This is a preprint of: “The limit cycles of discontinuous piecewise linear differential systems formed by centers and separated by irreducible cubic curves I”, Rebiha Benterki, Jaume Llibre, Dyn. Contin. Discrete Impuls. Syst. Ser A: Math Anal., vol. 28, 153–192, 2021. THE LIMIT CYCLES OF PIECEWISE LINEAR DIFFERENTIAL SYSTEMS FORMED BY CENTERS AND SEPARATED BY IRREDUCIBLE CUBIC CURVES REBIHA BENTERKI1 AND JAUME LLIBRE2 Abstract. In the qualitative theory of the planar discontinuous piecewise linear differential systems one of the main problems is the study of the number of crossing limit cycles that these systems can have. We study the number of crossing limit cycles of discontinuous piecewise linear differential systems formed by centers and separated by an irreducible algebraic cubic curve. We prove that these differential systems only can exhibit 0, 1, 2 or 3 crossing limit cycles having two intersection points with the cubic of separation. 1. Introduction One of the first works studying the discontinuous piecewise linear differential systems in the plane is due to Andronov, Vitt and Khaikin in [1]. Later on these systems became a topic of great interest in the mathematical community due to their applications for modeling real phenomena, see for instance the books [3, 19] and references there quoted. To determine the non–existence, the existence of limit cycles and their number is one of the big problems in the qualitative theory of the planar differential systems, and in particular of the planar discontinuous piecewise linear differential systems separated by a curve Σ. In this work we are considering that a crossing limit cycle is a periodic orbit isolated in the set of all periodic orbits of the system which has exactly two points on the discontinuity curve Σ. The problem of finding the best upper bound for the maximum number of limit cycles that a family of piecewise linear differential systems in the plane separated by a straight line can have, has been studied by many authors recently, see for instance [2, 5, 7, 18]. Lum and Chua[16, 17] in 1990 conjectured that the continuous (but non–smooth) piecewise linear systems in the plane separated by one straight line have at most one limit cycle. This conjecture was proved by Freire et al [6] in 1998, for a shorter proof see [11]. Han and Zhang [8] in 2010 conjectured that discontinuous piecewise linear differential systems in the plane separated by a straight line have at most two crossing limit cycles. Huan and Yang [9] in 2012 provided a negative answer to this conjecture exhibiting a numerical example with three crossing limit cycles. Llibre and Ponce in [12] proved the existence of these three limit cycles analytically. Nowadays it remains as an open problem to know 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 AND J. LLIBRE if three is the maximum number of crossing limit cycles that this class of systems can have. In the paper [10] the authors considered the problem of Lum and Chua restricted to the class of discontinuous piecewise linear differential centers in the plane sepa- rated by a straight line, and they proved that those systems has no crossing limit cycles. But in [14, 15] were studied planar discontinuous piecewise linear differential centers with a curve of discontinuity different from a straight line, and then those systems can exhibit crossing limit cycles. For this reason it is interesting to study the role which plays the shape of the discontinuity curve in the number of crossing limit cycles that planar discontinuous piecewise linear differential centers can have. The objective of this paper is to study the maximum number N of crossing limit cycles of the discontinuous piecewise linear differential centers in R2 separated by an irreducible algebraic cubic curve. 1.1. Classification of the irreducible cubic polynomials. An algebraic cubic curve or simple 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 real 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. 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 of the curve C intersects C at least three times. 