 4 10 2 5 0 -2 0 -4 -2 0 2 4 6 -2 -1 0 1 2 (a) (b) 6 20 4 18 15 16 10 14 2 12 5 10 0 8 0 6 -6 -4 -2 0 2 4 6 -4 -2 0 2 4 6 3.0 3.5 4.0 4.5 (c) (d) Figure 3.3: (a) Three limit cycles with Conf 6+ of system (3.9). (b) Three limit cycles with Conf 7 of system (3.11). (c) Three limit cycles with Conf 8 of system (3.13). (d) Three limit cycles with Conf 9+ of system (3.14). To have a limit cycle with Conf 6− which intersects Σ+ in four different points, these points must satisfy the system H2 ( x1 , x12 ) − H2 ( x2 , k ) = 0, H3 ( x2 , k ) − H3 ( x3 , x32 ) = 0, (3.8) H5 ( x3 , x32 ) − H5 ( x4 , k ) = 0, H1 ( x4 , k ) − H1 ( x1 , x12 ) = 0, for k > 0. We provide an example of a piecewise linear differential system with three limit cycles with Conf 6+ . We observed that the upper bound found does not depend of the value of the parameter k > 0, then we can consider without loss of generality that k = 4. We consider the discontinuous piecewise linear differential system defined by the following four linear Limit cycles of DPWLHS without equilibria separated by reducible cubics 21 Hamiltonian systems 3 ẋ = −4.133710.. + 6.015251..x − y, ẏ = −5.920817.. + 24.122170..x − 6.015251..y, 2 17 ẋ = 5.325742.. + 2.017936..x − y, ẏ = 4.036253.. + 1.197666..x − 2.017936..y, 5 (3.9) ẋ = −8.981178.. + 3.946297..x − y, ẏ = −15.942643.. + 15.573265..x − 3.946297..y, 3 ẋ = −2.454956.. + 4.664679..x + y, ẏ = 6.613677.. − 14.506158..x − 4.664679..y, 2 in the zones ZΣ1 4 , ZΣ3 4 , ZΣ4 4 and ZΣ5 4 , respectively. For the discontinuous piecewise differential (i ) (i ) (i ) (i ) system (3.9), system (3.7) has three real solutions ( x1 , x2 , x3 , x4 ), i = 1, 2, 3 given by 4, − 25 , 21 , 5 , 193 31 13 483 7 3 83 441    50 , − 100 , 20 , 100 , 2 , − 25 , 100 , 100 . These solutions provide the three limit cycles with Conf 6+ shown in Figure 3.3a. Crossing limit cycles with Conf 7: In order to have a limit cycle with Conf 7 which intersects Σ+ in the four different points ( x1 , k ), ( x2 , k ), ( x3 , x32 ) and ( x4 , x42 ), they must satisfy the system H1 ( x1 , k ) − H1 ( x2 , k ) = 0, H5 ( x2 , k ) − H5 ( x3 , x32 ) = 0, (3.10) H3 ( x3 , x32 ) − H3 ( x4 , x42 ) = 0, H5 ( x4 , x42 ) − H5 ( x1 , k ) = 0, with k > 0. We can suppose without loss of generality that k = 4. We consider the discontinuous piece- wise linear differential system defined by the following three linear Hamiltonian systems 3 ẋ = −2 − 6x − y, ẏ = −5.491482.. + 24x + 6y, 2 47 ẋ = 22.645454.. − 36.659999..x − y, ẏ = −35.463636.. + 142.973999..x + 36.659999..y, 5 11 ẋ = 5.300000.. − 8.579999..x − y, ẏ = −8.300000.. + 33.461999..x + 8.579999..y, 5 (3.11) in the zones ZΣ1 3 , ZΣ3 3 and ZΣ5 3 , respectively. For the discontinuous piecewise differential system (i ) (i ) (i ) (i ) (3.11), system (3.10) has three real solutions ( x1 , x2 , x3 , x4 ), i = 1, 2, 3 given by  0.502842.., −1.545218.., −0.572025.., 0.848539.. ,  0.442709.., −1.485086.., −0.427227.., 0.781483.. ,  0.378567.., −1.420944.., −0.276975.., 0.700080.. . These solutions provide the three limit cycles with Conf 7 shown in 3.3b. Crossing limit cycles with Conf 8: In order to have a limit cycle with Conf 8 which intersects Σ+ in four different points ( x1 , x12 ), ( x2 , x22 ), ( x3 , k ) and ( x4 , k ), they must satisfy H1 ( x1 , x12 ) − H1 ( x2 , x22 ) = 0, H2 ( x2 , x22 ) − H2 ( x3 , k ) = 0, (3.12) H3 ( x3 , k ) − H3 ( x4 , k ) = 0, H4 ( x4 , k ) − H4 ( x1 , x12 ) = 0, with k > 0. 22 R. Benterki, J. Jimenez and J. Llibre We can consider without loss of generality that k = 2. We consider the discontinuous piece- wise linear differential system defined by the following four linear Hamiltonian systems 9 19 19 17 19 ẋ = + x − y, ẏ = + 0.076000..x − y, 2 50 10 10 50 11 ẋ = 10.930108.. + 7.204668..x − y, ẏ = 99.090506.. + 23.594202.x − 7.204668..y, 5 69 89 93 4361 ẋ = − − 6.229999..x − y, ẏ = − + x + 6.229999..y, 2 10 10 1000 17 ẋ = 32.954952.. − 16.575663..x − y, ẏ = −277.274017.. + 80.809593..x + 16.575663..y, 5 (3.13) in the pieces ZΣ1 2 , ZΣ2 2 , ZΣ3 2 and ZΣ4 2 , respectively. For the discontinuous piecewise differential (i ) (i ) (i ) (i ) system (3.13), system (3.12) has three real solutions ( x1 , x2 , x3 , x4 ), i = 1, 2, 3 given by  2.514526.., −2.427396.., −6.114467.., 4.665258.. ,  2.449236.., −2.371782.., −6.028697.., 4.579488.. ,  2.374832.., −2.310077.., −5.941517.., 4.492308.. . These solutions provide the three limit cycles with Conf 8 shown in Figure 3.3c. Crossing limit cycles with Conf 9+ : In order to have a limit cycle with Conf 9+ which intersects Σ+ in the four different points ( x1 , x12 ), ( x2 , x22 ), ( x3 , k ) and ( x4 , k ), they must satisfy system (3.5) with k > 0. Without loss of generality we can suppose that k = 4. We consider the discontinuous piecewise linear differential system defined by the following three linear Hamiltonian systems ẋ = −170.859539.. + 99.779168..x − 15y, ẏ = −1139.726782.. + 663.725497..x − 99.779168..y, 9 148 148 ẋ = + x − 4y, ẏ = −779.664000.. + 219.040000..x − y, 10 5 5 23 ẋ = 116.632274.. − 30.946111..x − y, ẏ = −1635.644521.. + 416.374692..x + 30.946111..y, 10 (3.14) in