This is a preprint of: “Limit cycles of planar discontinuous piecewise linear Hamiltonian systems without equilibria separated by reducible cubics”, Rebiha Benterki, Johana Jimenez, Jaume Llibre, vol. 2021(69), 1–38, 2021. DOI: [10.14232/ejqtde.2021.1.69] LIMIT CYCLES OF PLANAR DISCONTINUOUS PIECEWISE LINEAR HAMILTONIAN SYSTEMS WITHOUT EQUILIBRIA SEPARATED BY REDUCIBLE CUBICS REBIHA BENTERKI1 , JOHANA JIMENEZ2 AND JAUME LLIBRE3 Abstract. Due to their applications to many physical phenomena during these last decades the interest for studying the discontinuous piecewise dierential systems has in- creased strongly. The limit cycles play a main role in the study of any planar dierential system, but to determine the maximum number of limits cycles that a class of planar dierential systems can have is one of the main problems in the qualitative theory of the planar dierential systems. Thus in general to provide a sharp upper bound for the number of crossing limit cycles that a given class of piecewise linear dierential system can have is a very dicult problem. In this paper we characterize the existence and the number of limit cycles for the piecewise linear dierential 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 dierential systems. 1. Introduction and statement of the main results Andronov, Vitt and Khaikin [1] started around 1920's the study of the piecewise dif- ferential 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 dierential systems are widely used to model processes appearing in mechanics, elec- tronics, economy, etc., see for instance the books [6] and [28], and the survey [25], as well as the hundreds of references cited there. A limit cycle is a periodic orbit of the dierential system isolated in the set of all periodic orbits of the system. Limit cycles are important in the study of the dierential 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 [2, 29], etc. The extended 16th Hilbert problem, that is, to nd an upper bound for the maximum number of limit cycles that a given class of dierential systems can exhibit, is in general an unsolved problem. Only for very few classes of dierential system this problem has been solved. For the class of discontinuous piecewise dierential systems here studied, we can obtain its solution by using the rst integrals provided by the Hamiltonians of the systems which form the discontinuous piecewise dierential systems. For the statement of the classical 16th Hilbert problem see [14, 16, 20]. Of course in order that a discontinuous piecewise dierential system be dened on the dis- continuous line, which separates the dierent dierential systems forming the discontinuous piecewise dierential system, we follow the rules of Filippov, see [9]. Date : 2010 Mathematics Subject Classication. Primary 34C29, 34C25, 47H11. Key words and phrases. limit cycles, discontinuous piecewise linear Hamiltonian systems, reducible cubic curves. 1 2 R. BENTERKI, J. JIMENEZ AND J. LLIBRE The discontinuous piecewise dierential systems formed by linear dierential systems can exhibit two kinds of limit cycles, the crossing and the sliding limit cycles, the rst 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 dierential systems are the planar ones formed by two pieces separated by a straight line having a linear dierential system in each piece. Several authors have tried to determine the maximum number of crossing limit cycles for this class of discontinuous piecewise dierential systems. Thus, in one of the rst papers dedicated to this problem, Giannakopoulos and Pliete [12] in 2001, showed the existence of discontinuous piecewise linear dierential systems with two crossing limit cycles. Then, in 2010 Han and Zhang [13] found other discontinuous piecewise linear dierential systems with two crossing limit cycles and they conjectured that the maximum number of crossing limit cycles for discontinuous piecewise linear dierential systems with two pieces separated by a straight line is two. But in 2012 Huan and Yang [15] provided numerical evidence of the existence of three crossing limit cycles in this class of discontinuous piecewise linear dierential systems. In 2012, Llibre and Ponce [22] inspired by the numerical example of Huan and Yang, proved for the rst time that there are discontinuous piecewise linear dierential 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 dierential systems with two pieces separated by a straight line, see Braga and Mello [7] in 2013, Buzzi, Pessoa and Torregrosa [8] in 2013, Liping Li [20] in 2014, Freire, Ponce and Torres [11] in 2014, and Llibre, Novaes and Teixeira [21] in 2015. But proving that discontinuous piecewise linear dierential systems separated by a straight line have at most three crossing limit cycles is an open problem. Also some discontinuous piecewise dierential systems separated by two parallel straight lines have been studied, see for instance [23, 24]. Recently, in [4, 17, 18] the authors have studied the extended 16th Hilbert problem to discontinuous piecewise linear dierential centers separated by either conics, or cubics. However for the discontinuous piecewise linear Hamiltonian systems without equilibrium points, it was proven in [10] 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 [3] the 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 dierential Hamiltonian systems without equilibrium points separated by two dierent 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 dierential Hamil- tonian 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 tangent 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 [10, 5], 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. LIMIT CYCLES OF DPWLHS WITHOUT EQUILIBRIA SEPARATED BY REDUCIBLE CUBICS 3 Doing an ane 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 + y 2 − 1) = 0, k ≥ 0}, formed by the circle x2 + y 2 = 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 [10]. Lemma 1. An arbitrary linear dierential 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 (1) H(x, y) = − λ2 bx2 + λbxy − y 2 + σx − µy. 2 2 Of course H(x, y) is a rst 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 without equilibrium points separated by Γk with k > 1. Let C2 be the class of planar discontinuous piecewise linear Hamiltonian systems without equilibrium points separated by Γk with k = 1. For these two classes we get the following three zones Z 1 = {(x, y) ∈ R2 : x2 + y 2 < 1}, (2) Z 2 = {(x, y) ∈ R2 : x2 + y 2 > 1 and x < k}, Z 3 = {(x, y) ∈ R2 : x2 + y 2 > 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 Z 1 = {(x, y) ∈ R2 : x2 + y 2 > 1 and x > k}, Z 2 = {(x, y) ∈ R2 : x2 + y 2 > 1 and x < k}, (3) Z 3 = {(x, y) ∈ R2 : x2 + y 2 < 1 and x < k}, Z 4 = {(x, y) ∈ R2 : x2 + y 2 < 1 and x > k}. We have three dierent congurations of crossing limit cycles for the class C3 . The rst one which will be denoted by Conf 1 are the limit cycles formed by four pieces of orbits, such that in each zone of (3) we have one piece of orbit of the linear Hamiltonian system without equilibria leaving in that zone, see Figure 4a. The second conguration of limit cycles denoted by Conf 2 are the limit cycles formed by pieces of orbits belonging to the three zones either Z 1 , Z 2 and Z 4 , or Z 1 , Z 2 and Z 3 . We are going to consider only the three zones Z 1 , Z 2 and Z 4 , because by a similar analysis we obtain the crossing limit cycles intersecting the three zones Z 1 , Z 2 and Z 3 , for this conguration see Figure 4b. Finally the third conguration namely Conf 3 is formed by pieces of orbits belonging to the three zones either Z 1 , Z 3 and Z 4 , or Z 2 , Z 3 and Z 4 . For the same reason as in 4 R. BENTERKI, J. JIMENEZ AND J. LLIBRE 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. (A) The three zones for the class C1 . (B) The four zones for the class C3 . the second conguration, we are going to consider only the three zones Z 1 , Z 3 and Z 4 , see Figure 4c. We notice that we can obtain two new congurations by combining the three previous ones, such as Conf 1 and Conf 2, Conf 1 and Conf 3. Note that we cannot have the conguration 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. Theorem 1. The following statements hold for the discontinuous piecewise linear Hamil- tonian systems without equilibria when the discontinuity line is formed by a circle and a straight line. The maximum number of crossing limit cycles 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 Figure 3a and Figure 3b, respectively; (ii) C3 with Conf 1 is three and this maximum is reached in Example 3, see Figure 4a; (iii) C3 with Conf 2 is three and this maximum is reached in Example 4, see Figure 4b; (iv) C3 with Conf 3 is three and this maximum is reached in Example 5, see Figure 4c; (v) C3 with Conf 1 and Conf 2 simultaneously is six and this maximum is reached in Example 6, see Figure 5; (vi) C3 with Conf 1 and Conf 3 simultaneously is six and this maximum is reached in Example 7, see Figure 6. Theorem 1 is proved in section 2. 1.2. The line of discontinuity is a parabola and a straight line parallel to the tangent at the vertex of the parabola. Let CΣk− be the class of discontinuous piecewise linear Hamiltonian systems without equilibria separated by Σk− with k < 0. In this case we have following three zones in the plane ZΣ1 ={(x, y) ∈ R2 : y > x2 }, k− ZΣ2 ={(x, y) ∈ R2 : y < x2 and y > k}, k− ZΣ3 ={(x, y) ∈ R2 : y < x2 and y < k}. k− 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 following four LIMIT CYCLES OF DPWLHS WITHOUT EQUILIBRIA SEPARATED BY REDUCIBLE CUBICS 5 1.0 1.0 2.0 0.8 0.8 1.5 1 0.6 ZSk- 1 1 ZSk + ZS0 0.6 4 ZSk 2 0.4 1.0 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 ZSk + - 0.4 ZSk ZS0 0.0 - 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 2. (A) Three zones for the class Ck− . (B) Four zones for the class C0 . (C) Five zones for the class Ck+ . 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}. In this class we have two congurations of crossing limit cycles, rst crossing limit cycles with Conf 4 which are constituted by pieces of orbits of the four Hamiltonian systems considered, see Figure 8a. Second crossing limit cycles with Conf 5 which intersect only three zones, in this case we have two options, rst 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 rst case because the study of the second is the same, see Figure 8b. 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 2 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 2 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 equilibria separated by Σk with k > 0, in this case we have the following ve zones in the plane ZΣ1 + ={(x, y) ∈ R2 : y > x2 and y > k}, k √ ZΣ2 ={(x, y) ∈ R2 : y < x2 and y > k, x < − k}, k+ ZΣ3 ={(x, y) ∈ R2 : y > x2 and y < k}, k+ √ ZΣ4 ={(x, y) ∈ R2 : y < x2 and y > k, x > k}, k+ ZΣ5 ={(x, y) ∈ R2 : x2 < y < k}. k+ In this class we have six dierent congurations 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 k+ , ZΣ5 k+ , ZΣ3 k+ and ZΣ4 k+ , or crossing limit cycles 6 R. BENTERKI, J. JIMENEZ AND J. LLIBRE formed by pieces of orbits of the four Hamiltonian systems in the zones ZΣ1 k+ , ZΣ2 k+ , ZΣ3 k+ and ZΣ5 k+ , namely crossing limit cycles with Conf 6+ and crossing limit cycles with Conf 6− , respectively, see Figure 11. In statement (ii) of Theorem 2 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 ZΣ3 + , see Figure 9b. Third we have the crossing limit cycles with Conf k k k 8, which intersect the zones ZΣ , ZΣ2 k+ , ZΣ3 k+ and ZΣ4 k+ , see Figure 9c. And nally we 1 k+ have the crossing limit cycles formed by pieces of orbits of the three Hamiltonian systems in the zones ZΣ1 k+ , ZΣ3 k+ and ZΣ4 k+ , or crossing limit cycles formed by pieces of orbits of the three Hamiltonian systems in the zones ZΣ1 k+ , ZΣ2 k+ and ZΣ3 k+ , namely crossing limit cycles with Conf 9+ and crossing limit cycles with Conf 9− , respectively, see Figure 9d. Without loss of generality in statement (ii) of Theorem 2 we study the crossing limit cycles with Conf 9+ because the study by the crossing limit cycles with Conf 9− is the same. We observe that there are no crossing limit cycles that intersect the ve zones ZΣi k+ for i = 1, 2, 3, 4, 5. Then in statement (ii) of Theorem 2 we study the crossing limit cycles with Conf 6 , Conf 7, Conf 8 and Conf 9 + + separately. In statements (iii)-(ix) of Theorem 2 we study the discontinuous piecewise linear Hamiltonian systems without equilibria in the class CΣk+ which have crossing limit cycles with two congurations 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 dierent congurations 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 2. The following statements hold for the discontinuous piecewise linear Hamil- tonian systems 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 for the class (i) CΣk− is three and this maximum is reached, see Figure 7; (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 Figure 8a-Figure 9d; (iii) CΣk+ with Conf 4 and Conf 5 simultaneously is six, see Figure 10; (iv) CΣk+ with Conf 6+ and Conf 6− simultaneously is six, see Figure 11; (v) CΣk+ with Conf 6− and Conf 7 simultaneously is six, see Figure 12a; (vi) CΣk+ with Conf 6+ and Conf 8 simultaneously is six, see Figure 12b; (vii) CΣk+ with Conf 6+ and Conf 9+ simultaneously is six, see Figure 13; (viii) CΣk+ with Conf 7 and Conf 8 simultaneously is six, see Figure 14; (ix) CΣk+ with Conf 8 and Conf 9+ simultaneously is six, see Figure 15; (x) CΣk+ with Conf 6− , Conf 7 and Conf 8 simultaneously is nine, see Figure 16; (xi) CΣk+ with Conf 6+ , Conf 8 and Conf 9+ simultaneously is nine, see Figure 17; (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 18 (resp. 19). Theorem 2 is proved in section 3. LIMIT CYCLES OF DPWLHS WITHOUT EQUILIBRIA SEPARATED BY REDUCIBLE CUBICS 7 2 1 1 0 0 -1 -1 -2 -2 -1 0 1 2 3 4 -1 0 1 2 3 4 (a) (b) Figure 3. (A) The three limit cycles of the discontinuous piecewise dier- ential system (5). (B) The three limit cycles of the discontinuous piecewise dierential system (6). 