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. (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 2 2 c4 (x, y) = y − x (x − 1) = 0, or c5 (x, y) = y 2 − x2 (x + 1) = 0. See the graphics of the irreducible cubic curves ci = 0 for i = 1, . . . , 5 in Figures 1 to 5, respectively. 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]. 3 Figure 1. c1 (x, y) = 0. Figure 2. c2 (x, y) = 0. Figure 3. c3 (x, y) = 0. Figure 4. c4 (x, y) = 0. 1.2. Statement of the main results. 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. We recall that a limit cycle is a periodic solution of the discontinuous piecewise linear differential system isolated in the set of all periodic solutions of the system. In this paper we study the crossing limit cycles having exactly two points on the discontinuous cubic curve ck (x, y) = 0. Our first result provides the number of crossing limit cycles of the discontinuous piecewise linear differential systems formed by centers and separated by the irre- ducible cubic curves c1 , c3 and c4 . We note that such piecewise systems are formed by two pieces in each one there is a linear differential center. Theorem 2. The following statements hold. (a) There are systems of the classes C1 , C3 and C4 without crossing limit cycles. 4 R. BENTERKI AND J. LLIBRE Figure 5. c5 (x, y) = 0. (b) There are systems of the classes C1 , C3 and C4 exhibiting exactly one cross- ing limit cycle, see Figures 6, 7 and 8, respectively. (c) There are systems of the classes C1 , C3 and C4 exhibiting exactly two cross- ing limit cycles, see Figures 9, 10 and 11, respectively. These classes can’t exhibit the configurations 12, 13 and 14, respectively, because the ellipses of every center are concentric. (d) There are systems of the classes C1 , C3 and C4 exhibiting exactly three crossing limit cycles, see Figures 15, 16 and 17, respectively. (e) Every system of the class Ck for k = 1, 3, 5 can exhibit at most three crossing limit cycles. Theorem 2 is proved in section 2. Now we give our second main result which provides the number of crossing limit cycles of the discontinuous piecewise linear differential systems formed by centers and separated by the irreducible cubic curves c2 and c5 . We note that such piecewise systems are formed by three pieces in each one there is a linear differential center. Theorem 3. The following statements hold. (a) There are systems of the classes C2 and C5 without crossing limit cycles. (b) There are systems of classes C2 and C5 exhibiting exactly one crossing limit cycle. Each class has three possible different configurations for the limit cycle, see Figures 18, 19 and 20 for the class C2 , and Figures 21, 22 and 23 for the class C5 . (c) There are systems of classes C2 and C5 exhibiting exactly two crossing limit cycles. We get six possible configurations for the class C2 , see Figures 24, 25, 26, 27, 28 and 29. And six possible configurations for the class C5 , see Figures 30, 31, 32, 33, 34 and 35. The class C2 can’t exhibit the configurations 36 and 37, and the class C5 can’t exhibit the configurations 38 and 39, because the ellipses of every center are concentric. (d) There are systems of the classes C2 and C5 exhibiting exactly three crossing limit cycles, see Figures 40 and 41, respectively. 5 (e) The systems of the classes C2 and C5 can exhibit at most three crossing limit cycles. Theorem 3 is proved in section 3. Figure 6. The unique limit cycle of the discontinuous piecewise linear differential system (1)–(2). Figure 7. The unique limit cycle of the discontinuous piecewise linear differential system (4)–(5). 2. Proof of Theorem 2 We separate the proof of Theorem 2 by its statements. We remark that since the proofs of statements (a) and (e) of Theorems 2 and 3 are the same, we provide them in this section. Proof of statement (a) of Theorems 2 and 3. It is sufficient to take in each piece the same linear differential center, for instance the center ẋ = −y, ẏ = x. Then the discontinuous piecewise linear differential system (in this case continuous but separated by the cubic ck (x, y) = 0) has a continuum of periodic orbits and no limit cycles.  