the zones ZΣ1 4 , ZΣ3 4 , and ZΣ4 4 , respectively. For the discontinuous piecewise differential (i ) (i ) (i ) (i ) system (3.14), system (3.5) with k = 4, has three real solutions ( x1 , x2 , x3 , x4 ), i = 1, 2, 3 given by 4, 3, 16 141 303 517   47  5 , 5 , 4.109491.., 50 , 100 , 100 , 10 , 2.053733.., 2.068270.., 6.131729.. . These solutions provide the three limit cycles with Conf 9+ shown in Figure 3.3d. This completes the proof of statement (ii). Proof of statement (iii) of Theorem 1.3. In order to have limit cycles with Conf 4 and Conf 5 simultaneously, the points of intersection of the limit cycles with Conf 4 with Σ0 must satisfy system (3.2) with k = 0, and the points of intersection of the limit cycles with Conf 5 with Σ0 must satisfy system (3.5). In statement (ii) we proved that the maximum number of limit cycles with Conf 4 and Conf 5 is three, then we have that the upper bound of maximum number of limit cycles with both configurations is six. We provide an example of a piecewise linear Limit cycles of DPWLHS without equilibria separated by reducible cubics 23 25 20 4 80 15 3 60 10 2 40 1 5 20 0 0 0 -1 0 10 20 30 -5 0 5 10 0.0 0.5 1.0 1.5 2.0 2.5 Figure 3.4: Three limit cycles with Conf 4, and three limit cycles with Conf 5 of system (3.15) simultaneously. differential system in CΣ0 such that have six limit cycles with three limit cycles with Conf 4 and Conf 5, respectively. This is the upper bound is reached. We consider the discontinuous piecewise linear differential system defined by the following four linear Hamiltonian systems ẋ = −154.076990.. − 28.017658..x + 15y, ẏ = −387.181918.. − 52.332611..x + 28.017658..y, 3 ẋ = −0.400024.. + 0.532848..x − y, ẏ = 0.058205.. + 0.946425..x − 0.532848..y, 10 9 28 28 ẋ = + x − 4y, ẏ = −8.101333.. + 7.839999..x − y, 10 5 5 11 ẋ = −3.005265.. + 2.848936..x − y, ẏ = −7.616106.. + 7.378583..x − 2.848936..y, 10 (3.15) in the zones ZΣ1 0 , ZΣ2 0 , ZΣ3 0 and ZΣ4 0 , respectively. For the discontinuous piecewise differential system (3.15), system (3.2) with k = 0, has the following three real solutions 27 1 69   10 , − 10 , −0.166825.., 2.233491.. , 25 , −0.147032.., −0.232725.., 2.299392.. , 14  5 , − 0.177898.., − 0.278200.., 2.344866.. , and system (3.5) with k = 0, has the three real solutions 2, 21 , 25 , 35 , 93 47 479    50 , 0.628914.., 100 , 300 , 1.393438.., 1.075216.., 0.604474.., 1.462192.. . These solutions provide the three limit cycles with Conf 4 and Conf 5 shown in Figure 3.4. This completes the proof of statement (iii). Proof of statement (iv) of Theorem 1.3. In order to have limit cycles with Conf 6− and Conf 6+ simultaneously, the points of intersection of the limit cycles with Conf 6+ and Σk+ must satisfy system (3.7), and the points of intersection of the limit cycles with Conf 6− and Σk+ must satisfy system (3.8). In statement (ii) we proved that the maximum number of limit cycles with each configuration is three, then we have that the upper bound of maximum number of limit cycles with both configurations is six. We provide an example of a piecewise linear 24 R. Benterki, J. Jimenez and J. Llibre 20 10 26 14 24 12 0 22 10 20 - 10 8 18 - 20 16 6 14 -5 -4 -3 -2 -1 0 10 20 30 3.0 3.5 4.0 4.5 5.0 5.5 Figure 3.5: Three limit cycles with Conf 6− and Conf 6+ of system (3.16). differential system in CΣk+ such that have six limit cycles with three limit cycles with each configuration. This is the upper bound is reached. Without loss of generality we can suppose that k = 4. We consider the discontinuous piecewise linear differential system defined by the following five linear Hamiltonian systems 9 ẋ = − 23138.489410.. + 403.676452..x + y, ẏ = 2942.120325.. − 36212.150741..x 2 − 403.676452..y, 3 ẋ = 4.276633.. + 1.873985..x − y, ẏ = 4.991226.. + 11.706072..x − 1.873985..y, 10 17 (3.16) ẋ = 15.472057.. − 3.117904..x − y, ẏ = −13.354567.. + 2.859213..x + 3.117904..y, 5 ẋ = 48.158492.. − 6.082779..x − y, ẏ = −31.590984.. + 37.000210..x + 6.082779..y, 3 ẋ = − 151.854124.. − 136.354901..x + y, ẏ = −10611.949690.. − 12395.106180..x 2 + 136.354901..y, in the pieces ZΣ1 4 , ZΣ2 4 , ZΣ3 4 , ZΣ4 4 and ZΣ5 4 , respectively. For the discontinuous piecewise differ- ential system (3.16), (3.7), has the three real solutions 9 23 5, 21 , 20 9 19 91 7 41    , 5 , 2 , 20 , 100 , 2 , 10 , 1.196150.., 1.163297.., 2.719447.. , and system (3.8), has the following three real solutions − 18 9 49   5 , − 2 , − 50 , −1 , − 3, −3.411586.., −1.557354.., −1.546135.. , 31  − 2.809209.., − 10 , −1.671884.., −1.662653.. . These solutions provide the three limit cycles with Conf 6− and Conf 6+ shown in Figure 3.5. This completes the proof of statement (iv). Proof of statement (v) of Theorem 1.3. In order to have limit cycles with Conf 6− and Conf 7 simultaneously, the points of intersection of the limit cycles with Conf 6− and Σk+ must satisfy Limit cycles of DPWLHS without equilibria separated by reducible cubics 25 system (3.8), and the points of intersection of the limit cycles with Conf 7 and Σk+ must satisfy system (3.10). In statement (ii) we proved that the maximum number of limit cycles