This is a preprint of: “The limit cycles of discontinuous piecewise linear differential systems formed by centers and separated by irreducible cubic curves II”, Rebiha Benterki, Loubna Damene, Jaume Llibre, Differ. Equ. Dyn. Syst., 2021. DOI: [10.1007/s12591-021-00564-w] THE LIMIT CYCLES OF DISCONTINUOUS PIECEWISE LINEAR DIFFERENTIAL SYSTEMS FORMED BY CENTERS AND SEPARATED BY IRREDUCIBLE CUBIC CURVES II REBIHA BENTERKI1 , LOUBNA DAMENE2 AND JAUME LLIBRE3 Abstract. In this paper we provide a lower bound for the maximum number of crossing limit cycles of some class of planar discontinuous piecewise linear differential systems formed by centers and separated by an irreducible algebraic cubic curve. First we prove that the systems constituted by three zones can exhibit 0, 1, 2, 3 or 4 crossing limit cycles having four intersection points with the cubic of separation. Second we prove that the systems constituted by two zones can exhibit 0, 1, or 2 crossing limit cycles having four intersection points with the cubic of separation. 1. Introduction 1.1. Classification of the irreducible cubic polynomials. A cubic curve is the set of points (x, y) ∈ R2 satisfying P (x, y) = 0 for some polynomial P (x, y) of degree three. This cubic is irreducible (respectively reducible) if the polynomial P (x, y) is irreducible (respectively reducible) in the ring of all real polynomials in the variables x and y. A point (x0 , y0 ) of a cubic P (x, y) = 0 is singular if Px (x0 , y0 ) = 0 and Py (x0 , y0 ) = 0. A cubic curve is singular if it has some singular point, as usual here Px and Py denote the partial derivaties of P with respect to the variables x and y respectively. A flex of an algebraic curve C is a point p of C such that C is nonsingular at p and the tangent at p intersects C at least three times. The next theorem characterizes all the irreducible cubic algebraic curves. Theorem 1. The following statements classify all the irreducible cubic algebraic curves. (a) A cubic curve is nonsingular and irreducible if and only if it can be trans- formed with affine transformations into one of the following two curves; c1 (x, y) = y 2 − x(x2 + bx + 1) = 0 with b ∈ (−2, 2), or 2 c2 (x, y) = y − x(x − 1)(x − r) = 0 with r > 1. 2010 Mathematics Subject Classification. Primary 34C29, 34C25, 47H11. Key words and phrases. limit cycles, discontinuous piecewise linear differential systems, linear differential centers, irreducible cubic curves. 1 2 R.BENTERKI, L. DAMENE AND J. LLIBRE (b) A cubic curve is singular and irreducible if and only if it can be transformed with affine transformations into one of the following three curves: c3 (x, y) = y 2 − x3 = 0, or c4 (x, y) = y 2 − x2 (x − 1) = 0, or c5 (x, y) = y 2 − x2 (x + 1) = 0. Statement (a) of Theorem 1 is proved in Theorem 8.3 of the book [4] under the additional assumption that the cubic has a flex, but in section 12 of that book it is shown that every nonsingular irreducible cubic curve has a flex. While statement (b) of Theorem 1 follows directly from Theorem 8.4 of [4]. 1.2. Crossing limit cycles. For k = 1, . . . , 5 let Ck be the five classes of planar discontinuous piecewise linear differential systems formed by centers and separated by the irreducible cubic curve ck (x, y) = 0, or simply the irreducible cubic curve ck . Piecewise linear differential systems appear in the characterization of many real processes such as switches in electronic circuits, see for instance [2, 12, 13]. In the qualitative theory of piecewise linear differential systems one of the impor- tant and difficult problems is the determination of the existance and the number of limit cycles. We recall that a crossing periodic orbit, is a periodic orbit of a piecewise linear differential system with a cubic separation curve ck with at least two points in ck . If this periodic orbit is isolated we call it a crossig limit cycle. In recent years, much progress has been made in studying the existance and lower and upper bounds of crossing limit cycles of piecewise linear differential systems, see for example [1, 5, 6, 7, 8, 9, 15, 17, 18, 19]. There are previous results on the number of crossing limit cycles of discontinuous piecewise linear differential centers separated by a curve Σ. More precisely, if Σ is one straight line then such systems have no crossing limit cycles see [14, 16]. If Σ is a conic the number of crossing limit cycles have been studied in [10], and if Σ is a reducible cubic curve the number of crossing limit cycles have been analyzed in [11]. In [3] we started the study of crossing limit cycles of the discontinuous piecewise linear differential centers in R2 separated by an irreducible algebraic cubic curve. We proved that these differential systems only can exhibit at most three crossing limit cycles having two intersection points with the cubic of separation. The objective of this paper is to provide lower bounds for the maximum number N of crossing limit cycles for the planar discontinuous piecewise linear differential centers which intersect the irreducible cubic curves ci , with i = 1 . . . 5, and have four points of intersection with the cubic of separation. First, we study the number N when the crossing limit cycles intersect the curves c2 and c5 , in four points in three different zones. Second, we give N when the crossing limit cycles intersect ci with i = 1 . . . 