6 R. BENTERKI AND J. LLIBRE Figure 8. The unique limit cycle of the discontinuous piecewise linear differential system (7)–(8). Figure 9. Two limit cycles of the discontinuous piecewise linear dif- ferential system (10)–(11). Proof of statement (b) of Theorem 2. First we prove the statement for class C1 when b = 1. We consider the first linear differential center x y 3 y (1) ẋ = − − + 1 + √ , ẏ = x + − 2, 2 2 2 14 2 in the region c1 (x, y) > 0. This system has the first integral  √  ! y 2 6 + 4 14 y H1 (x, y) = 4 x + + 8 −2x − √ + y2 . 2 4 14 Now we consider the second linear differential center x y 13 y 1 (2) ẋ = − − + 1 + √ , ẏ = x + + , 2 2 2 14 2 2 in the region c1 (x, y) < 0. This differential system has the first integral y x 13 H2 (x, y) = 4(x + )2 + 8( − (1 + √ )y) + y 2 . 2 2 2 14 7 Figure 10. Two limit cycles of the discontinuous piecewise linear dif- ferential system (12)–(13). Figure 11. Two limit cycles of the discontinuous piecewise linear dif- ferential system (14)–(15). The discontinuous piecewise linear differential system formed by the linear differ- ential centers (1)–(2) has exactly one crossing limit cycle, because the system of equations Hl (α, β) − Hl (γ, δ) = 0, Hk (α, β) − Hk (γ, δ) = 0, (3) β 2 − α(α2 + α + 1) = 0, δ 2 − γ(γ 2 + γ + 1) = 0, √ when l = 1 and k = 2, has a unique real solution (α, β, γ, δ) = (0, 0, 2, 14). We prove the statement for the class of discontinuous piecewise linear differential systems C3 . We define two different linear differential centers, the first one is 1  √  y 1 (4) ẋ = −72x − 90y + 295 3 + 216 , ẏ = x + + , 288 4 6 8 R. BENTERKI AND J. LLIBRE Figure 12 Figure 13 Figure 14 Figure 15. Three limit cycles of the discontinuous piecewise linear differential system (16)–(17). 9 Figure 16. Three limit cycles of the discontinuous piecewise linear differential system (18)–(19). Figure 17. Three limit cycles of the discontinuous piecewise linear differential system (20)–(21). in the region c3 (x, y) > 0. It has the first integral  √  ! y 2 x 36 3 − 78 y y2 H1 (x, y) = 4 x − +8 + √ + . 2 2 24 3 9 The second linear differential center is 1  √  y 1 (5) ẋ = −72x − 90y + 295 3 + 216 , ẏ = x + + , 288 4 6 in the region c3 (x, y) < 0. It has the first integral      y 2 x 1 295 √ H2 (x, y) = 4 x + +8 + √ − − 18 3 y + y 2 . 4 6 24 3 4 10 R. BENTERKI AND J. LLIBRE For the discontinuous piecewise linear differential system (4)–(5) the unique solution of the system of equations Hl (α, β) − Hl (γ, δ) = 0, Hk (α, β) − Hk (γ, δ) = 0, (6) β 2 − α3 = 0, δ 2 − γ 3 = 0, √ when l = 1 and k = 2, is (α, β, γ, δ) = (0, 0, 3, 3 3). This proves the statement for the class C3 . Finally we prove the statement for the class C4 and we define two different linear differential centers, the first one is (7) ẋ = −3 + x − (37y)/36, ẏ = 1 + x − y in the region c4 (x, y) < 0. It has the first integral y2 H1 (x, y) = 4(x − y)2 + 8(x + 3y) + . 9 The second linear differential center is x 5y 3 y (8) ẋ = − − + , ẏ = x + − 1, 2 4 2 2 in the region c4 (x, y) > 0. It has the first integral    y 2 3y H2 (x, y) = 4 x + + 8 −x − + 4y 2 . 2 2 For the piecewise linear differential system (7)–(8) the unique real solution of the system of equations Hl (α, β) − Hl (γ, δ) = 0, Hk (α, β) − Hk (γ, δ) = 0, (9) β 2 − α2 (α − 1) = 0, δ 2 − γ 2 (γ − 1) = 0, √ √ when l = 1 and k = 2, is (α, β, γ, δ) = (3, 3 2, 3, −3 2). This completes the proof of the statement (b) of Theorem 2.  Proof of statement (c) of Theorem 2. For the class C1 we consider the linear dif- ferential center 5y 1  √  (10) ẋ = −x − + 3, ẏ = x + y + 8 14 − 86 , 4 16 in the region c1 (x, y) > 0. This differential system has the first integral     1 √ H3 (x, y) = 4(x + y)2 + 8 8 14 − 86 x − 3y + y 2 . 16 The other linear differential center is   x 5y 3 y 1 √ 67 (11) ẋ = − − + , ẏ = x + + 4 14 − , 2 16 2 2 16 2 in the region c1 (x, y) < 0. It has the first integral      y 2 1 √ 67 3y y2 H4 (x, y) = 4 x + +8 4 14 − x− + . 2 16 2 2 4 11 This discontinuous piecewise differential system formed by the linear differential centers (10)–(11) has exactly two crossing limit cycles, √ because √ the real solutions √ of the system (3), when l = 3 and k = 4 are (3, − 39, 3, 39) and (0, 0, 2, 14). For the class C3 we define the linear differential center √ 1  √  55 3 1071 (12) ẋ = −52x − 65y − 55 3 + 376 , ẏ = x + y + − , 52 13 104 in the region c3 (x, y) < 0, which has the first integral    1  √  1  √ H3 (x, y) = 4(x + y)2 + 8 440 3 − 1071 x + 55 3 − 376 y + y 2 . 