with each configuration is three, then we have that the maximum number of limit cycles with both configurations is six. Moreover this upper bound is reached. Without loss of generality we can suppose that k = 3. We consider the discontinuous piecewise linear differential system defined by the following four linear Hamiltonian systems 3 ẋ = −0.567977.. − 5.151614..x − y, ẏ = −6.233588.. + 17.692757..x + 5.151614..y, 2 2 ẋ = 11.250254.. + 0.637407..x − y, ẏ = −35.085985.. + 1.015720..x − 0.637407..y, 5 47 ẋ = 22.645454.. − 36.659999..x − y, ẏ = −35.463636.. + 142.973999..x + 36.659999..y, 5 11 ẋ = 5.300000.. − 8.579999..x − y, ẏ = −8.300000.. + 33.461999..x + 8.579999..y, 5 (3.17) in the pieces ZΣ1 3 , ZΣ2 3 , ZΣ3 3 and ZΣ5 3 , respectively. For the discontinuous piecewise differential 20 15 10 10 0 5 0 - 10 -5 - 20 -4 -2 0 2 4 6 -4 -2 0 2 4 6 (a) (b) Figure 3.6: (a) Three limit cycles with Conf 6− and Conf 7 of system (3.17). (b) Three limit cycles with Conf 6+ and Conf 8 of system (3.18). system (3.17), system (3.8), has the following three real solutions − 41 5 − 106 263   10 , − 2 , 1.569412.., 1.457623.. , 25 , − 100 , 1.647799.., 1.587623.. , − 199  50 , −2.401954.., 1.508473.., 1.359577.. , and (3.10), has the three real solutions  0.502842.., −1.545218.., −0.572025.., 0.848539.. ,  0.442709.., −1.485086.., −0.427227.., 0.781483.. ,  0.378567.., −1.420944.., −0.276975.., 0.700080.. . These solutions provide the three limit cycles with Conf 6− and Conf 7 shown in Figure 3.6a. This completes the proof of statement (v). 26 R. Benterki, J. Jimenez and J. Llibre Proof of statement (vi) of Theorem 1.3. In order to have limit cycles with Conf 6+ and Conf 8 simultaneously, the points of intersection of the limit cycles with Conf 6+ and Σk+ must satisfy system (3.8), and the points of intersection of the limit cycles with Conf 8 and Σk+ must satisfy system (3.12). In statement (ii) we proved that the maximum number of limit cycles with each configuration is three, then we have that the maximum number of limit cycles with both configurations is six. Moreover this upper bound is reached. Without loss of generality we can consider that k = 4. We consider the discontinuous piecewise linear differential system defined by the following five linear Hamiltonian systems 3 ẋ = −0.325270.. − 1.247316..x − y, ẏ = −35.808990.. + 1.037199..x + 1.247316..y, 2 ẋ = −13.295856.. − 8.394370..x − 2y, ẏ = 47.825295.. + 35.232724..x + 8.394370..y, 17 ẋ = 33.366737.. − 7.961636..x − y, ẏ = −44.896945.. + 18.643428..x + 7.961636..y, 5 ẋ = 65.056521.. − 17.010074..x − y, ẏ = −1052.5380642.. + 289.342621..x + 17.010074..y, 3 ẋ = 74.167422.. + 12.003227..x + y, ẏ = −187.420662.. − 96.051650..x − 12.003227..y, 2 (3.18) in the pieces ZΣ1 4 , ZΣ2 4 , ZΣ3 4 , ZΣ4 4 and ZΣ5 4 , respectively. For the discontinuous piecewise differ- ential system (3.18), system (3.7), has the following three real solutions 7 6 2 19 71 143 31 389 343    2, −5, 5, 5 , 20 , − 100 , 100 , 100 , 100 , −0.893313.., 0.533583.., 3.654678.. , and (3.12), has the three real solutions 4, −3, − 16 23 589   5, 5 , 4.073407.., −3.179377.., −3.311999.., 125 , 241  4.144187.., −3.341881.., −3.420000.., 50 . These solutions provide the three limit cycles with Conf 6+ and Conf 8 shown in Figure 3.6b. This completes the proof of statement (vi). Proof of statement (vii) of Theorem 1.3. In order to have limit cycles with Conf 6+ and Conf 9+ simultaneously, the points of intersection of the limit cycles with Conf 6+ and Σk+ must satisfy system (3.7), and the points of intersection of the limit cycles with Conf 9+ and Σk+ must satisfy system (3.5) with k > 0. In statement (ii) we proved that the maximum number of limit cycles with each configuration is three, then we have that the maximum number of limit cycles with both configurations is six. Moreover this upper bound is reached. We can suppose without loss of generality that k = 4. We consider the discontinuous piecewise linear differential system defined by the following four linear Hamiltonian systems 3 ẋ = −17.085953.. + 9.977916..x − y, ẏ = −113.972678.. + 66.372549..x − 9.977916..y, 2 7 ẋ = 34.897550.. − 4.677048..x − y, ẏ = −44.332934.. + 6.249936..x + 4.677048..y, 2 13 ẋ = 65.922589.. − 17.491280..x − y, ẏ = −924.494729.. + 235.342217..x + 17.491280..y, 10 9 ẋ = 13.883036.. + 3.280745..x − y, ẏ = −44.382913.. + 2.391842..x − 3.280745..y, 2 (3.19) Limit cycles of DPWLHS without equilibria separated by reducible cubics 27 in the pieces ZΣ1 4 , ZΣ3 4 , ZΣ4 4 and ZΣ5 4 , respectively. For the discontinuous piecewise differential system (3.19), system (3.7), has the following three real solutions 6, 1.209968.., 57 , 8.457532.. , 156 5   25 , 1.006799.., 4 , 8.915579.. , 117  20 , 1.328327.., 1.486618.., 8.175706.. , and (3.5), has the three real solutions 4, 3, 16 4.109491.., 141 303 517 47    5 ,5 , 50 , 100 , 100 , 10 , 2.053733.., 2.068270.., 6.131729.. . These solutions provide the three limit cycles with Conf 6+ and Conf 9+ shown in Figure 3.7. This completes the proof of statement (vii). 