5, in four points in two different zones. Figures 1 and 2 show the different regions separated by the cubic curves ci , with i = 1 . . . 5. 3 3 2 2 R2 R3 1 1 0 0 -1 -2 -1 -3 -4 -2 -2 0 2 4 6 -2 -1 0 1 2 Figure 1. The three regions R1 , R2 and R3 of the plane separated by the curves c2 on the left and c5 on the right. 2 2 2 1 1 1 R2 0 0 0 R2 R2 -1 -1 -1 -2 -2 -2 -2 -1 0 1 2 -2 -1 0 1 2 -1 0 1 2 3 4 Figure 2. The two regions R1 and R2 of the plane separated by the curves c1 on the left, c3 on the middle and c4 on the right. 1.3. Crossing limit cycles contained in three regions intersecting the two cubic curves c2 and c5 in four points. In this subsection we are interessted to provide lower bounds for the maximum number of crossing limit cycles of piecewise linear differential centers separated by the irreducible cubic curves c2 or c5 , having four points of intersection with the cubic of separation. We note that such piecewise systems are formed by three pieces in each one there is a linear differential center. Our first main result is the following. Theorem 2. We study the crossing limit cycles contained in three regions of the discontinuous piecewise linear centers in the class C2 or C5 and having four points of intersection with the cubic of separation. (a) There are systems in C2 and in C5 exhibiting exactly one crossing limit cycle which intersect c2 or c5 in four points. The class C2 has one possible configuration see Figure 3, while the class C5 has two possible configurations see (C51 ) and (C52 ) of Figure 4. (b) There are systems in C2 and in C5 exhibiting exactly two crossing limit cycles which intersect c2 or c5 in four points. The class C2 has one possible configuration see Figure 5, while the class C5 has two possible configurations see (C51 ) and (C52 ) of Figure 6. 4 R.BENTERKI, L. DAMENE AND J. LLIBRE (c) There are systems in C2 and in C5 exhibiting exactly three crossing limit cycles which intersect c2 or c5 in four points. The class C2 has one possible configuration see Figure 7, while the class C5 has two possible configurations see (C51 ) and (C52 ) of Figure 8. (d) There are systems in C2 and in C5 exhibiting exactly four crossing limit cycles which intersect c2 or c5 in four points see (C2 ) and (C5 ) of Figure 9 for the classes C2 and C5 , respectively. Theorem 2 is proved in section 2. 1.4. Crossing limit cycles contained in two regions intersecting the irre- ducible cubic curves ci in four points. Now we give our second main result which provides information on the number of crossing limit cycles of the discontin- uous piecewise linear differential systems formed by two centers and intersect the cubic curves ci , with i = 1 . . . 5 in four points. Our second main result is the following. Theorem 3. We study the crosssing limit cycles contained only in two regions of the discontinuous piecewise linear centers in the class Ck for k = 1 . . . 5, and having four intersection points with the cubic of separation. (a) There are systems in Ck exibiting exactly one crossing limit cycle intersect- ing the cubic curves ci in four points. The classes C1 , C3 and C4 have one possible configuration, see (C1 ), (C3 ) of Figure 10 and (C4 ) of Figure 12, re- spectively. The classes C2 and C5 have two possible different configurations see (C21 ) and (C22 ) of Figure 11, and (C51 ) and (C52 ) of Figure 13. (b) There are systems in Ck exibiting exactly two crossing limit cycles inter- secting the cubic curves ci in four points. The classes C1 , C3 and C4 have one possible configuration see (C1 ), (C3 ) and (C4 ) of Figure 14, respectively. The classes C2 and C5 have two possible different configurations see (C21 ) and (C22 ) of Figure 15 for the class C2 , and (C51 ) and (C52 ) of Figure 16 for the class C5 . Theorem 3 is proved in section 3. 2. Proof of Theorem 2 Proof of statement (a) of Theorem 2. First we prove the statement for the class C2 . We consider the first linear differential center in the region R1 x 5y 5 y 1 (1) ẋ = − − + , ẏ = x + + , 4 16 4 4 2 this system has the first integral    y 2 x 5y H1 (x, y) = 4 x + +8 − + y2 . 4 2 4 The second linear differential center in the region R2 is 1 y 211 (2) ẋ = (14x − 7y − 79), ẏ = x − − , 56 4 64 5 10 5 -6 -4 -2 2 4 6 -5 -10 -15 -20 -25 Figure 3. The unique limit cycle of the discontinuous piecewise linear differential system (1)–(3) contained in three zones. 1 6 -3 -2 -1 1 2 3 4 4 -1 2 -2 -2 2 4 6 -3 -2 -4 -4 -5 -6 -6 (C51 ) (C52 ) Figure 4. The unique limit cycle of the discontinuous piecewise linear differential system (C51 ) for (5)–(7), and (C52 ) for (9)–(11) contained in three zones. this differential system has the first integral   211 1 H2 (x, y) = 4x2 + x −2y − + y(7y + 158). 8 14 Now we consider the third linear differential center in the region R3 5y 1 (3) ẋ = −x − + , ẏ = x + y − 2, 4 2 this differential system has the first integral  y H3 (x, y) = 4(x + y)2 + 8 −2x − + y2 . 2 The discontinuous piecewise linear differential system formed by the linear differen- tial centers (1), (2) and (3) has exactly one crossing limit cycle, because the system 6 R.BENTERKI, L. DAMENE AND J. LLIBRE 1.0 0.5 -2 -1 1 2 3 4 5 -0.5 -1.0 -1.5 -2.0 -2.5 (C2 ) Figure 5. The two limit cycles of the discontinuous piecewise linear differential system (12)–(14) contained in three zones. 