104 52 The second linear differential center is 1  √  y 5  √  (13) ẋ = 13x − 13y − 5 3 − 19 , ẏ = x − + 8 3 − 45 , 26 2 52 in the region c3 (x, y) > 0, which has the first integral     y 2 5 √  1  √  H4 (x, y) = 4 x − +8 8 3 − 45 x + 19 + 5 3 y + y 2 . 2 52 26 This discontinuous piecewise differential system formed by the linear differential centers (12)–(13) has exactly two crossing limit cycles, because the system of equa- √ tions (6), when √ l = 3 and k = 4 has exactly two real different solutions (0, 0, 3, 3 3) and (3, −3 3, 4, 8). Hence, the statement holds for this class. Now we prove the statement for the class C4 . We define the linear differential center √ √ √ x 41y −72096 + 6009 2 + 17224 7 + 3072 14 ẋ = − + , 10 1600 150400 (14) √ √ √ y −444982 + 18657 2 + 35592 7 + 2892 14 ẏ = x − + , 10 75200 in the region c4 (x, y) < 0, which has the first integral  √ √ √ y 2 −444982 + 18657 2 + 35592 7 + 2892 14 H3 (x, y) = 4 x − + x 10 9400 √ √ √ −72096 + 6009 2 + 17224 7 + 3072 14 + . 18800 The second linear differential center is √ √ x 5y 9475 − 216 2 − 3584 7 ẋ = − + √ √ , 8 64 64 −160 + 9 2 + 56 7 (15) √ √ √ y 51355 − 3417 2 − 18548 7 + 480 14 ẏ = x − + √ √  , 8 32 −160 + 9 2 + 56 7 12 R. BENTERKI AND J. LLIBRE in the region c4 (x, y) > 0, which has the first integral  √ √ √ y 2 (51355 − 3417 2 − 18548 7 + 480 14)x H4 (x, y) = 4 x − +8 √ √ 8 32(−160 + 9 2 + 56 7) √ √ ! (9475 − 216 2 − 3584 7)y y2 − √ √ + . 64(−160 + 9 2 + 56 7) 4 The discontinuous piecewise linear differential system formed by the linear differ- ential centers (14)–(15) has exactly two crossing limit cycles, because the system of equations √ (9), when l = 3 √ and k = 4 has exactly two real different solutions (3, 3 2, 10, 30) and (5, 10, 8, 8 7). This completes the proof of statement (c) of Theorem 2.  Proof of statement (d) of Theorem 2. For the class C1 we consider in the region c1 (x, y) > 0 the linear differential center (16) √ √ √ √ √ √ √ √ x (14 3 − 18 7 − 14 14 + 28 21 − 14 39 + 42 + 6 91 + 546)y ẋ = + √ √ √ √ + 14 140 3 − 98 14 + 196 21 − 140 39 A + √ √ √ √ √ √ √ √ , 7 3( 13 − 1)( 3 + 14)(10 3 − 7 14 + 14 21 − 10 39) y B ẏ = x − + √ √  √ √ √ √ 14 14 3 13 − 1 (10 3 − 7 14 + 14 21 − 10 39) √ √ √ √ −420 91 + 21 182 − 258 273 + 28 546 + √ √  √ √ √ √ , 14 3 13 − 1 (10 3 − 7 14 + 14 21 − 10 39) √ √ √ √ √ √ √ where √ A = 63 √ + 588 2√+ 502 3 − √63 13 + 579 14 + 84 √ 21 − 588 √26 + 38√ 39 + 21 √42 − 39 √182 − 84 √273 − 21√ 546 and √ B = 63 √+ 588 2√+ 502 3 √ − 63 13 + 579 14 + 84 21 − 588 26 + 38 39 + 21 42 − 39 182 − 84 273 − 21 546. This differential system has the first integral H5 (x, y) equal to C √ √ √ √ √ 420 + 42 7 − 60 13 − 7 42 − 42 91 + 7 546 √ √ √ √ √ √ √ y((−42 − 6 7 + 6 13 + 42 + 6 91 − 546)y − 6 3( 13 − 1)) + √ √ √ √ √ 420 + 42 7 − 60 13 − 7 42 − 42 91 + 7 546 √ √ √ √ √ √ √ √ 8x((−294 2 − 322 3 − 399 14 − 42 21 + 294 26 − 38 39 + 39 182 + 42 273)y + √ √ √ √ √ √ √ √ 14 3( 13 − 1)( 3 + 14)(−10 3 + 7 14 − 14 21 + 10 39) √ √ √ √ √ √ √ −3(3286 + 1722 6 + 1046 7 + 439 42) + 3 13(−38 + 798 6 + 650 7 + 13 42)) + √ √ √ √ √ √ √ √ 14 3( 13 − 1)( 3 + 14)(−10 3 + 7 14 − 14 21 + 10 39) √ √ √ √ √ √ √ √ +16y(3(42 + 56 6 + 24 7 + 3 42 − 14 78 − 6 91)y − 84 273 − 39 182 + 38 39 + √ √ √ √ √ √ √ √ 14 3( 13 − 1)( 3 + 14)(−10 3 + 7 14 − 14 21 + 10 39) √ √ √ √ √ −588 26 + 84 21 + 579 14 + 502 3 + 588 2) + √ √ √ √ √ √ √ √ , 14 3( 13 − 1)( 3 + 14)(−10 3 + 7 14 − 14 21 + 10 39) 13 √ √ √ √ √ √ where √ C= √ 4((420 + √42 7√− 60 13 − 7 42 − 42 91 + 7 546)x2 + 4(−204 − 15 7 + 33 13 + 42 + 15 91 − 546)x. The other linear differential center in the region c1 (x, y) < 0 is (17) √ √ √ √ √ √ √ √  x 6 3 − 18 7 − 6 14 + 12 21 − 6 39 + 42 + 6 91 + 546 y ẋ = + √ √ √ √ 6 60 3 − 42 14 + 84 21 − 60 39 D + √ √ √ √ √ √ √ , 3(−322 − 98 6 − 42 7 − 38 13 − 133 42 + 98 78 + 42 91 + 13 546) y E ẏ = x − + √ √ √ √ √ √ 6 2 3( 13 − 1)(10 3 − 7 14 + 14 21 − 10 39) √ √ √ √ −60 91 + 7 182 − 86 273 + 4 546 + √ √ √ √ √ √ , 2 3( 13 − 1)(10 3 − 7 14 + 14 21 − 10 39) √ √ √ √ √ √ √ where √ D = √ 502 + 9 √3 + 196 √6 + 84 7 + 38 13 + 9 14√ − 9 39√+ 193 √42 − 196√78 − 84 √91 − 9 182 √ − 13 546 √ and E√= 816 + √ 196 2 + 