40 20 20 0 15 - 20 10 5 - 40 0 10 20 30 1 2 3 4 5 6 Figure 3.7: Three limit cycles with Conf 6+ and Conf 9+ of system (3.19). Proof of statement (viii) of Theorem 1.3. In order to have limit cycles with Conf 7 and Conf 8 simultaneously, the points of intersection of the limit cycles with Conf 7 and Σk+ must satisfy system (3.10), and the points of intersection of the limit cycles with Conf 8 and Σk+ must satisfy system (3.12). In statement (ii) we proved that the maximum number of limit cycles with each configuration is three, then we have that the maximum number of limit cycles with both configurations is six. Moreover this upper bound is reached. Without loss of generality we can suppose that k = 3. We consider the discontinuous piecewise linear differential system defined by the following five linear Hamiltonian systems 3 ẋ = −453.807220.. − 20.414445..x − y, ẏ = 83.559977.. + 277.833055..x + 20.414445..y, 2 2 ẋ = −29.218386.. − 5.465711..x + y, ẏ = −414.702614.. − 74.685013..x + 5.465711..y, 5 47 ẋ = 22.645454.. − 36.659999..x − y, ẏ = −35.463636.. + 142.973999..x + 36.659999..y, 5 6 ẋ = 3.918325.. − 3.744301..x + y, ẏ = 33.556264.. − 11.683162..x + 3.744301..y, 5 11 ẋ = 5.300000.. − 8.579999..x − y, ẏ = −8.300000.. + 33.461999..x + 8.579999..y, 5 (3.20) 28 R. Benterki, J. Jimenez and J. Llibre in the zones ZΣ1 3 , ZΣ2 3 , ZΣ3 3 , ZΣ4 3 and ZΣ5 3 , respectively. For the discontinuous piecewise differ- ential system (3.20), system (3.10), has the following three real solutions  0.502842.., −1.545218.., −0.572025.., 0.848539.. ,  0.442709.., −1.485086.., −0.427227.., 0.781483.. ,  0.378567.., −1.420944.., −0.276975.., 0.700080.. , and (3.12), has the three real solutions 84 79 88 3297 387 69   25 , − 20 , −4.562376.., 25 , 1000 , − 100 , −4.492376.., 20 , 34301 179  10000 , −4.039424.., −4.622376.., 50 . These solutions provide the three limit cycles with Conf 7 and Conf 8 shown in Figure 3.8. This completes the proof of statement (viii). 20 0 20 - 20 14 15 12 - 40 10 10 5 8 - 60 6 0 4 - 80 -5 -5 0 5 10 15 20 -4 -2 0 2 4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 Figure 3.8: Three limit cycles with Conf 7 and Conf 8 of system (3.20). Proof of statement (ix) of Theorem 1.3. In order to have limit cycles with Conf 8 and Conf 9+ simultaneously, the points of intersection of the limit cycles with Conf 8 and Σk+ must satisfy system (3.12), and the points of intersection of the limit cycles with Conf 9+ and Σk+ must satisfy system (3.5) with k > 0. In statement (ii) we proved that the maximum number of limit cycles with each configuration is three, then we have that the maximum number of limit cycles with both configurations is six. Moreover this upper bound is reached. Without loss of generality we can suppose that k = 4. We consider the discontinuous piecewise linear differential system defined by the following four linear Hamiltonian systems 3 ẋ = −17.085953.. + 9.977916..x − y, ẏ = −113.972678.. + 66.372549..x − 9.977916..y, 2 3 ẋ = −23.136372.. + 2.354826..x − y, ẏ = 81.642102.. + 18.484031..x − 2.354826..y, 10 431 7 3087 ẋ = + 14.700000..x − y, ẏ = −194.334000.. + x − 14.700000..y, 10 2 50 13 ẋ = 65.922589.. − 17.491280..x − y, ẏ = −924.494729.. + 235.342217..x + 17.491280..y, 10 (3.21) Limit cycles of DPWLHS without equilibria separated by reducible cubics 29 in the zones ZΣ1 4 , ZΣ2 4 , ZΣ3 4 and ZΣ4 4 , respectively. For the discontinuous piecewise differential system (3.21), system (3.12), has the following three real solutions 457   9, −2.341532.., −6.604799.., 14.804799.. , 50 , −2.481840.., −6.933823.., 15.133823.. , 187  20 , −2.692262.., −7.432829.., 15.632829.. , and (3.5), has the three real solutions 4, 3, 16 4.109491.., 141 303 517 47    5 ,5 , 50 , 100 , 100 , 10 , 2.053733.., 2.068270.., 6.131729.. . These solutions provide the three limit cycles with Conf 8 and Conf 9+ shown in Figure 3.9. This completes the proof of statement (ix). 80 60 40 20 20 15 0 10 - 20 5 - 40 - 10 -5 0 5 10 15 1 2 3 4 5 6 7 Figure 3.9: Three limit cycles with Conf 8 and Conf 9+ of system (3.21). 20 10 0 - 10 - 20 - 30 - 40 -5 0 5 10 Figure 3.10: Three limit cycles with Conf 6− , Conf 7 and Conf 8 of system (3.22). Proof of statement (x) of Theorem 1.3. In order to have limit cycles with Conf 6− , Conf 7 and Conf 8 simultaneously, the points of intersection of the limit cycles with Conf 6− and Σk+ 30 R. Benterki, J. Jimenez and J. Llibre 50 0 80 -50 20 60 -100 15 40 10 -150 20 5 -200 0 0 -250 0 50 100 150 - 10 -5 0 5 10 15 20 2 3 4 5 6 Figure 3.11: Three limit cycles with Conf 6+ , Conf 8 and Conf 9+ of system (3.23). must satisfy system (3.8), the points of intersection of the limit cycles with Conf 7 and Σk+ must satisfy system (3.10) and the points of intersection of the limit cycles with Conf 8 and Σk+ must satisfy system (3.12). In statement (ii) we proved that the maximum number of limit cycles with each configuration is three, then we have that the maximum number of limit cycles with the three configurations simultaneously, is nine. Moreover this upper bound is reached. Without loss of generality we can suppose that k = 3. We consider the discontinuous piecewise linear