2. Proof of Theorem 1 Proof of statement (i) of Theorem 1. We have to prove that the maximum number of cross- ing 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 . By Lemma 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. with bi 6= 0 and σi 6= λi µi , and the three zones Zi are dened in (2). Their corresponding Hamiltonian rst integrals are as (1) Hi (x, y) = −(λ2i bi /2)x2 + λi bi xy − (bi /2)y 2 + 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, fi ) and Di = (k, hi ), where Ai and Bi are points on the circle x2 + y 2 − 1 = 0 and i = 1, 2, 3, these points must satisfy the following system e1 = H1 (xi , yi ) − H1 (zi , wi ) = 0, e2 = H2 (xi , yi ) − H2 (k, fi ) = 0, e3 = H2 (zi , wi ) − H2 (k, hi ) = 0, (4) e4 = H3 (k, fi ) − H3 (k, hi ) = 0, x2i + yi2 − 1 = 0, zi2 + wi2 − 1 = 0. The points Ai and Bi can take the form Ai = (cos ri , sin ri ), Bi = (cos si , sin si ), and due to the fact that they satisfy equation e1 , then by solving e1 = 0 when i = 1 for the parameter σ1 , 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 ) . 8 R. BENTERKI, J. JIMENEZ AND J. LLIBRE 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 4. (A) The three limit cycles of Conf 1 of the discontinuous piece- wise dierential system (7). (B) The three limit cycles of Conf 2 of the discontinuous piecewise dierential system (9). (C) The three limit cycles of Conf 3 of the discontinuous piecewise dierential system (11). Since the points A2 = (cos r2 , sin r2 ), B2 = (cos s2 , sin s2 ) also satisfy equation e1 = 0, we obtain the following expression of the parameter µ1 1    µ1 =      b1 csc r1 −s 2 1 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 . Likewise the points A3 = (cos r3 , sin r3 ), B3 = (cos s3 , sin s3 ) satisfy equation  e3 = 0, by √ solving it for the parameter λ1 and we get the two values λ1 = (A±2 2 sin 2 (r1 −r2 +s1 − 1,2 1 √ s2 ) B)/C given in the appendix. Now if we suppose that the points A4 = (cos r4 , sin r4 ), LIMIT CYCLES OF DPWLHS WITHOUT EQUILIBRIA SEPARATED BY REDUCIBLE CUBICS 9 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 5. Three limit cycles of Conf 1 and three limit cycles of Conf 2 for the class of the discontinuous piecewise dierential system (12). 2.0 1.5 1.0 0.5 0.0 -0.5 -1.0 -1 0 1 2 3 4 Figure 6. Three limit cycles of Conf 1 and three limit cycles of Conf 3 for the class of the discontinuous piecewise dierential system (13). B4 = (cos s4 , sin s4 ) satisfy equation e3 = 0, then we obtain b1 = 0. This 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 dierential 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 C1 . Here we consider the three zones dened in (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 Z 1 , (5) ẋ = 10.8..x + 18y − 3, ẏ = −6.48..x − 10.8..y in Z 2 , ẋ = 5.x − 3y − 1.38889.., ẏ = 8.33333..x − 5y in Z 3 . 10 R. BENTERKI, J. JIMENEZ AND J. LLIBRE The rst integrals of the linear Hamiltonian systems (5) are H1 (x, y) = −0.001..x2 + 0.02..xy − 0.1..y 2 − 0.316667..y, H2 (x, y) = −3.24..x2 − 10.8..xy − 9y 2 + 3y, H3 (x, y) = 4.16667..x2 − 5xy + 1.5y 2 + 1.38889..y, respectively. The discontinuous piecewise linear dierential system formed by the linear Hamiltonian systems (5) has exactly three crossing limit cycles, because the system of equations (4) has the three real solutions Si = (xi , yi , zi , wi , fi , 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 Figure 3a. Example 2: Three crossing limit cycles for the class C2 . We consider the three zones dened in (2) with k = 1. We consider the Hamiltonian systems ẋ = 15x − 3y − 11.25.., ẏ = 75.x − 15.y + 22.5 in Z 1 , (6) ẋ = 4x + 20y − 3, ẏ = −0.8x − 4y + 6 in Z 2 , ẋ = −0.4x + 4y + 0.6, ẏ = −0.04x + 0.4y − 1 in Z 3 . The rst integrals of the Hamiltonian systems (6) are 3y 2 H1 (x, y) = 37.5..x2 − 15xy + 22.5..x + + 11.25..y, 2 2 2 H2 (x, y) = −0.4x − 4xy + 6x − 10y + 3y, H3 (x, y) = −0.02..x2 + 0.4..xy − x − 2y 2 − 0.6..y, respectively. The discontinuous piecewise linear Hamiltonian system (6) has exactly three crossing limit cycles, because the system of equations (4) has the three real solutions Si = (xi , yi , zi , wi , fi , 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 dierential Hamil- tonian system (4), which are illustrate in Figure 3b. This completes the proof of statement (i).  To complete the proof of statements (ii)-(iv) of Theorem 1 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 C3 . For this class we consider the four zones dened in (3). We consider the Hamiltonian systems ẋ = −6.8..x + 4y − 2, ẏ = −11.56x + 6.8y − 2 in Z 1 , ẋ = 1.06216..x + 2y − 1.28925.., ẏ = −0.564089..x − 1.06216..y + 3.92358.. in Z 2 , (7) ẋ = −4x + 2y − 2.8.., ẏ = −8x + 4y − 1 in Z 3 , ẋ = 121.33..x + 3y + 508.239.., ẏ = −4907.01..x − 121.33..y + 611.017.. in Z 4 . LIMIT CYCLES OF DPWLHS WITHOUT EQUILIBRIA SEPARATED BY REDUCIBLE CUBICS 11 The linear Hamiltonian systems in (7) have the rst integrals H1 (x, y) = −5.78..x2 + 6.8..xy − 2x − 2y 2 + 2y, H2 (x, y) = −0.282045..x2 − 1.06216..xy + 3.92358..x − y 2 + 1.28925..y, H3 (x, y) = −4x2 + 4xy − x − y 2 + 2.8..y, H4 (x, y) = −2453.5..x2 − 121.33..xy + 611.017..x − 1.5y 2 − 508.239..y, respectively. The discontinuous piecewise linear Hamiltonian system (7) has exactly three crossing limit cycles intersecting Γk in the points Ai = (xi , yi ), Bi = (zi , wi ), Ci = (k, fi ) and D4 = (k, hi ) for i = 1, 2, 3, where Ai and Bi are points on the circle x2 + y 2 − 1 = 0, because the system of equations H1 (xi , yi ) − H1 (k, fi ) = 0, H2 (zi , wi ) − H2 (k, fi ) = 0, H3 (zi , wi ) − H3 (k, hi ) = 0, (8) H4 (xi , yi ) − H4 (k, hi ) = 0, x2i + yi2 − 1 = 0, zi2 + wi2 − 1 = 0, with k = 0, has only three real solutions Si = (xi , y, i, zi , wi , fi , 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 4a. This completes the proof of statement (ii). Example 