260 3 + 60 7− 132 13 + 179 14 + 218 21 − 196 26 − 20 39 − 4 42. It has the first integral H6 (x, y) equal to 1 √ √ √ √ √ (12(−47421 − 44394 6 + 3612 7 − 31207 42 + 7 13(−531 + 3066 6 F √ √ √ √ √ +852 7 + 703 42))x2 ) + 4x((47421 + 44394 6 − 3612 7 + 3717 13 √ √ √ √ √ +31207 42 − 21462 78 − 5964 91 − 4921 546)y − 3(−98442 − 264782 2 √ √ √ √ √ √ −330296 3 − 89628 6 − 15774 7 + 5562 13 − 170752 14 − 46296 21 √ √ √ √ √ √ +122066 26 + 45578 39 − 53461 42 + 37968 78 + 13038 91 + 28726 182 √ √ √ √ √ +30528 273 + 9697 546) + 4y(−3(−5643 + 15162 2 + 7791 3 − 5082 6 √ √ √ √ √ √ √ +1056 7 − 567 13 + 5481 14 + 1088 21 − 4326 26 − 1743 39 − 3751 42 √ √ √ √ √ √ +2562 78 + 636 91 − 1413 182 − 464 273 + 559 546)y + 15128 546 √ √ √ √ √ √ +378 273 + 1971 182 + 21672 91 + 73206 78 − 103196 42 + 1278 39 √ √ √ √ √ √ +2646 26 − 2646 21 − 4077 14 + 1494 13 + 7056 7 − 142002 6 √ √ +774 3 − 18522 2 − 210258), √ √ √ √ √ √ √ where √ F =√ 816 + 196 2 + 260 3 + 60 7 − 132 13 + 179 14 + 218 21 − 196 26 − 20 39 − 4 42. This discontinuous piecewise differential system formed by the linear differential centers (16)–(17) has exactly three crossing limit cycles, because the real solutions of the√system√ (3), when√l = 5 √ and k = 6 has √ exactly√three real different solutions (1, − 3, 2, 14), (2, − 14, 3, 39) and (3, − 39, 4, 84). See this configuration in Figure 15. 14 R. BENTERKI AND J. LLIBRE For the class C3 we consider the linear differential center (18) √ √ √ √ √ √  −2153 + 2068 2 + 2283 3 + 1704 6 x 61 + 4 2 + 9 3 + 6 6 y ẋ = + − √ √  , 7430 5 45 −9 + 4 2 + 3 3 √ √ √ y −46751 + 11159 2 + 14823 3 + 10803 6 ẏ = x − + ,, 5 66870 in the region c3 (x, y) > 0. This differential system has the first integral H5 (x, y) equal to 4 √ √ √ √ √ √ √ (45(26 − 11 2 − 15 3 + 9 6)x2 + (−2079 + 967 2 45(26 − 11 2 − 15 3 + 9 6) √ √ √ √ √ √ +1413 3 − 633 6)x − 18(26 − 11 2 − 15 3 + 9 6)xy + (18 + 86 2 √ √ √ √ √ p √ +99 3 + 12 6)y 2 + 9(151 − 74 2 − 108 3 + 42 6 − 5 35 + 12 6)y). The other linear differential center is (19) √ √ √  √ √ √ −7 + 28 2 + 21 3 + 6 6 y −2549 + 1460 2 + 1767 3 + 1272 6 ẋ = x − √ √  + , 9 −9 + 4 2 + 3 3 1486 √ √ √ −26171 + 7723 2 + 11379 3 + 8127 6 ẏ = x − y + , 13374 in the region c3 (x, y) < 0. It has the first integral H6 (x, y) equal to 4 √ √ √ √ √ √ (9(26 − 11 2 − 15 3 + 9 6)x2 + (−1243 9(26 − 11 2 − 15 3 + 9 6) √ √ √ √ √ √ +579 2 + 873 3 − 357 6)x − 18(26 − 11 2 − 15 3 + 9 6)xy + (154 √ √ √ √ √ √ p √ +6 2 − 9 3 + 60 6)y 2 + 9(147 − 74 2 − 108 3 + 42 6 − 35 + 12 6)y). This discontinuous piecewise differential system formed by the linear differential centers (18)–(19) has exactly three crossing limit cycles, because the real solutions of the system (6), when √ l =√ 5 and k = 6, has √ exactly three real different solutions are (0, 0, 1, 1), (2, −2 2, 3, 3 3) and (3, −3 3, 4, 8). See Figure 16. For the class C4 we consider the linear differential center 2x 1 √ √ 4 ẋ = + (33 − 114 2 + 32 3)y − , 9 963 9 (20) √ √ 2y −5751 − 2136 2 + 4384 3 ẏ = x − + , 9 1926 in the region c4 (x, y) > 0. This differential system has the first integral 4 8 √ √ H5 (x, y) = (x(107x − 539) − 7y 2 ) − (2x(107y − 1096 3 + 534 2 + 225) 107 963 √ √ −y((15 + 57 2 − 16 3)y + 428)). 15 The other linear differential center is x 1 √ √ 2 ẋ = + (34 − 57 2 + 16 3)y − , 7 749 7 (21) √ √ y −4223 − 1068 2 + 2192 3 ẏ = x − + , 7 1498 in the region c4 (x, y) < 0. It has the first integral H6 (x, y) equal to 4 4 √ √ (x(107x − 539) − 7y 2 ) − (2x(107y − 1096 3 + 534 2 + 225) 107 749 √ √ −y((15 + 57 2 − 16 3)y + 428)). This discontinuous piecewise differential system formed by the linear differential centers (20)–(21) has exactly three crossing limit cycles, because the system of equations 9, when l√= 5 and √ k = 6, has exactly √ three real different solutions (2, −2, 2, −2), (3, −3 2, 4, 4 3) and (5, −10, 9, 18 2). See Figure 17.  