2 2.0 1.5 1 1.0 0.5 -2 -1 1 2 3 -2.0 -1.5 -1.0 -0.5 0.5 1.0 1.5 -1 -0.5 -2 -1.0 (C51 ) (C52 ) Figure 6. The two limit cycles of the discontinuous piecewise linear differential system (C51 ) for (15)–(17), and (C52 ) for (18)–(20) contained un three zones. of equations H1 (α, β) − H1 (γ, δ) = 0, H2 (α, β) − H2 (f, g) = 0, H2 (γ, δ) − H2 (h, k) = 0, H3 (f, g) − H3 (h, k) = 0, (4) β 2 − α(α − 1)(α − 3) = 0, δ 2 − γ(γ − 1)(γ − 3) = 0, g 2 − f (f − 1)(f − 3) = 0, k 2 − h(h − 1)(h − 3) = 0, √ √ has√a unique √ √ (α, β, γ, δ, f, g, h, k) = (5, −2 10, 5, 2 10, 1/2, real√solution − 5/2 2, 1/2, 5/2 2). This completes the proof of statement (a) for the class C2 . 7 1 2 4 6 -1 -2 -3 (C2 ) Figure 7. The three limit cycles of the discontinuous piecewise linear differential system (21)–(23) contained in three zones. 1.5 2.0 1.0 1.5 0.5 1.0 -1.0 -0.5 0.5 1.0 1.5 2.0 -0.5 0.5 -1.0 -1.5 -1.0 -0.5 0.5 1.0 1.5 -1.5 -2.0 -0.5 (C51 ) (C52 ) Figure 8. The three limit cycles of the discontinuous piecewise linear differential system (C51 ) for (24)–(26), and (C52 ) for (27)–(29) contained in three zones. Now we prove the existence of two different configurations of one crossing limit cycle for the class C5 . For the first possible configuration we consider the linear differential center x 17y 3 y (5) ẋ = − − + , ẏ = x + − 1, 8 64 8 8 in the region R1 , with its first integral    y 2 3y H1 (x, y) = 4 x + + 8 −x − + y2 . 8 8 8 R.BENTERKI, L. DAMENE AND J. LLIBRE 0.5 2 1 2 4 6 -0.5 -1.0 -0.5 0.5 1.0 1.5 2.0 -1.0 -1 -1.5 -2 -2.0 -3 -2.5 -4 (C2 ) (C5 ) Figure 9. The four limit cycles of the discontinuous piecewise linear differential system (C2 ) for (30)–(32), and (C5 ) for (33)–(35) contained in three zones. In the region R2 we consider the linear center √ √ x 101y 3800 2 + 3584 5 + 2203113 ẋ = − + √ √  , 4 16 320 95 2 + 56 5 − 4380 (6) √ √ √ y −448604 2 − 315662 5 + 48 10 + 22560621 ẏ = x − + √ √  , 4 160 95 2 + 56 5 − 4380 with its first integral 1 H2 (x, y) = √ √ 40(95 2 + 56 5 − 4380) √ √ √ √ 4380)x2 √ (160(95 √2 + 56 5 − √ + x(−80(95 2 + 56 5 − 4380)y √ −897208 √ 2 − 631324 5√+ 96 10 +√ 45121242) + y(1010(95 2 +56 5 − 4380)y − 3800 2 − 3584 5 − 2203113). In the region R3 we consider the following linear differential center 1 (7) ẋ = −x − 10y − , ẏ = x + y − 2, 2 this differential system has the first integral     2 5 √ 81 288 H3 (x, y) = 4(x + y) + 8 −3 2 + − √ x − y + 16y 2 . 12 50 25 5 The discontinuous piecewise linear differential system formed by the linear differen- tial centers (5), (6) and (7) has exactly one crossing limit cycle, because the system 9 of equations H1 (α, β) − H1 (γ, δ) = 0, H2 (α, β) − H2 (f, g) = 0, H2 (γ, δ) − H2 (h, k) = 0, H3 (f, g) − H3 (h, k) = 0, (8) β 2 − α2 (α + 1) = 0, δ 2 − γ 2 (γ + 1) = 0, g 2 − f 2 (f + 1) = 0, k 2 − h2 (h + 1) = 0, √ has a√unique real solution (α, β, γ, δ, f, g, h, k) = (3, −6, 3, 6, −1/2, −1/(2 2), −4/5, 4/(5 5)). For the second configuration. In the region R1 we consider the linear differential center p √ x 785y 70 50 − 8 6 + 40083 y 1 (9) ẋ = − + √ √  , ẏ = x − + , 7 49 490 2 − 2 3 7 5 which has the first integral  √ √  ! y 2 x 70 2 − 280 3 − 40083 y H1 (x, y) = 4 x − +8 + √ √  + 64y 2 . 7 5 490 2 − 2 3 In the region R2 we consider the linear differential center (10) √ √ √ 3x 373y −23556784 2 − 26814823 3 + 14402472 6 + 83393133 ẋ = − − + , 7 1764 80960544 √ √ √ 3y −4861464 2 + 3092483 3 + 1548288 6 − 1763763 ẏ = x+ + √ √  , 7 56448 88 2 − 91 3 + 15 with its first integral  2 3y y2 A H2 (x, y) = 4 x + + + √ √  √ , 7 9 7056 88 2 − 91 3 + 15 16 3 − 3 where √ √ √ √ A = 88902216q 2 − 37497657 3 − 82428288 6 + 153730473)x − 8(7597532 3 √ +266112 6(259 − 32 3) − 23982207)y. In the region R3 we consider the linear differential system √ x 577y −10025 3 − 29787 y 1 (11) ẋ = − + , ẏ = x − − , 8 64 9216 8 3 which has the first integral √  !  y 2 10025 3 + 29787 y x H3 (x, y) = 4 x − +8 − + 36y 2 . 8 9216 3 The real solutions of the system of equations (8) with the values of Hi (x,√ y) with i = 1, 2, 3 given for this second configuration is (α, β, γ, δ, f, g, h, k) = (1, − 2, 2, 10 R.BENTERKI, L. DAMENE AND J. LLIBRE √ √ − 2 3, −1/4, − 3/8, −3/4, −3/8), then the discontinuous piecewise linear differ- ential system (9), (10) and (11) has one crossing limit cycle (C51 ) of Figure 4. This completes the proof of statement (a) for the class C5 .  Proof of statement (b) of Theorem 2. First we prove the statement for the class C2 . We consider the first linear differential center in the region R1 (12) ẋ = −x − 5y − 1.8035, ẏ = x + y − 0.664282, this system has the first integral H1 (x, y) = 4(x + y)2 + 8(1.8035y − 0.664282x) + 16y 2 . The second linear differential center in the region R2 is x 10y y 3 (13) ẋ = − − − 1, ẏ = x + − , 2 4 2 2 this differential system has the first integral    y 2 3x H2 (x, y) = 4 x + +8 y− + 9y 2 . 2 2 The third linear differential center in the region R3 is (14) ẋ = −2x − 8y + 0.837903, ẏ = x − 2y − 0.169396, this differential system has the first integral H3 (x, y) = 4(x − 2y)2 + 8(−0.169396x − 0.837903y) + 16y 2 . The discontinuous piecewise linear differential system formed by the linear differ- ential centers (12), (13) and (14) have exactly two crossing limit cycles, because the system of equations (4) has two real solutions (α1 , β1 , γ1 , δ1 , f1 , g1 , h1 , k1 ) = (3.00039.., 0.0485802.., 3.43695.., −1.91305.., 0.860569.., 0.506666.., 0.00442503.., 0.114878..) and (α2 , β2 , γ2 , δ2 , f2 , g2 , h2 , k2 ) = (3.00805.., 0.220494.., 3.50419.., − 2.1034.., 0.735716.., 0.663523.., 0.0779996.., 0.458408..). This completes the proof of statement (b) for the class C2 . Now we prove the existence of two different configurations of two crossing limit cycles for the class C5 . For the first configuration we consider the linear differential center 6x 369y 6y (15) ẋ = − − + 0.152456, ẏ = x + − 0.365572, 5 100 5 in the region R1 , with its first integral  2 6y H1 (x, y) = 4 x + + 8(−0.365572x − 0.152456y) + 9y 2 . 