4: Three crossing limit cycles of Conf 2 for the class C3 . In (3), we work only with the three zones Z 1 , Z 2 and Z 4 , with k = 0, and we consider the Hamiltonian systems ẋ = 19 − 18x − 3y, ẏ = −68 + 108x + 18y in Z 1 , (9) ẋ = −3.88389x − 2y + 5.99641.., ẏ = 7.54231..x + 3.88389..y − 7.99048.. in Z 2 , ẋ = 6 + 2x − 2y, ẏ = −2 + 2x − 2y in Z 4 . The rst integrals of the Hamiltonian systems (9) are 3y 2 H1 (x, y) = 54x2 + 18xy − 68x + − 19y, 2 H2 (x, y) = 3.77115..x + 3.88389..xy − 7.99048..x + y 2 − 5.99641..y, 2 H4 (x, y) = x2 − 2xy − 2x + y 2 − 6y, respectively The discontinuous piecewise linear dierential system formed by the linear Hamiltonian systems (9) has exactly three crossing limit cycles, because the system of equations H1 (xi , yi ) − H1 (k, fi ) = 0, H1 (zi , wi ) − H1 (k, hi ) = 0, H2 (zi , wi ) − H2 (k, fi ) = 0, (10) H4 (xi , yi ) − H4 (zi , wi ) = 0, x2i + yi2 − 1 = 0, zi2 + wi2 − 1 = 0, for k = 0 has the three real solutions Si = (xi , yi , zi , wi , fi , 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 4b. This completes the proof of statement (iii). 12 R. BENTERKI, J. JIMENEZ AND J. LLIBRE Example 5: Three crossing limit cycles of Conf 3 for the class C3 . Here we consider the three zones Z 1 , Z 3 and Z 4 dened in (3) with k = 0. 43x 43x 43y ẋ = − + 43y + 6, ẏ = − + − 2, in Z 1 , 2 4 2 (11) ẋ = −5.01788..x + 10y + 1.37209.., ẏ = −2.51792..x + 5.01788..y −0.356396.., in Z 4 , ẋ = −5.2..x + 13y + 1.78427.., ẏ = −2.08..x + 5.2..y + 7, in Z 3 . The rst integrals of the Hamiltonian systems (11) are 43x2 43xy 43y 2 H1 (x, y) = − + − 2x − − 6y, 8 22 2 H2 (x, y) = −1.25896..x + 5.01788..xy − 0.356396..x − 5y 2 − 1.37209..y, 13y 2 H3 (x, y) = −1.04..x2 + 5.2xy + 7x − − 1.78427..y, 2 respectively. The discontinuous piecewise linear dierential system formed by the linear Hamiltonian systems (11) has exactly three crossing limit cycles, because the system of equations (4) has the solutions Si = (xi , yi , zi , wi , fi , 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 4c. This completes the proof of statement (iv). Proof of statement (v) of Theorem 1. In order to have limit cycles with Conf 1 and Conf 2 simultaneously, the points of intersection of the limit cycles with Conf 1 with Γk must satisfy system (8) with k = 0, and the points of intersection of the limit cycles with Conf 2 with Γk must satisfy system (10). In statement (ii) and (iii) of Theorem 1 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 congurations is six. Example 6: Six crossing limit cycles for the class C3 , with three limit cycles of Conf 1 and three limit cycles of Conf 2 . Here we consider the four zones dened in (3). (12) ẋ = −3.37125..x − y + 3.95604.., ẏ = 11.3653..x + 3.37125..y − 11.7972.. in Z 1 , 1 ẋ = −0.121473..x − y + 2.02017.., ẏ = 0.0295115..x + 0.121473..y − 0.684232.. in Z 2 , 2 ẋ = 0.328515..x + y + 3, ẏ = −0.107922..x − 0.328515..y − 1.29868.. in Z 3 , ẋ = −9.2x − 2.3y + 17, ẏ = 36.8x + 9.2y − 56 in Z 4 . The rst integrals of the Hamiltonian systems (12) 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 + y 2 − 2.02017..y, 2 4 H3 (x, y) = −0.0539609x2 − 0.328515xy − 1.29868x − y2 − 3y, H4 (x, y) = 18.4x2 + 9.2xy − 56x + 1.15y 2 − 17y, respectively. LIMIT CYCLES OF DPWLHS WITHOUT EQUILIBRIA SEPARATED BY REDUCIBLE CUBICS 13 For the discontinuous piecewise dierential system (13), system (8) 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 (10), 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 5. This completes the proof of statement (v).  Proof of statement (vi) of Theorem 1. To get limit cycles with Conf 1 and Conf 3 simul- taneously, the points of intersection of the limit cycles with Conf 1 and Conf 3 with Γk must satisfy system (8) and (4), respectively, with k = 0. In statement (ii) and (iv) of Theorem 1 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 congurations is six. Example 7: Six crossing limit cycles for the class C3 , with three limit cycles of Conf 1 and three others of Conf 3 . Here we consider the four zones dened in (3) with k = 0 with the following Hamiltonian systems (13) ẋ = −8.8x + 22y − 3, ẏ = −3.52x + 8.8y − 4 in Z 1 , ẋ = 30.9637..x + 30y + 0.9.., ẏ = −31.9584..x − 30.9637..y + 24.1071.. in Z 2 , ẋ = 0.713131..x + 0.9y − 0.162525.., ẏ = −0.565063..x − 0.713131..y + 0.620587.. in Z 3 , ẋ = −8.37872..x + 22y − 3.97708.., ẏ = −3.19104..x + 8.37872..y − 3.05205.. in Z 4 . The rst integrals of the Hamiltonian systems (13) are H1 (x, y) = −1.76x2 + 8.8xy − 4x − 11y 2 + 3y, H2 (x, y) = −13.0028..x2 − 27.9315..xy + 24.0252..x − 15y 2 − 0.9y, H3 (x, y) = −0.282531..x2 − 0.713131..xy + 0.620587..x − 0.45..y 2 + 0.162525..y, H4 (x, y) = −1.59552..x2 + 8.37872..xy − 3.05205..x − 11y 2 + 3.97708..y, respectively. For the discontinuous piecewise dierential system (13), system (8) 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 (4), 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 Figure 6. This completes the proof of statement (vi).  14 R. BENTERKI, J. JIMENEZ AND J. LLIBRE 8 6 4 2 0 -2 -4 -2 0 2 4 Figure 7. Three limit cycles of system (16) intersecting Σ−1 . 4 20 3 5 15 2 4 10 3 1 2 5 0 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 8. (A) Three limit cycles with Conf 4 of system (17). (B) Three limit cycles with Conf 5 of system (19). 3. Proof of Theorem 2 We will prove the statement (i). For the other statements the proof is completely analo- gous. Proof of statement (i) of Theorem 2. From Lemma 1 we can consider an arbitrary piece- wise linear dierential Hamiltonian system in CΣk− formed by the following three linear Hamiltonian systems without equilibrium points (14) ẋ = −λi bi x + bi y + µi , ẏ = −λ2i bi x + λi bi y + σi in ZΣi k− , 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 − y 2 + σi x − µi y, in ZΣi − for i = 1, 2, 3. 2 2 k LIMIT CYCLES OF DPWLHS WITHOUT EQUILIBRIA SEPARATED BY REDUCIBLE CUBICS 15 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 9. (A) Three limit cycles with Conf 6+ of system (22). (B) Three limit cycles with Conf 7 of system (24). (C) Three limit cycles with Conf 8 of system (26). (D) Three limit cycles with Conf 9 + of system (27). In order to have a limit cycle which intersects Σk− in four dierent points (x1 , x21 ), (x2 , x22 ), (x3 , k) and (x4 , k), these points must satisfy the system H1 (x1 , x21 ) − H1 (x2 , x22 ) = 0, H2 (x2 , x22 ) − H2 (x3 , k) = 0, (15) H3 (x3 , k) − H3 (x4 , k) = 0, H4 (x4 , k) − H4 (x1 , x21 ) = 0, k < 0. Assume that the discontinuous piecewise linear dierential system (14) has four limit cycles. For this we must suppose that system (15) has four real solutions, namely (x(i) 1 , x2 , x3 , x4 ), (i) (i) (i) with i = 1, 2, 3, 4. We consider that (x(1) 1 , x2 , x3 , x4 ) satises system (15), solving the (1) (1) (1) rst and the third equations we obtain the expressions of the parameters µ1 and σ3 , respec- tively (1) (1) (1) (1) (1) (1) 2σ1 − b1 (x1 + x2 − λ1 )((x1 )2 + (x2 )2 − (x1 + x2 )λ1 ) µ1 = (1) (1) , 2(x1 + x2 ) 1 (1) (1) σ3 = b3 λ3 (−2k + (x3 + x4 )λ3 ). 