Proof of statement (e) of Theorems 2 and 3. We note that for the irreducible cubic curves with c2 (x, y) = 0 and c5 (x, y) = 0 which separated the plane into three connected components, the crossing limit cycles having only two points on the cubic curve only can be contained in two of these connected components, so in their definition only appear two linear differential centers. Therefore the proof of statement (e) is the same for all classes of discontinuous piecewise linear differential systems Ci for i = 1, . . . , 5. Due to Lemma 1 of [13] any linear differential center in R2 can be written into the form 1  (22) ẋ = −bx − y 4b2 + ω 2 + d, ẏ = x + by + c, 4 with ω > 0. This differential system has the first integral H7 (x, y) = 4(by + x)2 + 8(cx − dy) + ω 2 y 2 . We consider a second arbitrary linear differential center 1  (23) ẋ = −Bx − y 4B 2 + Ω2 + D, ẏ = x + By + C, 4 with Ω > 0. This differential system has the first integral H8 (x, y) = 4(By + x)2 + 8(Cx − Dy) + Ω2 y 2 . In order that the discontinuous piecewise linear differential system formed by the linear differential centers (22)–(23), separated by some irreducible cubic curve ck (x, y) = 0 with k ∈ {1, 2, 3, 4, 5}, has three crossing limit cycles intersecting the curve ck (x, y) = 0 in two different points (αi , βi ) and (γi , δi ), for i = 1, 2, 3, these points must satisfy the system of equations H7 (α, β) = H7 (γ, δ), H8 (α, β) = H8 (γ, δ), (24) ck (α, β) = 0, ck (γ, δ) = 0. 16 R. BENTERKI AND J. LLIBRE From the fact that the points (α1 , β1 ) and (γ1 , δ1 ) satisfy system (24), we can obtain the following values for the parameters d and D: 4α12 + 4b2 β12 − 4b2 δ12 + 8α1 bβ1 − 8bγ1 δ1 + β12 ω 2 + 8α1 c − 8cγ1 − 4γ12 − δ12 ω 2 d= , 8(β1 − δ1 ) and we get the same expression for D by changing (b, c, ω) in the expression of d by (B, C, Ω). Now since the points (α2 , β2 ) and (γ2 , δ2 ) also satisfy the system (24) we get the following values for the parameters c and C: −1 c= 8(−α1 β2 + α1 δ2 + α2 β1 − α2 δ1 − β1 γ2 + β2 γ1 − γ1 δ2 + γ2 δ1 ) (−4α12 β2 + 4α12 δ2 + 4α22 β1 − 4α22 δ1 − 4b2 β12 β2 + 4b2 β12 δ2 + 4b2 β1 β22 −4b2 β1 δ22 − 4b2 β22 δ1 + 4b2 β2 δ12 − 4b2 δ12 δ2 + 4b2 δ1 δ22 − 8α1 bβ1 β2 +8α1 bβ1 δ2 + 8α2 bβ1 β2 − 8α2 bβ2 δ1 − 8bβ1 γ2 δ2 + 8bβ2 γ1 δ1 −8bγ1 δ1 δ2 + 8bγ2 δ1 δ2 − β12 β2 ω 2 + β12 δ2 ω 2 + β1 β22 ω 2 − 4β1 γ22 −β1 δ22 ω 2 − β22 δ1 ω 2 + 4β2 γ12 + β2 δ12 ω 2 − 4γ12 δ2 + 4γ22 δ1 d − δ12 δ2 ω 2 + δ1 δ22 ω 2 ). If instead of (ω, b) in the expression of c we replace them by (Ω, B), then we get the expression of C. The last step is to use that (α3 , β3 ) and (γ3 , δ3 ) satisfy system (24), then we obtain the expressions of ω 2 = G/H and Ω2 , where G= −4(b2 β3 α2 β12 − b2 δ3 α2 β12 − b2 β2 α3 β12 + b2 δ2 α3 β12 − b2 β3 γ2 β12 +b2 δ3 γ2 β12 + b2 β2 γ3 β12 − b2 δ2 γ3 β12 − a2 α2 α32 β1 − 2abδ2 α3 γ2 β1 +a2 α2 γ32 β1 − a2 γ2 γ32 β1 − b2 β32 α2 β1 + b2 δ32 α2 β1 + 2abβ3 α1 α2 β1 −2abδ3 α1 α2 β1 + b2 β22 α3 β1 − b2 δ22 α3 β1 + a2 α22 α3 β1 − 2abβ2 α1 α3 β1 +2abδ2 α1 α3 β1 + 2abβ2 α2 α3 β1 − 2abβ3 α2 α3 β1 + b2 β32 γ2 β1 − b2 δ32 γ2 β1 +a2 α32 γ2 β1 − 2abβ3 α1 γ2 β1 + 2abδ3 α1 γ2 β1 + 2abβ3 α3 γ2 β1 − a2 α3 γ22 β1 −b2 β22 γ3 β1 + b2 δ22 γ3 β1 − a2 α22 γ3 β1 + a2 γ22 γ3 β1 + 2abβ2 α1 γ3 β1 −2abδ2 α1 γ3 β1 − 2abβ2 α2 γ3 β1 + 2abδ3 α2 γ3 β1 + 2abδ2 γ2 γ3 β1 −a2 β3 α1 α22 + a2 δ3 α1 α22 + a2 β2 α1 α32 − a2 δ2 α1 α32 + a2 δ1 α2 α32 −a2 β3 α2 γ12 + a2 δ3 α2 γ12 + a2 β2 α3 γ12 − a2 δ2 α3 γ12 + 2abβ2 δ3 α1 α2 −a2 δ3 α1 γ22 + a2 δ1 α3 γ22 − a2 β3 γ1 γ22 + a2 δ3 γ1 γ22 − a2 β2 α1 γ32 + a2 β3 α1 γ22 17 +a2 δ2 α1 γ32 − a2 δ1 α2 γ32 + a2 β2 γ1 γ32 − a2 δ2 γ1 γ32 + a2 δ1 γ2 γ32 + 2abβ2 δ1 α3 γ1 +b2 β2 β32 α1 + b2 β3 δ22 α1 − b2 β2 δ32 α1 + b2 δ2 δ32 α1 − 2abδ3 γ2 γ3 β1 −b2 β32 δ2 α1 + b2 β22 δ3 α1 − b2 δ22 δ3 α1 − b2 β3 δ12 α2 − b2 δ1 δ32 α2 − b2 β22 β3 α1 +a2 β3 α12 α2 − a2 δ3 α12 α2 + b2 β32 δ1 α2 + b2 δ12 δ3 α2 − 2abβ2 β3 α1 α2 +b2 β2 δ12 α3 + b2 δ1 δ22 α3 − a2 β2 α12 α3 + a2 δ2 α12 α3 − a2 δ1 α22 α3 − a2 δ3 α22 γ1 −b2 β22 δ1 α3 − b2 δ12 δ2 α3 + 2abβ2 β3 α1 α3 − 2abβ3 δ2 α1 α3 − 2abβ2 δ1 α2 α3 −b2 β2 β32 γ1 − b2 β3 δ22 γ1 + b2 β2 δ32 γ1 − b2 δ2 δ32 γ1 + a2 β3 α22 γ1 + 2abβ3 δ1 α2 α3 −a2 β2 α32 γ1 + a2 δ2 α32 γ1 + b2 β22 β3 γ1 + b2 β32 δ2 γ1 − b2 β22 δ3 γ1 + b2 δ22 δ3 γ1 +2abβ2 β3 α2 γ1 − 2abβ3 δ1 α2 γ1 − 2abβ2 δ3 α2 γ1 + 2abδ1 δ3 α2 γ1 − 2abβ2 β3 α3 γ1 +2abβ3 δ2 α3 γ1 − 2abδ1 δ2 α3 γ1 + b2 β3 δ12 γ2 + b2 δ1 δ32 γ2 − a2 β3 α12 γ2 + a2 δ3 α12 γ2 −a2 δ1 α32 γ2 + a2 β3 γ12 γ2 − a2 δ3 γ12 γ2 − b2 β32 δ1 γ2 − b2 δ12 δ3 γ2 + 2abβ3 δ2 α1 γ2 −2abδ2 δ3 α1 γ2 − 2abβ3 δ1 α3 γ2 + 2abδ1 δ2 α3 γ2 + 2abβ3 δ1 γ1 γ2 − 2abβ3 δ2 γ1 γ2 +2abδ2 δ3 γ1 γ2 − b2 β2 δ12 γ3 − b2 δ1 δ22 γ3 + a2 β2 