5 In the region R3 we consider the linear center x 29y y (16) ẋ = − − , ẏ = x + , 5 100 5 with its first integral  y 2 H2 (x, y) = 4 x + + y2 . 5 In the region R2 we consider the following linear differential center (17) ẋ = −y − 0.251587, ẏ = x + 1.93017, 11 this differential system has the first integral H3 (x, y) = 4x2 + 8(1.93017x + 0.251587y) + 4y 2 . The discontinuous piecewise linear differential system formed by the linear differ- ential centers (12), (13) and (14) have exactly two crossing limit cycles, because the system of equations (8) has two real solutions (α1 , β1 , γ1 , δ1 , f1 , g1 , h1 , k1 ) = (0.286549.., 0.325022.., 0.400999.., −0.474638.., −0.416749.., 0.318275.., −0.313035.., − 0.259454..) and (α2 , β2 , γ2 , δ2 , f2 , g2 , h2 , k2 ) = (0.484398.., 0.590171.., 0.681486.., − 0.883698.., −0.719407.., 0.381077.., −0.569542.., −0.373673..). For the second configuration. In the region R1 we consider the linear differential center 25y (18) ẋ = −2x − + 6.3854, ẏ = x + 2y + 0.134162, 4 which has the first integral H1 (x, y) = 4(x + 2y)2 + 8(0.134162x − 6.3854y) + 9y 2 . In the region R3 we consider the linear differential center 3 25y 1 (19) ẋ = − , ẏ = x + , 2 16 5 with its first integral   x 3 25y 2 H2 (x, y) = 4x2 + 8 + + . 5 2 4 In the region R2 we consider the linear differential system x 5y y (20) ẋ = − − + 0.902952, ẏ = x + + 0.234228, 2 2 2 which has the first integral  y 2 H3 (x, y) = 4 x + + 8(0.234228x − 0.902952y) + 9y 2 . 2 The discontinuous piecewise linear differential system formed by the linear differ- ential centers (18), (19) and (20) have exactly two crossing limit cycles, because the system of equations (8) has two real solutions (α1 , β1 , γ1 , δ1 , f1 , g1 , h1 , k1 ) = (0.844854.., 1.14753.., 0.137908.., 0.14711.., −0.884123.., 0.300962.., −0.112734.., 0.106189..) and (α2 , β2 , γ2 , δ2 , f2 , g2 , h2 , k2 ) = (0.783948.., 1.04708.., 0.219822.., 0.242783.., −0.824415.., 0.345453.., −0.186379.., 0.168115..). This completes the proof of statement (b) for the class C2 .  Proof of statement (c) of Theorem 2. First we prove the statement for the class C2 . We consider the first linear differential center in the region R1 (21) ẋ = −3.95435x − 19.6368y − 6.39482, ẏ = x + 3.95435y + 4.13635, this system has the first integral H1 (x, y) = 4(x + 3.95435y)2 + 8(4.13635x + 6.39482y) + 16y 2 . The second linear differential center in the region R2 x 5y y 3 (22) ẋ = − − − 1, ẏ = x + − , 2 2 2 2 12 R.BENTERKI, L. DAMENE AND J. LLIBRE this differential system has the first integral    y 2 3x H2 (x, y) = 4 x + +8 y− + 9y 2 . 2 2 Now we consider the third linear differential center in the region R3 (23) ẋ = −0.241343x − 1.00852y − 0.358353, ẏ = x + 0.241343y − 0.869754, this differential system has the first integral H3 (x, y) = 4(x + 0.241343y)2 + 8(0.358353y − 0.869754x) + 4y 2 . The discontinuous piecewise linear differential system formed by the linear differ- ential centers (21), (22) and (23) have exactly three crossing limit cycles, because the system of equations (4) has three real solutions (α1 , β1 , γ1 , δ1 , f1 , g1 , h1 , k1 ) = (3.00039.., 0.0485802.., 3.43695.., −1.91305.., 0.860569.., 0.506666.., 0.00442503.., 0.114878..), (α2 , β2 , γ2 , δ2 , f2 , g2 , h2 , k2 ) = (3.00318.., 0.138419.., 3.47181.., −2.01219.., 0.803882.., 0.588414.., 0.0319264.., 0.302877..) and (α3 , β3 , γ3 , δ3 , f3 , g3 , h3 , k3 ) = (3.00805.., 0.220494.., 3.50419.., −2.1034.., 0.735716.., 0.663523.., 0.0779996.., 0.458408..). Now we prove the existence of two different configurations of three crossing limit cycles for the class C5 . For the first configuration we consider the linear differential center (24) ẋ = −0.345578x − 1.11942y − 0.128163, ẏ = x + 0.345578y − 0.440337, in the region R1 , with its first integral H1 (x, y) = 4(x + 0.345578y)2 + 8(0.128163y − 0.440337x) + 4y 2 . In the region R3 we consider the linear center x 29y y (25) ẋ = − − , ẏ = x + , 5 100 5 with its first integral  y 2 H2 (x, y) = 4 x + + y2 . 5 In the region R2 we consider the following linear differential center (26) ẋ = − − 0.0923038x − 1.00852y − 0.0805185, ẏ = x + 0.0923038y + 0.46371, this differential system has the first integral H3 (x, y) = 4(x + 0.0923038y)2 + 8(0.46371x + 0.0805185y) + 4y 2 . The discontinuous piecewise linear differential system formed by the linear differ- ential centers (24), (25) and (26) has exactly three crossing limit cycles, because the system of equations (8) has three real solutions (α1 , β1 , γ1 , δ1 , f1 , g1 , h1 , k1 ) = (0.286549.., 0.325022.., 0.681486.., −0.883698.., −0.416749.., 0.318275.., −0.569542.., − 0.373673..), (α2 , β2 , γ2 , δ2 , f2 , g2 , h2 , k2 ) = (0.366248.., 0.428095.., 0.749792.., −0.991822.., −0.538507.., 0.365825.., −0.639545.., −0.383969..) and (α3 , β3 , γ3 , δ3 , f3 , g3 , h3 , k3 ) = (0.429988.., 0.514188.., 0.81153.., −1.09226.., −0.636378.., 0.383743.., − 0.706638.., −0.382736..). For the second configuration. In the region R1 we consider the linear differential center (27) ẋ = −0.121241x − 2.2647y + 1.93257, ẏ = x + 0.121241y + 0.00611219, 13 which has the first integral H1 (x, y) = 4(x + 2y)2 + 8(0.134162x − 6.3854y) + 9y 2 . In the region R3 we consider the linear differential center 3 25y 1 (28) ẋ = − , ẏ = x + , 2 16 5 with its first integral   2 x 3y 25y 2 H2 (x, y) = 4x + 8 − + . 