2 16 R. BENTERKI, J. JIMENEZ AND J. LLIBRE When we suppose the second solution of system (15) we xed the points (x(2) 1 , x2 , x3 ) and (2) (2) by third equation we get that x(2) 4 = x3 + x4 − x3 . Moreover from the rst equation of (1) (1) (2) system (15) we get (2) (2)  b1 (x1 − x2 ) (2) (2) (2) 2 (2) 2 σ1 =  (2) (2)  (x1 + x2 − λ1 )((x1 ) + ((x2 ) (2) (2) −(x1 )2 + (x2 )2 2 x1 − x2 + (1) (1) x1 + x2 (2) (2) (1) (1) (1) (1) (1) (1)  (2) (2) (x1 + x2 )(x1 + x2 − λ1 )((x1 )2 + (x2 )2 − (x1 + x2 )λ1 ) − (x1 + x2 )λ1 ) − (1) (1) . x1 + x2 If we suppose a third solution, we xed the points (x(3) 1 , x2 , x3 ), and from the third (3) (3) equation we get that x(3) 4 = x3 + x4 − x3 . By rst equation we obtain (1) (1) (3) (1) (2) (2) (3) (3) (1) (1) (2) (2) (3) (3) (1) (2) λ1 = (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) (3) (3) + x2 (−(x1 )3 − (x1 )2 x2 − x1 (x2 )2 − (x2 )3 + (x1 )3 + (x1 )2 x2 + x1 (x2 )2 (3) (1) (2) (2) (2) (2) (2) (2) (3) (1) (2) + (x2 )3 ) + x1 (−(x1 )3 − (x1 )2 x2 − x1 (x2 )2 − (x2 )3 + (x1 )3 + (x2 )2 (x1 (2) (3) (3) (3) (3) (3) (3) (3)  (2) (3) (2) (2) (3) + x2 − x1 − x2 ) + (x1 )2 x2 + x1 (x2 )2 + (x2 )3 ) / 2((x1 )2 x1 + x1 x2 x1 (2) (3) (2) (3) (2) (3) (1) (2) (2) (3) (3) (1) + (x2 )2 x1 − x1 (x1 )2 − x2 (x1 )2 + (x1 )2 (x1 + x2 − x1 − x2 ) + (x2 )2 (2) (2) (3) (3) (2) (3) (2) (2) (3) (2) (3) (2) (3) (3) (x1 + x2 − x1 − x2 ) + (x1 )2 x2 + x1 x2 x2 + (x2 )2 x2 − x1 x1 x2 (2) (3) (3) (2) (3) (2) (3) (1) (2) (2) (2) (2) (3) − x2 x1 x2 − x1 (x2 )2 − x2 (x2 )2 + x2 (−(x1 )2 − x1 x2 − (x2 )2 + (x1 )2 (3) (3) (3) (1) (2) (2) (2) (2) (3) (1) (2) (2) + x1 x2 + (x2 )2 ) + x1 (−(x1 )2 − x1 x2 − (x2 )2 + (x1 )2 + x2 (x1 + x2 (3) (3) (3) (3) (3)  − x1 − x2 ) + x1 x2 + (x2 )2 )) . When we suppose the fourth solution (x(4) 1 , x2 , x3 , x3 +x4 −x3 ), from the rst equation (4) (4) (1) (1) (4) we have that b1 = 0 which is a contradiction, because from Lemma 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 observed that the upper bound found does not depend of value of parameter k < 0, then we can consider without loss of generality that k = −1. We consider the discontinuous piecewise linear dierential system dened by the following three linear Hamiltonian systems (16) 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, 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 (16), we have that system (15) has three real solutions (x(i)1 , x2 , x3 , x4 ), i = 1, 2, 3, namely (i) (i) (i)    3, −2, − 27 , 27 , 2.625658.., − 17 31 31 10 .., − 10 , 10 , 14 5 , −1.843412.., −3.287307.., 3.287307.. . These three real solutions provide the three limit cycles intersecting Σ−1 shown in Figure 7. This completes the proof of statement (i).  LIMIT CYCLES OF DPWLHS WITHOUT EQUILIBRIA SEPARATED BY REDUCIBLE CUBICS 17 Proof of statement (ii) of Theorem 2. The proof in this statement is similar to the proof of statement (i). For each conguration 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 dierential system in CΣ0 with three limit cycles with Conf 4 and Conf 5, respectively. And piecewise linear dierential 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 dierent points (x1 , x21 ), (x2 , x22 ), (x3 , 0) and (x4 , 0), these points must satisfy system (15) with k = 0. We consider the discontinuous piecewise linear dierential system dened by the following four linear Hamiltonian systems (17) 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 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 dierential system (17), system (15) has three real solutions (x(i) 1 , x2 , x3 , x4 ), i = 1, 2, 3 given by (i) (i) (i)  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 8a. Crossing limit cycles with Conf 5: In order to have a limit cycle with Conf 5 which intersects Σ0 in the four dierent points (x1 , x21 ), (x2 , x22 ), (x3 , 0) and (x4 , 0), they must satisfy H1 (x1 , x21 ) − H1 (x2 , x22 ) = 0, H4 (x2 , x22 ) − H4 (x3 , k) = 0, (18) H3 (x3 , k) − H3 (x4 , k) = 0, H4 (x4 , k) − H4 (x1 , x21 ) = 0, with k = 0. We consider the discontinuous piecewise linear dierential system dened by the following three linear Hamiltonian systems (19) 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, 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 dierential system (19), system (18) has three real solutions (x(i) 1 , x2 , x3 , x4 ), i = 1, 2, 3 given by (i) (i) (i)    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 8b. 18 R. BENTERKI, J. JIMENEZ AND J. LLIBRE Crossing limit cycles with Conf 6+ : In order to have a limit cycle with Conf 6+ which intersects Σ+ in four dierent points (x1 , x21 ), (x2 , k), (x3 , x23 ) and (x4 , k), these points must satisfy H1 (x1 , x21 ) − H1 (x2 , k) = 0, H5 (x2 , k) − H5 (x3 , x23 ) = 0, (20) H3 (x3 , x23 ) − H3 (x4 , k) = 0, for k > 0. H4 (x4 , k) − H4 (x1 , x21 ) = 0, To have a limit cycle with Conf 6− which intersects Σ+ in four dierent points, these points must satisfy the system H2 (x1 , x21 ) − H2 (x2 , k) = 0, H3 (x2 , k) − H3 (x3 , x23 ) = 0, (21) H5 (x3 , x23 ) − H5 (x4 , k) = 0, for k > 0. H1 (x4 , k) − H1 (x1 , x21 ) = 0, We provide an example of a piecewise linear dierential 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 dierential system dened by the following four linear Hamiltonian systems (22) 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 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 dierential system (22), system (20) has three real solutions (x(i) 1 , x2 , x3 , x4 ), i = 1, 2, 3 given by (i) (i) (i)    4, − 25 , 12 , 5 , 193 31 13 483 50 , − 100 , 20 , 100 7 3 83 441 2 , − 25 , 100 , 100 , . These solutions provide the three limit cycles with Conf 6 + shown in Figure 9a. Crossing limit cycles with Conf 7: In order to have a limit cycle with Conf 7 which intersects Σ+ in the four dierent points (x1 , k), (x2 , k), (x3 , x23 ) and (x4 , x24 ), they must satisfy the system H1 (x1 , k) − H1 (x2 , k) = 0, H5 (x2 , k) − H5 (x3 , x23 ) = 0, (23) H3 (x3 , x23 ) − H3 (x4 , x24 ) = 0, H5 (x4 , x24 ) − H5 (x1 , k) = 0, with k > 0. We can suppose without loss of generality that k = 4. We consider the discontinuous piecewise linear dierential system dened by the following three linear Hamiltonian systems (24) 