α12 γ3 − a2 δ2 α12 γ3 + a2 δ1 α22 γ3 −a2 β2 γ12 γ3 + a2 δ2 γ12 γ3 − a2 δ1 γ22 γ3 + b2 β22 δ1 γ3 + b2 δ12 δ2 γ3 − 2abβ2 δ3 α1 γ3 +2abδ2 δ3 α1 γ3 + 2abβ2 δ1 α2 γ3 − 2abδ1 δ3 α2 γ3 − 2abβ2 δ1 γ1 γ3 + 2abδ1 δ2 γ1 γ3 +2abβ2 δ3 γ1 γ3 − 2abδ2 δ3 γ1 γ3 − 2abδ1 δ2 γ2 γ3 + 2abδ1 δ3 γ2 γ3 − 2abδ1 δ3 γ1 γ2 ) and H = −α3 β12 β2 + γ3 β12 β2 + α2 β12 β3 − γ2 β12 β3 + α3 β12 δ2 − γ3 β12 δ2 − α2 β12 δ3 +γ3 β1 δ22 + γ2 β12 δ3 + α3 β1 β22 − γ3 β1 β22 − α2 β1 β32 + γ2 β1 β32 − α3 β1 δ22 +α2 β1 δ32 − γ2 β1 δ32 − α1 β22 β3 + γ1 β22 β3 − α3 β22 δ1 + γ3 β22 δ1 + α1 β22 δ3 −γ1 β22 δ3 + α1 β2 β32 − γ1 β2 β32 + α3 β2 δ12 − γ3 β2 δ12 − α1 β2 δ32 + γ1 β2 δ32 −γ2 β32 δ1 − α1 β32 δ2 + γ1 β32 δ2 − α2 β3 δ12 + γ2 β3 δ12 + α1 β3 δ22 + α2 β32 δ1 −γ1 β3 δ22 − α3 δ12 δ2 + γ3 δ12 δ2 + α2 δ12 δ3 − γ2 δ12 δ3 + α3 δ1 δ22 − γ3 δ1 δ22 −α2 δ1 δ32 + γ2 δ1 δ32 − α1 δ22 δ3 + γ1 δ22 δ3 + α1 δ2 δ32 − γ1 δ2 δ32 . Again we get Ω2 to changing b by B in the expression of ω 2 . Substituting the obtained values of d, c and ω into the first integral H7 (x, y) it takes the form H7 (x, y) = f (x, y, α1 , α2 , α3 , β1 , β2 , β3 , γ1 , γ2 , γ3 , δ1 , δ2 , δ3 ) +bg(x, y, α1 , α2 , α3 , β1 , β2 , β3 , γ1 , γ2 , γ3 , δ1 , δ2 , δ3 ). The expression of the first integral H8 (x, y) is the same than the expression of H7 (x, y) changing b by B. Now assume that we have a fourth limit cycle which intersect the cubic in the points (α4 , β4 ) and (γ4 , δ4 ). Then we must have (25) H7 (α4 , β4 ) = H7 (γ4 , δ4 ), and H8 (α4 , β4 ) = H8 (γ4 , δ4 ). From the expressions of H7 , H8 and the equations (25) it follows that b = B, and consequently the first integrals H7 and H8 are equal. Then the two linear centers defining the piecewise linear differential system are the same, and consequently the system is linear and has no limit cycles in contradiction, that we are assuming that it has four limit cycles. So the piecewise linear differential systems here studied have at most three limit cycles.  3. Proof of Theorem 3 This section is dedicated to prove statements (b), (c) and (d) of Theorem 3. 18 R. BENTERKI AND J. LLIBRE Figure 18. The unique limit cycle of the discontinuous piecewise lin- ear differential system (26)–(27). Figure 19. The unique limit cycle of the discontinuous piecewise lin- ear differential system (29)–(30). Figure 20. The unique limit cycle of the discontinuous piecewise lin- ear differential system (31)–(32). Proof of statement (b) of Theorem 3. First we prove the existence of three different configurations of one crossing limit cycle for the class C2 . 19 Figure 21. The unique limit cycle of the discontinuous piecewise lin- ear differential system (33)–(34). Figure 22. The unique limit cycle of the discontinuous piecewise lin- ear differential system (36)–(37). Figure 23. The unique limit cycle of the discontinuous piecewise lin- ear differential system (38)–(39). For the first possible configuration we consider the linear differential center x y 1 √ y 1 (26) ẋ = − − + (20 + 10), ẏ = x + − , 2 2 80 2 2 20 R. BENTERKI AND J. LLIBRE in the region c2 (x, y) > 0, with its first integral 1 H1 (x, y) = 4x2 + 4x(y − 1) + y(2y − √ − 2). 10 In the region c2 (x, y) < 0 we consider the linear center r 5 1 (27) ẋ = x − 2y + − , ẏ = 1 + x − y, 2 2 with its first integral √ H2 (x, y) = −4(2x + 10 − 1)y + 4x(x + 2) + 8y 2 . For the discontinuous piecewise linear differential system (26)–(27) the unique so- lution of the system of equations Hl (α, β) − Hl (γ, δ) = 0, Hk (α, β) − Hk (γ, δ) = 0, (28) β 2 − α(α − 1)(α − 3) = 0, δ 2 − γ(γ − 1)(γ − 3) = 0, √ √ when l = 1 and k = 2, is (α, β, γ, δ) = (0, 0, 1/2, 5/(2 2)). This proves the uniqueness of the crossing limit cycle. See this configuration in Figure 18. For the second possible configuration for the class C2 we consider in the region c2 (x, y) > 0 the linear differential center 1  √  y (29) ẋ = −20x − 25y + 9 10 + 100 , ẏ = x + − 6, 80 4 with its first integral √  !  y 2 −18 − 20 10 y H1 (x, y) = 4 x + +8 √ − 6x + y 2 . 