5 2 4 In the region R2 we consider the linear differential system (29) ẋ = −0.217737x − 1.04741y + 0.515416, ẏ = x + 0.217737y + 0.350398, which has the first integral H3 (x, y) = 4(x + 0.217737y)2 + 8(0.350398x − 0.515416y) + 4y 2 . The discontinuous piecewise linear differential system formed by the linear differ- ential centers (27), (28) and (29) have exactly three crossing limit cycles, because the system of equations (8) has three real solutions (α1 , β1 , γ1 , δ1 , f1 , g1 , h1 , k1 ) = (0.844854.., 1.14753.., 0.137908.., 0.14711.., −0.884123.., 0.300962.., −0.112734.., 0.106189..), (α2 , β2 , γ2 , δ2 , f2 , g2 , h2 , k2 ) = (0.816222.., 1.1.., 0.177162.., 0.192216.., − 0.856934.., 0.324128.., −0.147429.., 0.136128..) and (α3 , β3 , γ3 , δ3 , f3 , g3 , h3 , k3 ) = (0.783948.., 1.04708.., 0.219822.., 0.242783.., −0.824415.., 0.345453.., −0.186379.., 0.168115..). This completes the proof of statement (c) for the class C5 .  Proof of statement (d) of Theorem 2. First we prove the statement for class C2 . We consider the first linear differential center in the region R1 (30) ẋ = −3.95435x − 19.6368y − 6.39482, ẏ = x + 3.95435y + 4.13635, this system has the first integral H1 (x, y) = 4(x + 3.95435y)2 + 8(4.13635x − 6.39482y) + 16y 2 . The second linear differential center in the region R2 x 5y y 3 (31) ẋ = − − − 1, ẏ = x + − , 2 2 2 2 this differential system has the first integral    y 2 3x H2 (x, y) = 4 x + +8 y− + 9y 2 . 2 2 Now we consider the third linear differential center in the region R3 (32) ẋ = −0.241343x − 1.00852y − 0.358353, ẏ = x + 0.241343y − 0.869754, this differential system has the first integral H3 (x, y) = 4(x + 0.241343y)2 + 8(0.358353y − 0.869754x) + 4y 2 . The discontinuous piecewise linear differential system formed by the linear differ- ential centers (30), (31) and (32) have exactly four crossing limit cycles, because the system of equations (4) has four real solutions (α1 , β1 , γ1 , δ1 , f1 , g1 , h1 , k1 ) = (3.00039.., 0.0485802.., 3.43695.., −1.91305.., 0.860569.., 0.506666.., 0.00442503.., 0.114878..), (α2 , β2 , γ2 , δ2 , f2 , g2 , h2 , k2 ) = (3.00318.., 0.138419.., 3.47181.., −2.01219.., 0.803882.., 0.588414.., 0.0319264.., 0.302877..), (α3 , β3 , γ3 δ3 , f3 , g3 , h3 , k3 ) = (3.00805.., 14 R.BENTERKI, L. DAMENE AND J. LLIBRE 0.220494.., 3.50419.., −2.1034.., 0.735716.., 0.663523.., 0.0779996.., 0.458408..) and (α4 , β4 , γ4 , δ4 , f4 , g4 , h3 4, k4 ) = (3.01815.., 0.332542.., 3.5491.., −2.22883.., 0.593338.., 0.762036.., 0.192725.., 0.66088..). Now we prove the existence of two different configurations of four crossing limit cycles for the class C5 . For the first configuration we consider the linear differential center (33) ẋ = 0.897851x − 2.12174y − 0.62272, ẏ = x − 0.897851y − 0.620434, in the region R1 , with its first integral H3 (x, y) = 4(x − 0.897851y)2 + 8(0.62272y − 0.620434x) + 5.2624y 2 . In the region R3 we consider the linear center x 17y y (34) ẋ = − , ẏ = x − , 10 450 10 with its first integral  y 2 y 2 H2 (x, y) = 4 x − + . 10 9 In the region R2 we consider the following linear differential center (35) ẋ = −0.225709x − 1.26066y − 0.265909, ẏ = x + 0.225709y + 0.334228, this differential system has the first integral H3 (x, y) = 4(x + 0.225709y)2 + 8(0.334228x + 0.265909y) + 4.83887y 2 . The discontinuous piecewise linear differential system formed by the linear differ- ential centers (33), (34) and (35) have exactly four crossing limit cycles, because the system of equations (8) has four real solutions (α1 , β1 , γ1 , δ1 , f1 , g1 , h1 , k1 ) = (0.275692.., 0.311385.., 0.490206.., −0.598415.., −0.22775.., 0.200142.., −0.593285.., − 0.378363..), (α2 , β2 , γ2 δ2 , f2 , g2 , h2 , k2 ) = (0.391233.., 0.461461.., 0.535875.., −0.664112.., −0.324107.., 0.266457.., −0.647454.., −0.38443..) , (α3 , β3 , γ3 , δ3 , f3 , g3 , h3 , k3 ) = (0.480359.., 0.584453.., 0.577721.., −0.72566.., −0.399001.., 0.309322.., −0.696708.., −0.383691..) and (α4 , β4 , γ4 δ4 , f4 , g4 , h3 4, k4 ) = (0.555792.., 0.693247.., 0.616543.., −0.783893.., −0.46287.., 0.339233.., −0.742.., −0.376889..).  3. Proof of Theorem 3 Proof of statement (a) of Theorem 3. First we prove the statement for the class C1 . We consider the first linear differential center in the region R2 √  √ √  x 113y 13 + 1 −187 3 + 192 13 − 64 ẋ = − − + , 7 3136 5376 (36) y B ẏ = x + + , 7 75264 √  √ √ √ √ √  with B = 13 + 1 −10651 13 + 896 14 − 1792 39 − 2618 42 + 896 182 + 18505 . This system has the first integral 15 10 4 2 5 -1 1 2 3 4 -2 -1 1 2 3 4 5 -2 -4 -5 -6 -10 (C1 ) (C3 ) Figure 10. The unique crossing limit cycle of the discontinuous piecewise linear differential system (C1 ) for (36)–(37), and (C3 ) for (43)–(44) contained in two zones. 4 5 2 2 4 6 8 10 1 2 3 4 -2 -5 -4 -10 (C21 ) (C22 ) Figure 11. The unique crossing limit cycle of the discontinuous piecewise linear differential system (C21 ) for (39)–(40), and (C22 ) for (41)–(42) contained in two zones. 