differential system defined by the following five linear Hamiltonian systems 3 ẋ = −0.567977.. − 5.151614..x − y, ẏ = −6.233588.. + 17.692757..x + 5.151614..y, 2 2 ẋ = 11.250254.. + 0.637407..x − y, ẏ = −35.085985.. + 1.015720..x − 0.637407..y, 5 47 ẋ = 22.645454.. − 36.659999..x − y, ẏ = −35.463636.. + 142.973999..x + 36.659999..y, 5 6 ẋ = −13.170507.. + 3.348185..x + y, ẏ = 36.547853.. − 9.341954..x − 3.348185..y, 5 11 ẋ = 5.300000.. − 8.579999..x − y, ẏ = −8.300000.. + 33.461999..x + 8.579999..y, 5 (3.22) in the zones ZΣ1 3 , ZΣ2 3 , ZΣ3 3 , ZΣ4 3 and ZΣ5 3 , respectively. For the discontinuous piecewise differ- ential system (3.22), system (3.8), has the following three real solutions 41 5 − 106 263   − 10 , − 2 , 1.569412.., 1.457623.. , 25 , − 100 , 1.647799.., 1.587623.. , − 199  50 , −2.401954.., 1.508473.., 1.359577.. , and (3.10), has the three real solutions  0.502842.., −1.545218.., −0.572025.., 0.848539.. ,  0.442709.., −1.485086.., −0.427227.., 0.781483.. ,  0.378567.., −1.420944.., −0.276975.., 0.700080.. , Limit cycles of DPWLHS without equilibria separated by reducible cubics 31 and system (3.12), has the three real solutions 1.910216.., −4.699349.., −3.192376.., 43   1.847758.., −4.593279.., −3.042376.., 2 , 20 , 57  1.962805.., −4.784530.., −3.322376.., 25 . These solutions provide the three limit cycles with Conf 6− , Conf 7 and Conf 8 shown in Figure 3.10. This completes the proof of statement (x). Proof of statement (xi) of Theorem 1.3. In order to have limit cycles with Conf 6+ ,Conf 8 and Conf 9+ simultaneously, the points of intersection of the limit cycles with Conf 6+ and Σk+ must satisfy system (3.7), the points of intersection of the limit cycles with Conf 8 and Σk+ must satisfy system (3.12) and the points of intersection of the limit cycles with Conf 9+ and Σk+ must satisfy system (3.5) with k > 0. In statement (ii) we proved that the maximum number of limit cycles with each configuration is three, then we have that the maximum number of limit cycles with the three configurations, is nine. Moreover this upper bound is reached. Without loss of generality we can suppose that k = 3. We consider the discontinuous piecewise linear differential system defined by the following five linear Hamiltonian systems 3 ẋ = −17.085953.. + 9.977916..x − y, ẏ = −113.972678.. + 66.372549..x − 9.977916..y, 2 3 ẋ = −2.306102.. − 0.633078..x − y, ẏ = 2.662449.. + 1.335961..x + 0.633078..y, 10 7 ẋ = 34.897550.. − 4.677048..x − y, ẏ = −44.332934.. + 6.249936..x + 4.677048..y, 2 13 ẋ = 65.922589.. − 17.491280..x − y, ẏ = −924.494729.. + 235.342217..x + 17.491280..y, 10 9 ẋ = 13.883036.. + 3.280745..x − y, ẏ = −44.382913.. + 2.391842..x − 3.280745..y, 2 (3.23) 1 2 3 4 5 in the zones ZΣ4 , ZΣ4 , ZΣ4 , ZΣ4 and ZΣ4 , respectively. For the discontinuous piecewise differ- ential system (3.23), system (3.7), has the following three real solutions 6, 1.209968.., 57 , 8.457532.. , 156 5   25 , 1.006799.., 4 , 8.915579.. , 117  20 , 1.328327.., 1.486618.., 8.175706.. , and (3.12), has the three real solutions 93   9, −2.341532.., −6.604799.., 14.804799.. , 10 , −2.642166.., −7.313423.., 15.513423.. , 943  100 , −2.772412.., −7.624652.., 15.824652.. , and system (3.5), has the three real solutions 4, 3, 16 4.109491.., 141 303 517 47    5 ,5 , 50 , 100 , 100 , 10 , 2.053733.., 2.068270.., 6.131729.. . These solutions provide the three limit cycles with Conf 6+ , Conf 8 and Conf 9+ shown in 3.11. This completes the proof of statement (xi). Proof of statement (xii) of Theorem 1.3. In order to have limit cycles with Conf 6+ ,Conf 6− and Conf 8 simultaneously, the points of intersection of the limit cycles with Conf 6+ and Σk+ must satisfy system (3.7), the points of intersection of the limit cycles with Conf 6− and 32 R. Benterki, J. Jimenez and J. Llibre 20 30 0 20 10 - 20 0 - 10 - 40 - 20 - 30 - 10 0 10 20 30 40 50 60 0 10 20 30 Figure 3.12: Three limit cycles with Conf 6+ , two limit cycles with Conf 6− and one limit cycle with Conf 8 of system (3.24). 40 20 20 0 15 - 20 10 5 - 40 0 - 60 -6 -5 -4 -3 -2 -1 0 0 20 40 60 Figure 3.13: Two limit cycles with Conf 6+ , three limit cycles with Conf 6− and one limit cycle with Conf 8 of system (3.25). Σk+ must satisfy system (3.8) and the points of intersection of the limit cycles with Conf 8 and Σk+ must satisfy system (3.12). If we suppose that there is one solution for each system (3.7) and (3.8), then similar to statement (i) of Theorem 1.3, we obtain the value of the parameters γ1 , δ1 , γ2 , γ3 , δ3 , γ4 , γ5 , δ5 . Now we have two options, first we suppose that there is a solution of system (3.12), then we obtain the value of the parameters λ1 , δ2 , λ3 and δ4 , therefore we have two options, first we can suppose that there is a second solution of system (3.12) then we obtain the value of the parameters λ2 and λ4 , hence in the zones ZΣ1 + , ZΣ2 + , ZΣ3 + , ZΣ4 + we only have the parameters b1 , b2 , b3 , b4 as unknowns and in the zone ZΣ5 + we have λ5 , b5 as unknowns. Therefore we can obtain at most one solution either of system (3.7) or of system (3.8) and we cannot obtain more solutions