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 LIMIT CYCLES OF DPWLHS WITHOUT EQUILIBRIA SEPARATED BY REDUCIBLE CUBICS 19 in the zones ZΣ1 3 , ZΣ3 3 and ZΣ5 3 , respectively. For the discontinuous piecewise dierential system (24), system (23) has three real solutions (x(i) 1 , x2 , x3 , x4 ), i = 1, 2, 3 given by (i) (i) (i)  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 Figure 9b. Crossing limit cycles with Conf 8: In order to have a limit cycle with Conf 8 which intersects Σ+ in four dierent points (x1 , x21 ), (x2 , x22 ), (x3 , k) and (x4 , k), they must satisfy H1 (x1 , x21 ) − H1 (x2 , x22 ) = 0, H2 (x2 , x22 ) − H2 (x3 , k) = 0, (25) H3 (x3 , k) − H3 (x4 , k) = 0, H4 (x4 , k) − H4 (x1 , x21 ) = 0, with k > 0. We can consider without loss of generality that k = 2. We consider the discontinuous piecewise linear dierential system dened by the following four linear Hamiltonian systems (26) 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 in the pieces ZΣ1 2 , ZΣ2 2 , ZΣ3 2 and ZΣ4 2 , respectively. For the discontinuous piecewise dier- ential system (26), system (25) has three real solutions (x1(i) , x2(i) , x(i) 3 , x4 ), i = 1, 2, 3 given (i) 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 9c. Crossing limit cycles with Conf 9 : In order to have a limit cycle with Conf 9 which + + intersects Σ+ in the four dierent points (x1 , x21 ), (x2 , x22 ), (x3 , k) and (x4 , k), they must satisfy system (18) with k > 0. Without loss of generality we can suppose that k = 4. We consider the discontinuous piecewise linear dierential system dened by the following three linear Hamiltonian systems (27) 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 in the zones ZΣ1 4 , ZΣ3 4 , and ZΣ4 4 , respectively. For the discontinuous piecewise dierential system (27), system (18) with k = 4, has three real solutions (x(i) 1 , x2 , x3 , x4 ), i = 1, 2, 3 (i) (i) (i) given by    4, 3, 16 141 303 517 5 , 5 , 4.109491.., 50 , 100 , 100 , 47 10 , 2.053733.., 2.068270.., 6.131729.. . 20 R. BENTERKI, J. JIMENEZ AND J. LLIBRE These solutions provide the three limit cycles with Conf 9+ shown in Figure 9d . This completes the proof of statement (ii).  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 10. Three limit cycles with Conf 4, and three limit cycles with Conf 5 of system (28) simultaneously. Proof of statement (iii) of Theorem 2. 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 (15) with k = 0, and the points of intersection of the limit cycles with Conf 5 with Σ0 must satisfy system (18). 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 congurations is six. We provide an example of a piecewise linear dierential 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 dierential system dened by the following four linear Hamiltonian systems (28) 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 in the zones ZΣ1 0 , ZΣ2 0 , ZΣ3 0 and ZΣ4 0 , respectively. For the discontinuous piecewise dierential system (28), system (15) 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 (18) with k = 0, has the three real solutions    2, 12 , 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 10. This completes the proof of statement (iii).  LIMIT CYCLES OF DPWLHS WITHOUT EQUILIBRIA SEPARATED BY REDUCIBLE CUBICS 21 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 11. Three limit cycles with Conf 6− and Conf 6+ of system (29). Proof of statement (iv) of Theorem 2. 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 (20), and the points of intersection of the limit cycles with Conf 6− and Σk+ must satisfy system (21). In statement (ii) we proved that the maximum num- ber of limit cycles with each conguration is three, then we have that the upper bound of maximum number of limit cycles with both congurations is six. We provide an example of a piecewise linear dierential system in CΣk+ such that have six limit cycles with three limit cycles with each conguration. This is the upper bound is reached. Without loss of generality we can suppose that k = 4. We consider the discontinuous piecewise linear dierential system dened by the following ve linear Hamiltonian systems (29) 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, 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 dierential system (29), (20), has the three real solutions    5, 21 , 20 9 23 , 5 , 9 19 91 7 2 , 20 , 100 , 2 , 41 10 , 1.196150.., 1.163297.., 2.719447.. , and system (21), has the following three real solutions 18 9 49   − 5 , − 2 , − 50 , −1 , − 3, −3.411586.., −1.557354.., −1.546135.. ,  − 2.809209.., − 31 10 , −1.671884.., −1.662653.. . These solutions provide the three limit cycles with Conf 6− and Conf 6+ shown in Figure 11. This completes the proof of statement (iv).  22 R. BENTERKI, J. JIMENEZ AND J. LLIBRE Proof of statement (v) of Theorem 2. 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 Σ k+ must satisfy system (21), and the points of intersection of the limit cycles with Conf 7 and Σk+ must satisfy system (23). In statement (ii) we proved that the maximum number of limit cycles with each conguration is three, then we have that the maximum number of limit cycles with both congurations is six. Moreover this upper bound is reached. Without loss of generality we can suppose that k = 3. We consider the discontinuous piecewise linear dierential system dened by the following four linear Hamiltonian systems (30) 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 in the pieces ZΣ1 3 , ZΣ2 3 , ZΣ3 3 and ZΣ5 3 , respectively. For the discontinuous piecewise dier- 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 12. (A) Three limit cycles with Conf 6− and Conf 7 of system (30). (B) Three limit cycles with Conf 6+ and Conf 8 of system (31). ential system (30), system (21), 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 (23), 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 Fig- ure 12a. This completes the proof of statement (v).  Proof of statement (vi) of Theorem 2. 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 LIMIT CYCLES OF DPWLHS WITHOUT EQUILIBRIA SEPARATED BY REDUCIBLE CUBICS 23 Σk+ must satisfy system (21), and the points of intersection of the limit cycles with Conf 8 and Σk+ must satisfy system (25). In statement (ii) we proved that the maximum number of limit cycles with each conguration is three, then we have that the maximum number of limit cycles with both congurations is six. Moreover this upper bound is reached. Without loss of generality we can consider that k = 4. We consider the discontinuous piecewise linear dierential system dened by the following ve linear Hamiltonian systems (31) 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 in the pieces ZΣ1 4 , ZΣ2 4 , ZΣ3 4 , ZΣ4 4 and ZΣ5 4 , respectively. For the discontinuous piecewise dierential system (31), system (20), 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 (25), has the three real solutions   4, −3, − 16 23 589 5 , 5 , 4.073407.., −3.179377.., −3.311999.., 125 ,  4.144187.., −3.341881.., −3.420000.., 241 50 . These solutions provide the three limit cycles with Conf 6+ and Conf 8 shown in Fig- ure 12b. This completes the proof of statement (vi).  