1 √ √ √ H1 (x, y) = √ (3136( 13 − 1)x2 + y(113( 13 − 1)y + 14(187 3 784( √13 − 1) √ √ √ −192 √13 + 64)) +√x(896( 13√− 1)y − 10651 13 + 896 14 −1792 39 − 2618 42 + 896 182 + 18505)). The second linear differential center in the region R1 is (37) √ 751 −288 13 + √ + 96 x 25y 3 ẋ = − − − √ , 6 576 576( 13 − 1) √ √ √ √ √ y 6009 13 + 576 14 + 1152 39 − 1502 42 + 576 182 − 10515 ẏ = x + − √ , 6 3456( 13 − 1) 16 R.BENTERKI, L. DAMENE AND J. LLIBRE 10 5 -2 -1 1 2 3 4 5 -5 -10 (C4 ) Figure 12. The unique crossing limit cycle of the discontinuous piecewise linear differential system (45)–(46) contained in two zones. 1.0 4 2 0.5 -2 -1 1 2 3 4 -2 -1 1 2 -2 -4 -0.5 -6 -1.0 (C51 ) (C52 ) Figure 13. The unique crossing limit cycle of the discontinuous piecewise linear differential system (C51 ) for (47)–(48), and (C52 ) for (49)–(50) contained in two zones. this differential system has the first integral 1 √ √ √ H2 (x, y) = √ (1728( 13 − 1)x2 − x(−576( 13 − 1)y + 6009 13 432( √13 − 1) √ √ √ √ +576 14 + 1152 √ 39 −√1502 42 + 576 182 − 10515) + y(75( 13 −1)y + 2(751 3 − 864 13 + 288))). 17 6 6 4 4 2 2 2 4 6 2 4 6 -2 -2 -4 -4 -6 -6 (C1 ) (C3 ) 30 20 10 5 10 15 20 -10 -20 -30 (C4 ) Figure 14. The two crossing limit cycles of the discontinuous piecewise linear differential system (C1 ) for (51)–(52), (C3 ) for (58)– (59), and (C4 ) for (60)–(61) contained in two zones. For the piecewise linear differential system (36)–(37) the unique real solution of the system of equations H1 (α1 , β1 ) − H1 (γ1 , δ1 ) = 0, H1 (α2 , β2 ) − H1 (γ2 , δ2 ) = 0, H2 (α1 , β1 ) − H2 (α2 , β2 ) = 0, (38) H2 ((γ1 , δ1 ) − H2 (γ2 , δ2 ) = 0, ci (α1 , β1 ) = 0, ci (α2 , β2 ) = 0, ci (γ1 , δ1 ) = 0, ci (γ2 , δ2 ) = 0, √ √ √ √ when i = 1, is (α1 , β1 , α2 , β2 , γ1 , δ1 , γ2 , δ2 ) = (1, 3, 2, − 14, 2, 14, 3, − 39). Now we prove the statement for the class C2 . We consider the first linear differ- ential center in the region R1 x 5y y 1  √ √  (39) ẋ = − − 1, ẏ = x − + 12 3 + 6 10 − 151 , 6 18 6 18 this differential system has the first integral 18 R.BENTERKI, L. DAMENE AND J. LLIBRE 1.0 20 15 0.5 10 5 0.5 1.0 1.5 2.0 2.5 3.0 5 10 15 20 -0.5 -5 -10 -1.0 -15 (C21 ) (C22 ) Figure 15. The two crossing limit cycles of the discontinuous piecewise linear differential system (C21 ) for (54)–(55), and (C22 ) for (56)–(57) contained in two zones. 4 3 0.5 2 1 -1.5 -1.0 -0.5 0.5 1.0 -1.0 -0.5 0.5 1.0 1.5 2.0 -1 -0.5 -2 -1.0 -3 (C51 ) (C52 ) Figure 16. The two crossing limit cycles of the discontinuous piecewise linear differential system (C51 ) for (62)–(63), and (C52 ) for (64)–(65) contained in two zones. 2 √ √ H1 (x, y) =(18x2 + 2x(−3y + 12 3 + 6 10 − 151) + y(5y + 36)). 9 The second linear differential center in the region R2 is √ √ √ x 13y −82 3 − 205 10 − 96 30 − 5664 ẋ = − + , 6 144 4896 (40) √ q √  y −432 10 + 30 3936 10 + 24721 − 17964 ẏ = x − + , 6 2448 this system has the first integral 19 1 √ √ √ √ H2 (x, y) = √ √ (144(2 3 − 5 10)x2 − 4x(24 3y − 60 10y 36(2 √ 3 − 5 10) √ √ √ √ +1117√ 3 − 574√ − 2649 10 + 96 30 − 720) + y(26 3y − 65 10y +384 3 − 1632 10)). The unique real solution of the system of equations (38) for i = 2, for the√piecewise linear √ differential √ system √ (39)–(40) is (α1 , β1 , α2 , β2 , γ1 , δ1 , γ2 , δ2 ) = (4, 2 3, 5, − 2 10, 6, 3 10, 6, −3 10). For the second configuration of the class C2 , we consider the linear system √ √ √ x 41y 1440 3 − 640 10 − 160 33 + 2503 ẋ = − + √ √ √  , 5 400 3200 −3 3 + 2 10 + 33 (41) √ √ √2503 10 + 960 30 − 3200 160 10 + √ √ √ − 4390 y −3 3 + 2 10 + 33 ẏ = x − + , 5 6400 this differential system has the first integral 1 √ √ √ √ H1 (x, y) = √ √ √ (3200(−3 3 + 2 10 + 33)x2 + x(13170 3 800(−3√ 3 + 2 10 √ + 33) √ √ √ √ −6277 √ 10 + 480 30√− 4390√ 33 +√160 330 − 1280(−3 √ 3√+ 2 10+√ 33)y) + 2y(164(−3 3 + 2 10 + 33)y − 1440 3 + 640 10 + 160 33 −2503)). The second linear differential center in the region R2 is √ √ √ x 13y 360 3 − 160 10 − 40 33 + 579 ẋ = − + √ √ √  , 5 100 800 −3 3 + 2 10 + 33 (42) √ √ √ √ √ y −3873 3 + 4898 10 + 1440 11 + 480 30 + 2449 33 ẏ = x − − √ √ √  , 5 3200 −3 3 + 2 10 + 33 this system has the first integral 1 √ √ √ √ H2 (x, y) = √ √ √ 1600(−3 3 + 2 10 + 33)x2 − x(640(−3 3 400(−3 √ 3√+ 2 10 + 33) √ √ √ √ √ +2 10 + 33)y√ − 3873 √ 3√+ 4898 10 + √1440 11 √+ 480 √ 30 + 2449  33) +4y 52 −3 3 + 2 10 + 33 y − 360 3 + 160 10 + 40 33 − 579 . The unique real solution of the system of equations (38) for i = 2, for the√piecewise linear √ differential √ system √ (41)–(42) is (α1 , β1 , α2 , β2 , γ1 , δ1 , γ2 , δ2 ) = (1/4, 33/8, 1/2, − 5/(2 2), 3/4, 3 3/8, 1, 0). We prove the statement for the class C3 . We consider the first linear differential center in the region R1 √ √ x 25y −128 2 + 288 3 + 483 ẋ = − − + √ √ , 6 576 −384 2 + 576 3 + 1344 (43) √ q √ y −96618 2 + 9 3(58745128 2 + 89093337) − 500192 ẏ = x + + , 6 192384 this differential system has the first integral 20 R.BENTERKI, L. DAMENE AND J. LLIBRE 1 √ √ √ √ H1 (x, y) = √ √ (576(2 2 − 3 3 − 7)x2 + x(192(2 2 − 3 3 144(2 2 − 3√ 3 − 7) √ √ √ √ −7)y − 3770√ 2 + 6861√ 3 + 1152 6 + 14875) + y(25(2 2 − 3 3 −7)y − 768 2 + 1728 3 + 2898). The second linear differential center in the region R2 is √ √ √ x 145y 134932 2 − 69489 3 − 77682 6 + 250949 ẋ = − − + , 9 5184 577152 (44) √ √ √ y −23197 2 + 45834 3 − 1728 6 + 79814 ẏ = x + + √ √ , 9 5184(2 2 − 3 3 − 7) this system has the first integral 1 √ √ √ H2 (x, y) = √ √ √ (5184(− 2 − 9 3 + 3 6 − 10)x2 648(−2 2 − √ 18 3√+ 6 6√− 20) √ √ +x(1152(− √ 2 − 9 3 +√3 6 −√10)y + √7645 2 + 193608 √ 3 + 332692 √ −67863 √ 6) + y(145(− 2 − 9 3 + 3 6 − 10)y + 2304 2 + 10755 3 −6912 6 + 41343)). This piecewise linear differential centers has a unique real solution of the √ system√ of equations (38) for i = 3, which is (α1 , β1 , α2 , β2 , γ1 , δ1 , γ2 , δ2 ) = (1, 1, 2, −2 2, 3, 3 3, 4, −8). Now we prove the statement for the class C4 . We consider the linear differential center in the region R1 √ x 181y 2 y 2 2071 8 (45) ẋ = − − + , ẏ = x + + − + √ , 9 8100 9 9 3 540 3 3 this differential system has the first integral 2  √ √  y(181y − 3600) H1 (x, y) = 4x2 + x 60y + 180 2 + 480 3 − 2071 + . 135 2025 The second linear differential center in the region R2 is √ √ √ x y 66 2 − 62 3 − 18 6 + 525 ẋ = − − + , 8 32 1128 (46) √ q √  y −600 6 + 8 6 232 6 + 2555 − 12627 ẏ = x + + , 8 4512 this system has the first integral 1 √ √ √ H2 (x, y) = √ √ √ 32(−15 2 − 4 3 + 6 6 + 24)x2 + 8(−15 2√− 4 3√ + 6 6√ + 24) √ √ √ 2x(4(−15 2 − 4 √3 + 6 √6 + 24)y √ + 1521 2 + 556√ 3 − 666 √ 6 −2456) √ + y((−15 2 − 4 3 + 6 6 + 24)y − 8(−51 2 − 16 3 +24 6 + 76)). In this case the unique real solution of the system √ of√equations (38) for i = 4, is (α1 , β1 , α2 , β2 , γ1 , δ1 , γ2 , δ2 ) = (2, 2, 2, −2, 3, 3 2, 4, −4 3). 21 We prove the statement for the first configuration of the class C5 . We consider the linear differential center in the region R3 √ x 13y 53 2 + 1134 ẋ = − − + , 6 144 4896 (47) √ √ √ y −114 2 + 91 3 − 24 6 + 843 ẏ = x + + √ , 6 72( 2 − 6) this system has the first integral 1 √ √ √ √ H1 (x, y) = √ (144( 2 − 6)x2 − 4x(−12( 2 − 6)y + 114 2 − 91 3 36( √2 − 6) √ √ +24 6 − 843) + y(13( 2 − 6)y − 48 2 + 394)). The second linear differential center in the region R1 is x 5y 1 √  ẋ = − + −89 2 − 636 , 6 18 612 (48) q √ y 1 √ ẏ = x − + (432 3 + 178 3( 3 + 2) − 795), 6 306 this differential system has the first integral 1 √ √ √ √ H2 (x, y) = √ (36( 2 − 6)x2 − 4x(3( 2 − 6)y + 141 2 + 142 3 9( √2 − 6) √ √ +6 6 − 312) + 2y(5( 2 − 6)y + 6 2 − 214)). For this piecewise linear differential centers the √ unique real √ solution √ of system (38) when i = 5 is (α1 , β1 , α2 , β2 , γ1 , δ1 , γ2 , δ2 ) = (1, 2, 2, −2 3, 2, 2 3, 3, −6). Hence, the discontinuous piecewise linear differential system (47)–(48) has a unique crossing limit cycle, see (C51 ) of Figure 13. Finally we prove the statement for the second configuration of the class C5 .In the region R3 we consider the linear differential center x 5y 2 y 1 √ √ (49) ẋ = − + , ẏ = x − + (96 2 + 88 3 + 365), 2 4 5 2 640 this system has the first integral 1  √ √  1 H1 (x, y) = 4x2 + x −320y + 96 2 + 88 3 + 365 + y(25y − 16). 80 5 In the region R2 we consider the linear center √ √ √ x y −22000 2 − 3000 3 + 34816 5 + 24585 ẋ = − − + √ √ √ , 4 8 640(275 2 + 75 3 − 272 5 (50) p √ √ √ 4125(149 11 − 4 6 + 200) 2400 2 − 2200 3 − √ √ √ + 62593 y 275 2 + 75 3 − 272 5 ẏ = x + + , 4 108800 22 R.BENTERKI, L. DAMENE AND J. LLIBRE this system has the first integral 1 √ √ √ H2 (x, y) = √ √ √ (3200(275 2 + 75 3 − 272 5)x2 800(275 2 + 75 3 − 272 5) √ √ √ √ √ +x(1600(275√ 2 + 75 √3 − 272 5)y√ + 940225 √2 + 312300 3 √ −1001488 √ 5√− 25000 6 − √ 38400 10 √ + 35200 15) √ + 10y(40(275 2 +75 3 − 272 5)y + 22000 2 + 3000 3 − 34816 5 − 24585)). For this piecewise linear differential centers the unique √ real solution √of system (38) when √ i = 5 is (α1 , β1√, α2 , β2 , γ1 , δ1 , γ2 , δ2 ) = (−1/4, 3/8, −4/5, 4/(5 5), −1/2, − 1/(2 2), −4/5, −4/(5 5). Then the discontinuous piecewise linear differential system (49)–(50) has exactly one crossing limit cycle, see (C52 ) of Figure 13. This completes the proof of statement (a) of Theorem 3.  Proof of statement (b) of Theorem 3. First we prove the statement for class C1 . In the region R1 we consider the linear differential center (51) ẋ = −0.0327708x − 0.167012y + 0.202324, ẏ = x + 0.0327708y − 2.82026, its first integral is H1 (x, y) = 4(x + 0.0327708y)2 + 8(−2.82026x − 0.202324y) + 0.663752y 2 . The second linear differential center in the region R2 is x 13y 1 y 16 (52) ẋ = − − , ẏ = x − − , 10 50 5 10 5 its first integral is    y 2 y 16x H2 (x, y) = 4 x − +8 − + y2 . 10 5 5 For the piecewise linear differential system (51)–(52) the real solutions of the system of equations H1 (αi , βi ) − H1 (γi , δi ) = 0, H1 (fi , gi ) − H1 (hi , ki ) = 0, H2 (αi , βi ) − H1 (fi , gi ) = 0, (53) H2 (γi , δi ) − H1 (hi , ki ) = 0, cs (αi , βi ) = cs (γi , δi ) = 0, i = 1, 2, cs (fi , gi ) = cs (hi , ki ) = 0. when s = 1, is (α1 , β1 , γ1 , δ1 , f1 , g1 , h1 , k1 ) = (0.571172.., 1.04103.., 2.93153.., 6.0596.., 0.449711.., −0.861917.., 2.66907.., −5.36724..) and (α2 , β2 , γ2 , δ2 , f2 , g2 , h2 , k2 ) = (1.19066.., 2.07275.., 2.40283.., 4.69567.., 0.928007.., −1.60885.., 2.25189.., −4.32923..). Hence these linear differential centers have exactly two crossing limit cycles, see (C1 ) of Figure 14. Now we prove the statement for the class C2 . In the region R1 we consider the linear differential center (54) ẋ = −0.252669y, ẏ = x − 0.498272, its first integral is H1 (x, y) = 4x2 − 3.98617x + 1.01067y 2 . 23 The second linear differential center in the region R2 is y 1 (55) ẋ = − , ẏ = x − , 25 2 its first integral is 4y 2 H2 (x, y) = 4x2 − 4x + . 