of systems (3.7), (3.8) and (3.12), because we would have that bi = 0 for some i = 1, 2, 3, 4, 5. Hence we would have five limit cycles with two (resp. one) limit cycles with Conf 6+ , one (resp. two) limit cycle(s) with Conf 6− and two limit cycles with Conf 8. Second we can suppose that there is a second solution for each system (3.7) and (3.8), then we obtain Limit cycles of DPWLHS without equilibria separated by reducible cubics 33 the values of the parameters λ2 , λ4 , λ5 hence we cannot obtain more solutions of systems (3.7), (3.8) and (3.12), because we would have that bi = 0 for some i = 1, 2, 3, 4, 5. Therefore in this case we would obtain five limit cycles with two limit cycles with Conf 6+ , two limit cycles with Conf 6− and one limit cycle with Conf 8. Second, after considering the first solution of each system (3.7) and (3.8), we can suppose that there is a second solution for each system (3.7) and (3.8), then we obtain the values of λ1 , δ2 , λ3 and δ4 . Then in the zones ZΣ1 + , ZΣ3 + and ZΣ5 + we only have the parameters b1 , b3 and b5 as unknowns and in the zones ZΣ2 + and ZΣ4 + we have the parameters λ2 , b2 , λ4 , b4 unknowns. Hence can have two cases, first we can suppose that there is a solution of system (3.12), then we determine the value of parameters λ2 and λ4 , hence we cannot to have more limit cycles because we would have that bi = 0 for some i = 1, 2, 3, 4, 5. Therefore we would have five limit cycles with two limit cycles with Conf 6+ , two limit cycles with Conf 6− and one limit cycle with Conf 8. Second we can suppose that there is a third solution of system (3.7) (resp. (3.8)) and we obtain the value of parameter λ4 (res. λ2 ), then in the zone ZΣ4 + (resp. ZΣ2 + ) we only have the parameter b4 (resp. b2 ) as unknown and in the zone ZΣ2 + (resp. ZΣ4 + ) we have that the parameters λ2 , b2 (res. λ4 , b4 ) as unknowns. Now we suppose that there is one solution of system (3.12) and we obtain the parameter λ2 (res. λ4 ). We observe that we cannot obtain more solutions of systems (3.7), (3.8) and (3.12), because we would have that bi = 0 for some i = 1, 2, 3, 4, 5. Therefore we have at most six limit cycles with three (resp. two) limit cycles with Conf 6+ , two (resp. three) limit cycles with Conf 6− and one limit cycle with Conf 8. We observe that these six limit cycles can be either three limit cycles with Conf 6+ , two limit cycles with Conf 6− and one limit cycle with Conf 8 or two limit cycles with Conf 6+ , three limit cycles with Conf 6− and one limit cycle with Conf 8. We shall give an example of each case. We observe that without loss of generality we can suppose that k = 4. We consider the discontinuous piecewise linear differential system defined by the following five linear Hamiltonian systems 9 ẋ = − 23138.489410.. + 403.676452..x + y, ẏ = 2942.120325.. − 36212.150741..x 2 − 403.676452..y, 3 ẋ = 1.812606.. + 1.308936..x − y, ẏ = −25.828218.. + 5.711045..x − 1.308936..y, 10 17 (3.24) ẋ = 15.472057.. − 3.117904..x − y, ẏ = −13.354567.. + 2.859213..x + 3.117904..y, 5 ẋ = 48.158492.. − 6.082779..x − y, ẏ = −31.590984.. + 37.000210..x + 6.082779..y, 3 ẋ = − 151.854124.. − 136.354901..x + y, ẏ = −10611.949690.. − 12395.106180..x 2 + 136.354901..y, in the zones ZΣ1 3 , ZΣ2 3 , ZΣ3 3 , ZΣ4 3 and ZΣ5 3 , respectively. For the discontinuous piecewise differ- ential system (3.24), system (3.7) has the following three real solutions 5, 12 , 20 9 23 9 19 91 7 41    , 5 , 2 , 20 , 100 , 2 , 10 , 1.196150.., 1.163297.., 2.719447.. ; and (3.8) has the two real solutions 18 9 49   − 5 , − 2 , − 50 , −1 , − 3, −3.411586.., −1.557354.., −1.546135.. ; 34 R. Benterki, J. Jimenez and J. Llibre and system (3.12) has the real solution 5.688640.., −4.154651.., −5.682382.., 63  10 . These solutions provide the three limit cycles with Conf 6+ , the two limits cycles with Conf 6− and the limit cycle with Conf 8 shown in Figure 3.12. Now we consider the discontinuous piecewise linear differential system defined by the following five linear Hamiltonian systems 9 ẋ = − 23138.489410.. + 403.676452..x + y, ẏ = 2942.120325.. − 36212.150741..x 2 − 403.676452..y, 3 ẋ = 4.276633.. + 1.873985..x − y, ẏ = 4.991226.. + 11.706072..x − 1.873985..y, 10 17 ẋ = 15.472057.. − 3.117904..x − y, ẏ = −13.354567.. + 2.859213..x + 3.117904..y, (3.25) 5 ẋ = 293.931246.. − 44.804431..x − y, ẏ = −6905.938713.. + 2007.437106..x + 44.804431..y, 3 ẋ = − 151.854124.. − 136.354901..x + y, ẏ = −10611.949690.. − 12395.106180..x 2 + 136.354901..y, in the zones ZΣ1 3 , ZΣ2 3 , ZΣ3 3 , ZΣ4 3 and ZΣ5 3 , respectively. For the discontinuous piecewise differ- ential system (3.25), system (3.7) has the following two real solutions 9 23 5, 21 , 20 9 19 91 7   , 5 , 2 , 20 , 100 , 2 ; and (3.8) has the three real solutions − 18 9 49   5 , − 2 , − 50 , − 1 , − 3, − 3.411586.., − 1.557354.., − 1.546135.. , 31  − 2.809209.., − 10 , −1.671884.., −1.662653.. ; and system (3.12) has the real solution 667  100 , −4.419374.., −6.225080.., 6.842698.. . These solutions provide the two limit cycles with Conf 6+ , the three limits cycles with Conf 6− and the limit cycle with Conf 8 shown in Figure 3.13. This completes the proof of statement (xii). 