Proof of statement (vii) of Theorem 2. 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 (20), and the points of intersection of the limit cycles with Conf 9+ and Σk+ must satisfy system (18) with k > 0. In statement (ii) we proved that the maximum number of limit cycles with each conguration is three, then we have that the maximum number of limit cycles with both congurations is six. Moreover this upper bound is reached. We can suppose without loss of generality that k = 4. We consider the discontinuous piecewise linear dierential system dened by the following four linear Hamiltonian systems (32) 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 in the pieces ZΣ1 4 , ZΣ3 4 , ZΣ4 4 and ZΣ5 4 , respectively. For the discontinuous piecewise dieren- tial system (32), system (20), has the following three real solutions   6, 1.209968.., 75 , 8.457532.. , 156 5 25 , 1.006799.., 4 , 8.915579.. , 117  20 , 1.328327.., 1.486618.., 8.175706.. , 24 R. BENTERKI, J. JIMENEZ AND J. LLIBRE and (18), has the three real solutions    4, 3, 16 141 303 517 5 , 5 , 4.109491.., 50 , 100 , 100 , 47 10 , 2.053733.., 2.068270.., 6.131729.. . These solutions provide the three limit cycles with Conf 6+ and Conf 9+ shown in Figure 13. 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 13. Three limit cycles with Conf 6+ and Conf 9+ of system (32). Proof of statement (viii) of Theorem 2. 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 (23), and the points of intersection of the limit cycles with Conf 8 and Σk+ must satisfy system (25). In statement (ii) we proved that the maximum number of limit cycles with each conguration is three, then we have that the maximum number of limit cycles with both congurations is six. Moreover this upper bound is reached. Without loss of generality we can suppose that k = 3. We consider the discontinuous piecewise linear dierential system dened by the following ve linear Hamiltonian systems (33) 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 in the zones ZΣ1 3 , ZΣ2 3 , ZΣ3 3 , ZΣ4 3 and ZΣ5 3 , respectively. For the discontinuous piecewise dierential system (33), system (23), 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.. , LIMIT CYCLES OF DPWLHS WITHOUT EQUILIBRIA SEPARATED BY REDUCIBLE CUBICS 25 and (25), 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 14. 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 14. Three limit cycles with Conf 7 and Conf 8 of system (33). Proof of statement (ix) of Theorem 2. 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 (25), and the points of intersection of the limit cycles with Conf 9+ and Σk+ must satisfy system (18) with k > 0. In statement (ii) we proved that the maximum number of limit cycles with each conguration is three, then we have that the maximum number of limit cycles with both congurations is six. Moreover this upper bound is reached. Without loss of generality we can suppose that k = 4. We consider the discontinuous piecewise linear dierential system dened by the following four linear Hamiltonian systems (34) 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 in the zones ZΣ4 , ZΣ4 , ZΣ4 and ZΣ4 , respectively. For the discontinuous piecewise dierential 1 2 3 4 system (34), system (25), 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 (18), has the three real solutions    4, 3, 16 141 303 517 5 , 5 , 4.109491.., 50 , 100 , 100 , 47 10 , 2.053733.., 2.068270.., 6.131729.. . These solutions provide the three limit cycles with Conf 8 and Conf 9+ shown in Figure 15. This completes the proof of statement (ix).  26 R. BENTERKI, J. JIMENEZ AND J. LLIBRE 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 15. Three limit cycles with Conf 8 and Conf 9+ of system (34). 20 10 0 - 10 - 20 - 30 - 40 -5 0 5 10 Figure 16. Three limit cycles with Conf 6− , Conf 7 and Conf 8 of system (35). 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 17. Three limit cycles with Conf 6+ , Conf 8 and Conf 9+ of system (36). LIMIT CYCLES OF DPWLHS WITHOUT EQUILIBRIA SEPARATED BY REDUCIBLE CUBICS 27 Proof of statement (x) of Theorem 2. 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+ must satisfy system (21), the points of intersection of the limit cycles with Conf 7 and Σk + must satisfy system (23) and the points of intersection of the limit cycles with Conf 8 and Σk + must satisfy system (25). In statement (ii) we proved that the maximum number of limit cycles with each conguration is three, then we have that the maximum number of limit cycles with the three congurations 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 dierential system dened by the following ve linear Hamiltonian systems (35) 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 in the zones ZΣ1 3 , ZΣ2 3 , ZΣ3 3 , ZΣ4 3 and ZΣ5 3 , respectively. For the discontinuous piecewise dierential system (35), system (21), 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 (23), 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.. , and system (25), has the three real solutions  43  1.847758.., −4.593279.., −3.042376.., 2 , 1.910216.., −4.699349.., −3.192376.., 20 ,  1.962805.., −4.784530.., −3.322376.., 57 25 . These solutions provide the three limit cycles with Conf 6− , Conf 7 and Conf 8 shown in Figure 16. This completes the proof of statement (x).  Proof of statement (xi) of Theorem 2. 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 (20), the points of intersection of the limit cycles with Conf 8 and Σk + must satisfy system (25) and the points of intersection of the limit cycles with + and Σ Conf 9 k+ must satisfy system (18) with k > 0. In statement (ii) we proved that the maximum number of limit cycles with each conguration is three, then we have that the maximum number of limit cycles with the three congurations, is nine. Moreover this upper bound is reached. Without loss of generality we can suppose that k = 3. We consider the discontinuous piecewise linear dierential system dened by the following ve linear 28 R. BENTERKI, J. JIMENEZ AND J. LLIBRE Hamiltonian systems (36) 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 in the zones ZΣ4 , ZΣ4 , ZΣ4 , ZΣ4 and ZΣ5 4 , respectively. For the discontinuous piecewise 1 2 3 4 dierential system (36), system (20), 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 (25), 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 (18), has the three real solutions    4, 3, 16 141 303 517 5 , 5 , 4.109491.., 50 , 100 , 100 , 47 10 , 2.053733.., 2.068270.., 6.131729.. . These solutions provide the three limit cycles with Conf 6+ , Conf 8 and Conf 9+ shown in Figure 17. This completes the proof of statement (xi).  