25 For these centers the real solutions of system (53) when s = 2 are (0.898773.., 0.43723.., 0.169784.., 0.631617.., 0.898773.., −0.43723.., 0.169784.., −0.631617..) and (0.758688.., 0.640578.., 0.368993.., 0.782685.., 0.758688.., −0.640578.., 0.368993.., −0.782685..). Hence the discontinuous piecewise linear differential system (54)–(55) has two crossing limit cycles, see (C21 ) of Figure 15. For the second configuration of the class C2 we consider the linear center (56) ẋ = 0.057143x − 0.0550264y − 0.0521176, ẏ = x − 0.057143y − 0.805757, in the region R1 , its first integral is H1 (x, y) = 4(x − 0.057143y)2 + 8(0.0521176y − 0.805757x) + 0.207044y 2 . The second linear differential center in the region R2 is x 17y 1 y 11 (57) ẋ = − − , ẏ = x − − , 10 80 10 10 10 its first integral is    y 2 y 11x 81y 2 H2 (x, y) = 4 x − +8 − + . 10 10 10 100 The real solutions of system (53) for these centers are (0.253068.., 0.283285.., 1.26272.., 1.89942.., 0.204637.., −0.224601.., 1.16927.., −1.72215..) and (0.145203.., 0.155388.., 1.36881.., 2.10673.., 0.119104.., −0.125997.., 1.24855.., −1.87223..). Hence the dis- continuous piecewise linear differential system (56)–(57) has two crossing limit cy- cles, see (C22 ) of Figure 15. For the class C3 and in the region R1 we consider the center (58) ẋ = 0.0333015x − 0.132045y − 0.0410184, ẏ = x − 0.0333015y − 1.97694, its first integral is H1 (x, y) = 4(x − 0.0333015y)2 + 8(0.0410184y − 1.97694x) + 0.523744y 2 . The second linear differential center in the region R2 is (59) ẋ = 0.1x − 0.26y − 0.2, ẏ = x − 0.1y − 2.3, its first integral is H2 (x, y) = 4(x − 0.1y)2 + 8(0.2y − 2.3x) + y 2 . The real solutions of system (53) when s = 3 are (0.340737.., 0.198898.., 2.70162.., 4.44056.., 0.308678.., −0.171498.., 2.52616.., −4.01504..) and (0.862714.., 0.801309.., 2.24826.., 3.37109.., 0.727481.., −0.620487.., 2.17185.., −3.2007..). Hence the discon- tinuous piecewise linear differential system (58)–(59) has two crossing limit cycles, see (C3 ) of Figure 14. We prove the statement for the class C4 . In the region R1 we consider the center (60) ẋ = −0.142331x − 0.0908868y + 1.59716, ẏ = x + 0.142331y − 9.92891, 24 R.BENTERKI, L. DAMENE AND J. LLIBRE its first integral is H1 (x, y) = 4(x + 0.142331y)2 + 8(−9.92891x − 1.59716y) + 0.282515y 2 . We consider the second linear differential center in the region R2 x y 7 y 44 (61) ẋ = − − , ẏ = x − − , 10 10 10 10 5 its first integral is    y 2 7y 44x 9y 2 H2 (x, y) = 4 x − +8 − + . 10 10 5 25 The two real solutions of system (53) when s = 4 are (2.34486.., 2.71929.., 9.31289.., 26.851.., 1.97922.., −1.95854.., 8.44735.., −23.0527..) and (4.45798.., 8.2899.., 7.62439.., 19.6236.., 3.20672.., −4.7636.., 7.33329.., −18.455..). Hence the discontinuous piece- wise linear differential system (60)–(61) has two crossing limit cycles, see (C4 ) of Figure 14. For the first configuration of the class C5 . We consider the linear differential center (62) ẋ = −0.453205y, ẏ = x + 0.497252, in the region R3 with its first integral H1 (x, y) = 4x2 + 3.97802x + 1.81282y 2 . We consider the second linear differential center in the region R2 y 1 (63) ẋ = − , ẏ = x + , 10 2 its first integral is 2y 2 H2 (x, y) = 4x2 + 4x + . 5 The two real solutions of system (53) when s = 5 are (−0.0927079.., 0.088306.., − 0.837019.., 0.337912.., −0.0927079.., −0.088306.., −0.837019.., −0.337912..) and (−0.217826.., 0.192647.., −0.663868.., 0.38489.., −0.217826.., −0.192647.., −0.663868.., −0.38489..). Hence, the discontinuous piecewise linear differential system (62)–(63) has two crossing limit cycles, see (C5 ) of Figure 16. Finally we prove the statement for the second configuration of the class C5 . In the region R3 we consider the linear differential center (64) ẋ = 0.057143x − 0.0550264y − 0.0521176, ẏ = x − 0.057143y − 0.805757, its first integral is H1 (x, y) = 4(x − 0.057143y)2 + 8(0.0521176y − 0.805757x) + 0.207044y 2 . We consider the second linear differential center in the region R1 x 17y 1 y 11 (65) ẋ = − − , ẏ = x − − , 10 80 10 10 10 its first integral is    y 2 y 11x 81y 2 H2 (x, y) = 4 x − +8 − + . 10 10 10 100 25 The two real solutions of system (53) when s = 5 are (0.253068.., 0.283285.., 1.26272.., 1.89942.., 0.204637.., −0.224601.., 1.16927.., −1.72215..) and (0.145203.., 0.155388.., 1.36881.., 2.10673.., 0.119104.., −0.125997.., 1.24855.., −1.87223..). Hence the dis- continuous piecewise linear differential system (64)–(65) has two crossing limit cy- cles, see (C52 ) of Figure 16.  References [1] J.C. Artes, J. Llibre, J.C. Medrado and M.A. Teixeira, Piecewise linear differential systems with two real saddles, Math. Comput. Simulation 95 (2014), 13–22. [2] S. Banerjee and G. Verghese, Nonlinear phenomena in power electronics. Attractors, bifurcations chaos and nonlinear control, Wiley-IEEE Press, New York, 2001. [3] R.Benterki and J. 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Li, The qualitative analysis of a class of plana Filippov systems, Nonlinear Anal. 73 (2010), 1277–1288. 1 and 2 Département de Mathématiques, Université Mohamed El Bachir El Ibrahimi, Bordj Bou Arréridj 34265, El Anasser, Algeria Email address: r.benterki@univ-bba.dz and damine.loubna@univ-bba.dz 3 Departament de Matematiques, Universitat Autònoma de Barcelona, 08193 Bel- laterra, Barcelona, Catalonia, Spain Email address: jllibre@mat.uab.cat