4 Appendix Here we provide the values A, B and C  r − s  1  1  3 3 A = csc − cos (3r1 − r2 + 2r3 + s1 − s2 ) + cos (r1 − r2 + 4r3 + s1 − s2 ) 2 2 2 1  1  + cos (r1 − 3r2 − 2r3 + s1 − s2 ) − cos (r1 − r2 − 4r3 + s1 − s2 ) 2 2 1  1  − cos (r1 − r2 + 2r3 + 3s1 − s2 ) − cos (3r1 + r2 − 2r3 + s1 + s2 ) 2 2 Limit cycles of DPWLHS without equilibria separated by reducible cubics 35 1  1  + cos (r1 + 3r2 − 2r3 + s1 + s2 ) − cos (r1 + r2 − 2r3 + 3s1 + s2 ) 2 2 1  1  + cos (r1 + r2 − 2r3 + s1 + 3s2 ) + cos (r1 − r2 − 2r3 + s1 − 3s2 ) 2 2 1  1  + cos (3r1 − r2 + s1 − s2 + 2s3 ) + cos (r1 − r2 + 3s1 − s2 + 2s3 ) 2 2 1  1  − cos (r1 − r2 + s1 − s2 + 4s3 ) − cos (r1 − 3r2 + s1 − s2 − 2s3 ) 2 2 1  1  + cos (3r1 + r2 + s1 + s2 − 2s3 ) − cos (r1 + 3r2 + s1 + s2 − 2s3 ) 2 2 1  1  + cos (r1 + r2 + 3s1 + s2 − 2s3 ) − cos (r1 + r2 + s1 + 3s2 − 2s3 ) 2 2 1  1  − cos (r1 − r2 + s1 − 3s2 − 2s3 ) + cos (r1 − r2 + s1 − s2 − 4s3 ) , 2 2 B = − (cos(r1 − r2 ) + cos(r1 − r3 ) + cos(r2 − r3 ) − 2 cos(r1 − s1 ) + cos(r2 − s1 ) + cos(r3 − s1 ) + cos(r1 − s2 ) − 2 cos(r2 − s2 ) + cos(r3 − s2 ) + cos(s1 − s2 ) + 2 cos(r1 − r2 + s1 − s2 ) + cos(2r1 − 2r2 + s1 − s2 ) − cos(2r1 − r2 − r3 + s1 − s2 ) − cos(r1 − 2r2 + r3 + s1 − s2 ) + cos(r1 − 2r2 + 2s1 − s2 ) − cos(r1 − r2 − r3 + 2s1 − s2 ) + cos(2r1 − r2 + s1 − 2s2 ) − cos(r1 − r2 + r3 + s1 − 2s2 ) + cos(r1 − r2 + 2s1 − 2s2 ) + cos(r1 − s3 ) + cos(r2 − s3 ) − 2 cos(r3 − s3 ) + cos(s1 − s3 ) + 2 cos(r1 − r3 + s1 − s3 ) − cos(2r1 − r2 − r3 + s1 − s3 ) + cos(2r1 − 2r3 + s1 − s3 ) − cos(r1 + r2 − 2r3 + s1 − s3 ) − cos(r1 − r2 − r3 + 2s1 − s3 ) + cos(r1 − 2r3 + 2s1 − s3 ) − cos(2r1 − r2 + s1 − s2 − s3 ) − cos(2r1 − r3 + s1 − s2 − s3 ) + cos(r1 + r2 − r3 + s1 − s2 − s3 ) + cos(r1 − r2 + r3 + s1 − s2 − s3 ) − cos(r1 − r2 + 2s1 − s2 − s3 ) − cos(r1 − r3 + 2s1 − s2 − s3 ) + cos(s2 − s3 ) + 2 cos(r2 − r3 + s2 − s3 ) − cos(r1 + r2 − 2r3 + s2 − s3 ) + cos(2r2 − 2r3 + s2 − s3 ) + cos(r1 + r2 − r3 − s1 + s2 − s3 ) − cos(2r2 − r3 − s1 + s2 − s3 ) + cos(r1 − r2 − r3 + s1 + s2 − s3 ) − cos(r1 − 2r3 + s1 + s2 − s3 ) − cos(r2 − 2r3 + s1 + s2 − s3 ) + cos(r2 − 2r3 + 2s2 − s3 ) − cos(r2 − r3 − s1 + 2s2 − s3 ) − cos(r1 − 2r2 + r3 − s2 + s3 ) + cos(r1 − r2 + r3 − s1 − s2 + s3 ) − cos(r1 − 2r2 + s1 − s2 + s3 ) + cos(r1 − r2 − r3 + s1 − s2 + s3 ) − cos(r1 − r2 + r3 − 2s2 + s3 ) − cos(r1 − r2 + s1 − 2s2 + s3 ) + cos(2r1 − r3 + s1 − 2s3 ) − cos(r1 + r2 − r3 + s1 − 2s3 ) + cos(r1 − r3 + 2s1 − 2s3 ) − cos(r1 + r2 − r3 + s2 − 2s3 ) + cos(2r2 − r3 + s2 − 2s3 ) − cos(r1 − r3 + s1 + s2 − 2s3 ) − cos(r2 − r3 + s1 + s2 − 2s3 ) 1  r − s  3 3 + cos(r2 − r3 + 2s2 − 2s3 ) − 6) csc2 (r1 − r2 + s1 − s2 ) sin2 , 2 2  1  1  C =2 − cos (r1 − 3r2 − r3 + s1 − s2 − s3 ) + cos (r1 − r2 − 3r3 + s1 − s2 − s3 ) 2 2 1  1  − cos (3r1 + r2 − r3 + s1 + s2 − s3 ) + cos (r1 + 3r2 − r3 + s1 + s2 − s3 ) 2 2 1  1  − cos (r1 + r2 − r3 + 3s1 + s2 − s3 ) + cos (r1 + r2 − r3 + s1 + 3s2 − s3 ) 2 2 1  1  − cos (r1 − r2 − r3 + s1 − 3s2 − s3 ) + cos (3r1 − r2 + r3 + s1 − s2 + s3 ) 2 2 36 R. Benterki, J. Jimenez and J. Llibre 1  1  − cos (r1 − r2 + 3r3 + s1 − s2 + s3 ) + cos (r1 − r2 + r3 + 3s1 − s2 + s3 ) 2 2 1  1  − cos (r1 − r2 + r3 + s1 − s2 + 3s3 ) + cos (r1 − r2 − r3 + s1 − s2 − 3s3 ) . 2 2 Acknowledgements The third author is supported by the Ministerio de Ciencia, Innovación y Universidades, Agen- cia Estatal de Investigación grants MTM2016-77278-P (FEDER) and PID2019-104658GB-I00 (FEDER), the Agència de Gestió d’Ajuts Universitaris i de Recerca grant 2017SGR1617, and the H2020 European Research Council grant MSCA-RISE-2017-777911. References [1] A. Andronov, A. Vitt, S. Khaikin, Theory of oscillations, Pergamon Press, Oxford, 1996. https://doi.org/10.1002/zamm.19670470720; MR198734; Zbl 0188.56304 [2] A. Belfar, R. Benterki, J. Llibre, Limit cycles of planar discontinuous piecewise linear Hamiltonian systems without equilibrium points and separated by irreducible cubics, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms, accepted. [3] B. P. Belousov, A periodic reaction and its mechanism (in Russian), A Collection of Short Papers on Radiation Medicine for 1958, Moscow: Med. Publ., 1959. [4] R. Benterki, L. Damene, J. Llibre, The limit cycles of discontinuous piecewise linear differential systems formed by centers and separated by irreducible cubic curves II, Differ. Equ. Dyn. Syst., published online. https://doi.org/10.1007/s12591-021-00564-w [5] R. Benterki, J. LLibre, Crossing limit cycles of planar piecewise linear Hamiltonian sys- tems without equilibrium points, Mathematics 8(2020), No. 5, Article No. 755, 14 pp. https://doi.org/10.3390/math8050755 [6] R. Benterki, J. Llibre, The limit cycles of discontinuous piecewise linear differential systems formed by centers and separated by irreducible cubic curves I, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 28(2021), 153–192. MR4249577 [7] R. Benterki, J. Llibre, On the limit cycles of discontinuous piecewise linear differential systems formed by centers and separated by irreducible cubic curves III., submitted, 2021. [8] M. di Bernardo, C. J. Budd, A. R. Champneys, P. Kowalczyk, Piecewise-smooth dynam- ical systems. Theory and applications, Applied Mathematical Sciences, Vol. 163, Springer- Verlag, London, 2008. https://doi.org/10.1007/978-1-84628-708-4; MR2368310; Zbl 1146.37003 [9] D. C. Braga, L. F. Mello, Limit cycles in a family of discontinuous piecewise linear differential systems with two zones in the plane, Nonlinear Dynam. 