20 30 0 20 10 - 20 0 - 10 - 40 - 20 - 30 - 10 0 10 20 30 40 50 60 0 10 20 30 Figure 18. Three limit cycles with Conf 6+ , two limit cycles with Conf 6− and one limit cycle with Conf 8 of system (37). Proof of statement (xii) of Theorem 2. 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 (20), the points of intersection of the limit cycles with Conf 6− and Σk+ must satisfy system (21) and the points of intersection of the limit cycles with Conf 8 and Σk + must satisfy system (25). If we suppose that there is one solution for each LIMIT CYCLES OF DPWLHS WITHOUT EQUILIBRIA SEPARATED BY REDUCIBLE CUBICS 29 40 20 20 0 15 - 20 10 5 - 40 0 - 60 -6 -5 -4 -3 -2 -1 0 0 20 40 60 Figure 19. Two limit cycles with Conf 6+ , three limit cycles with Conf 6− and one limit cycle with Conf 8 of system (38). system (20) and (21), then similar to statement (i) of Theorem 2, we obtain the value of the parameters γ1 , δ1 , γ2 , γ3 , δ3 , γ4 , γ5 , δ5 . Now we have two options, rst we suppose that there is a solution of system (25), then we obtain the value of the parameters λ1 , δ2 , λ3 and δ4 , therefore we have two options, rst we can suppose that there is a second solution of system (25) 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 (20) or of system (21) and we cannot obtain more solutions of systems (20), (21) and (25), because we would have that bi = 0 for some i = 1, 2, 3, 4, 5. Hence we would have ve 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 (20) and (21), then we obtain the values of the parameters λ2 , λ4 , λ5 hence we cannot obtain more solutions of systems (20), (21) and (25), because we would have that bi = 0 for some i = 1, 2, 3, 4, 5. Therefore in this case we would obtain ve 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 rst solution of each system (20) and (21), we can suppose that there is a second solution for each system (20) and (21), 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, rst we can suppose that there is a solution of system (25), 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 ve 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 (20) (resp. (21)) 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 (25) and we obtain the parameter λ2 (res. λ4 ). We observe that we cannot obtain more solutions of systems (20), (21) and (25), 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 30 R. BENTERKI, J. JIMENEZ AND J. LLIBRE 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 dierential system dened by the following ve linear Hamiltonian systems (37) 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 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 dierential system (37), system (20) has the following three real solutions    9 23 5, 12 , 20 , 5 , 9 19 91 7 2 , 20 , 100 , 2 , 41 10 , 1.196150.., 1.163297.., 2.719447.. ; and (21) has the two real solutions 18 9 49   − 5 , − 2 , − 50 , −1 , − 3, −3.411586.., −1.557354.., −1.546135.. ; and system (25) 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 18. Now we consider the discontinuous piecewise linear dierential system dened by the following ve linear Hamiltonian systems (38) 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, 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 dierential system (38), system (20) has the following two real solutions   5, 12 , 20 9 23 , 5 9 19 91 7 2 , 20 , 100 , 2 ; LIMIT CYCLES OF DPWLHS WITHOUT EQUILIBRIA SEPARATED BY REDUCIBLE CUBICS 31 and (21) has the three real solutions 18 9 49   − 5 , − 2 , − 50 , −1 , − 3, −3.411586.., −1.557354.., −1.546135.. ,  − 2.809209.., − 31 10 , −1.671884.., −1.662653.. ; and system (25) 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 19. This completes the proof of statement (xii).  4. The Appendix Here we provide the values A, B and C      r3 −s3 A = csc − cos 12 (3r1 − r2 + 2r3 + s1 − s2 ) + cos 21 (r1 − r2 + 4r3 2     +s1 − s2 ) + cos 12 (r1 − 3r2 − 2r3 + s1 − s2 ) − cos 21 (r1 − r2 − 4r3     +s1 − s2 ) − cos 12 (r1 − r2 + 2r3 + 3s1 − s2 ) − cos 12 (3r1 + r2 − 2r3     +s1 + s2 ) + cos 21 (r1 + 3r2 − 2r3 + s1 + s2 ) − cos 21 (r1 + r2 − 2r3     1 +3s1 + s2 ) + cos 2 (r1 + r2 − 2r3 + s1 + 3s2 ) + cos 21 (r1 − r2 − 2r3     +s1 − 3s2 ) + cos 21 (3r1 − r2 + s1 − s2 + 2s3 ) + cos 12 (r1 − r2 + 3s1     −s2 + 2s3 ) − cos 21 (r1 − r2 + s1 − s2 + 4s3 ) − cos 12 (r1 − 3r2 + s1     −s2 − 2s3 ) + cos 21 (3r1 + r2 + s1 + s2 − 2s3 ) − cos 12 (r1 + 3r2 + s1     +s2 − 2s3 ) + cos 21 (r1 + r2 + 3s1 + s2 − 2s3 ) − cos 12 (r1 + r2 + s1     +3s2 − 2s3 ) − cos 21 (r1 − r2 + s1 − 3s2 − 2s3 ) + cos 12 (r1 − r2 + s1  −s2 − 4s3 ) , 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 ) + cos(r2 − r3 1 r3 −s3 +2s2 − 2s3 ) − 6) csc2 2 (r1 − r2 + s1 − s2 ) sin2 2 , 32 R. BENTERKI, J. JIMENEZ AND J. LLIBRE      C = 2 − cos 21 (r1 − 3r2 − r3 + s1 − s2 − s3 ) + cos 12 (r1 − r2 − 3r3 + s1 − s2 − s3 )     − cos 12 (3r1 + r2 − r3 + s1 + s2 − s3 ) + cos 12 (r1 + 3r2 − r3 + s1 + s2 − s3 )     − cos 12 (r1 + r2 − r3 + 3s1 + s2 − s3 ) + cos 12 (r1 + r2 − r3 + s1 + 3s2 − s3 )     − cos 21 (r1 − r2 − r3 + s1 − 3s2 − s3 ) + cos 12 (3r1 − r2 + r3 + s1 − s2 + s3 )     − cos 12 (r1 − r2 + 3r3 + s1 − s2 + s3 ) + cos 12 (r1 − r2 + r3 + 3s1 − s2 + s3 )     − cos 21 (r1 − r2 + r3 + s1 − s2 + 3s3 ) + cos 12 (r1 − r2 − r3 + s1 − s2 − 3s3 ) . Acknowledgements The third author is supported by the Ministerio de Ciencia, Innovación y Universi- dades, Agencia 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 and S. 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Zhabotinsky, Periodical oxidation of malonic acid in solution (a study of the Belousov reaction kinetics), Biozika 9 (1964), 306311. 1 Département de Mathématiques, Université Mohamed El Bachir El Ibrahimi, Bordj Bou Arréridj 34000, El Anasser, Algeria Email address : r.benterki@univ-bba.dz 2 Universidade Federal do Oeste da Bahia, 46470000 Bom Jesus da Lapa, Bahia, Brazil Email address : jjohanajimenez@gmail.com 3 Departament de Matematiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain Email address : jllibre@mat.uab.cat