73(2013), 1283–1288. https://doi.org/10.1007/s11071-013-0862-3; MR3083780; Zbl 1281.34037 [10] C. Buzzi, C. Pessoa, J. Torregrosa, Piecewise linear perturbations of a linear center, Discrete Contin. Dyn. Syst. 33(2013), 3915–3936. https://doi.org/10.3934/dcds.2013. 33.3915; MR3038046; Zbl 1312.37037 Limit cycles of DPWLHS without equilibria separated by reducible cubics 37 [11] A. F. Filippov, Differential equations with discontinuous right-hand sides, Mathematics and its Applications (Soviet Series), Vol. 18, Kluwer Academic Publishers Group, Dordrecht, 1988. https://doi.org/10.1007/978-94-015-7793-9; MR1028776; Zbl 1028776 [12] A. F. Fonseca, J. Llibre, L. F. Mello, Limit cycles in planar piecewise linear Hamil- tonian systems with three zones without equilibrium points, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 30(2020), No. 11, 2050157, 8 pp. https://doi.org/10.3390/math8050755; MR4152369; Zbl 07258121 [13] E. Freire, E. Ponce, F. Torres, A general mechanism to generate three limit cycles in planar Filippov systems with two zones, Nonlinear Dynam. 78(2014), 251–263. https: //doi.org/10.1007/s11071-014-1437-7; MR3266440; Zbl 1314.37031 [14] F. Giannakopoulos, K. Pliete, Planar systems of piecewise linear differential equations with a line of discontinuity, Nonlinearity 14 (2001), 1611–1632. https://doi.org/10.1088/ 0951-7715/14/6/311; MR1867095; Zbl 1003.34009 [15] M. Han, W. Zhang, On Hopf bifurcation in non-smooth planar systems, J. Differential Equations 248(2010), No. 9, 2399–2416. https://doi.org/10.1016/j.jde.2009.10.002; MR2595726; Zbl 1198.34059 [16] D. Hilbert, Mathematische probleme, Lecture, Second Internat. Congr. Math. (Paris, 1900), Nachr. Ges. Wiss. Göttingen Math. Phys. KL. (1900), 253–297; English transl., Bull. Amer. Math. Soc. 8(1902), 437–479; Bull. (New Series) Amer. Math. Soc. 37(2000), 407–436. https: //doi.org/10.1090/S0002-9904-1902-00923-3; MR1557926 [17] S. M. Huan, X. S. Yang, On the number of limit cycles in general planar piecewise linear systems, Discrete Contin. Dyn. Syst. 32(2012), No. 6, 2147–2164. https://doi.org/ 10.3934/dcds.2012.32.2147; MR2885803; Zbl 1248.34033 [18] Yu. Ilyashenko, Centennial history of Hilbert’s 16th problem, Bull. (New Series) Amer. Math. Soc. 39(2002), 301–354. https://doi.org/10.1090/S0273-0979-02-00946-1; MR1898209; Zbl 1004.34017 [19] J. J. Jimenez, J. Llibre, J. C. Medrado, Crossing limit cycles for a class of piecewise linear differential centers separated by a conic, Electron. J. Differential Equations 2020, No. 41, 1–36. MR4098332; Zbl 07244060 [20] J. J. Jimenez, J. Llibre, J. C. Medrado, Crossing limit cycles for piecewise linear differen- tial centers separated by a reducible cubic curve, Electron. J. Qual. Theory Differ. Equ. 2020, No. 19, 1–48. https://doi.org/10.14232/ejqtde.2020.1.19; MR4089921; Zbl 07254929 [21] J. Li, Hilbert’s 16th problem and bifurcations of planar polynomial vector fields, In- ternat. J. Bifur. Chaos Appl. Sci. Engrg. 13(2003), 47–106. https://doi.org/10.1142/ S0218127403006352; MR1965270; Zbl 1063.34026 [22] L. Li, Three crossing limit cycles in planar piecewise linear systems with saddle-focus type, Electron. J. Qual. Theory Differ. Equ. 2014, No. 70, 1–14. https://doi.org/10.14232/ ejqtde.2014.1.70; MR3304196; Zbl 1324.34025 [23] J. Llibre, D. D. Novaes, M. A. Teixeira, Maximum number of limit cycles for certain piecewise linear dynamical systems, Nonlinear Dyn. 82(2015), No. 3, 1159–1175. https: //doi.org/10.1007/s11071-015-2223-x; MR3412479; Zbl 1348.34065 38 R. Benterki, J. Jimenez and J. Llibre [24] J. Llibre, E. Ponce, Three nested limit cycles in discontinuous piecewise linear differ- ential systems with two zones, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms 19(2012), No. 3, 325–335. https://doi.org/10.1007/s11071-014-1437-7; MR2963277; Zbl 1268.34061 [25] O. Makarenkov, J. S. W. Lamb, Dynamics and bifurcations of nonsmooth systems: a survey, Phys. D 241(2012), 1826–1844. https://doi.org/10.1016/j.physd.2012.08.002; MR2994324 [26] B. van der Pol, A theory of the amplitude of free and forced triode vibrations, Radio Review (later Wireless World) 1(1920), 701–710. [27] B. van der Pol, On relaxation-oscillations, London Edinburgh Philos. Mag. J. Sci. 2(1926), No. 11, 978–992. https://doi.org/10.1080/14786442608564127 [28] D. J. W. Simpson, Bifurcations in piecewise-smooth continuous systems, World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises, Vol. 70, World Scientific, Singapore, 2010. https://doi.org/10.1142/7612; MR3524764; Zbl 05655816 [29] A. M. Zhabotinsky, Periodical oxidation of malonic acid in solution (a study of the Belousov reaction kinetics), Biofizika 9(1964), 306–311.