4 16 10 In the region c2 (x, y) < 0 we consider the linear differential center 1  √  y (30) ẋ = 5x − 26y + 36 10 − 25 , ẏ = x − , 25 5 with its first integral   √ 1152  16 10 − y 2 5 H2 (x, y) = 4 x − + 4y 2 + √ y. 5 2 10 For this case the unique √ real solution of system (28), when l = 1 and k = 2, is (α, β, γ, δ) = (3, 0, 5, 2 10). Hence the discontinuous piecewise linear differential system (29)–(30) has a unique crossing limit cycle, see Figure 19. Finally we prove the statement for the last configuration of the class C2 . We consider the linear differential center x 29y 4 y (31) ẋ = − − , ẏ = x − + 1, 5 100 5 5 in the region c2 (x, y) < 0, with its first integral y 4y H1 (x, y) = 4(x − )2 + 8(x + ) + y 2 . 5 5 21 The second linear differential center is 13y (32) ẋ = −x − + 4, ẏ = −2 + x + y, 4 in the region c2 (x, y) > 0, with its first integral H2 (x, y) = 8(−2x − 4y) + 9y 2 + 4(x + y)2 . Since system √ (28), when l = 1 and k = 2, has the unique real solution (α, β, γ, δ) = √ (4, −2 3, 4, 2 3), the discontinuous piecewise linear differential system (31)–(32) has a unique crossing limit cycle, see it in Figure 20. The piecewise linear differential systems of the class C2 having one crossing limit cycle can exhibit this limit cycle in three different configurations, see Figures 18, 19 and 20. In short statement (b) is proved for the class C2 . Now we prove the statement for the first configuration of the class C5 . In the region c5 (x, y) < 0 we consider the linear center x 5y 1 √ y (33) ẋ = − − + (770 + 117 2), ẏ = x + − 1, 2 4 136 2 which has the first integral y 1 √ H1 (x, y) = 4(x + )2 + 8(−x − (770 + 117 2)y) + 4y 2 . 2 136 In the region c5 (x, y) > 0 we consider the linear center x 5y 1 √ y (34) ẋ = − − + (934 + 133 2), ẏ = x + − 2, 4 16 544 4 with its first integral    y 2 1  √  H2 (x, y) = 4 x + + 8 −2x − 934 + 133 2 y + y 2 . 4 544 The discontinuous piecewise linear differential system (33)–(34) has exactly one crossing limit cycle, because the system Hl (α, β) − Hl (γ, δ) = 0, Hk (α, β) − Hk (γ, δ) = 0, (35) β 2 − α2 (α + 1) = 0, δ 2 − γ 2 (γ + 1) = 0, √ when l = 1 and k = 2, has the unique real solution (α, β, γ, δ) = (1, 2, 3, 6). This proves the statement for the first configuration, see Figure 21. Now we prove the statement for the second configuration of the class C5 . In the region c5 (x, y) > 0 we consider the linear differential center (36) ẋ = −y, ẏ = 1 + x, with its first integral H1 (x, y) = 4x2 + 8x + 4y 2 . In the region c5 (x, y) < 0 we consider the linear differential center x 17y 1 y (37) ẋ = − − − , ẏ = x + − 1, 4 16 8 4 22 R. BENTERKI AND J. LLIBRE with its first integral  y 2 y  H2 (x, y) = 4 x + +8 − x + 4y 2 . 4 8 The discontinuous piecewise linear differential system (36)–(37) has exactly one crossing limit cycle, because the system √ of equations (35), √ when l = 1 and k = 2, has exactly one real solution (−1/2, 1/(2 2), −1/2, −1/(2 2)). See this configuration in Figure 22. Finally we prove the statement for the last possible configuration for the class C5 . We consider the differential center (38) ẋ = −3 + x − 5y, ẏ = −4 + x − y, in the region c5 (x, y) > 0, with the first integral H1 (x, y) = 4(x − y)2 + 16y 2 + 8(−4x + 3y). For the second differential linear center we consider (39) ẋ = −y, ẏ = x + 3, in the region c5 (x, y) < 0, with its first integral H2 (x, y) = 24x + 4x2 + 4y 2 . Since the unique intersection point between the three curves H1 (x, y) = 0, H2 (x, y) = 0 and c5 (x, y) = 0 are the two points (3, −6) and (3, 6), the discontinuous piecewise linear differential system (38)–(39) has exactly one crossing limit cycle, the one of Figure 23. This completes the proof of statement (b) of Theorem 3.  Figure 24. Two crossing limit cycles of the discontinuous piecewise linear differential system (40)–(41). Proof of statement (c) of Theorem 3. First we prove this statement for the six pos- sible configurations of two crossing limit cycles of the class C2 . For the first config- uration for the class C2 , we consider the differential linear center x 13y y √ 523 (40) ẋ = − − 2, ẏ = x − + 4 10 − , 2 36 2 36 23 Figure 25. Two crossing limit cycles of the discontinuous piecewise linear differential system (42)–(43). Figure 26. Two crossing limit cycles of the discontinuous piecewise linear differential system (44)–(45). Figure 27. Two crossing limit cycles of the discontinuous piecewise linear differential system (46)–(47). in the region c2 (x, y) > 0. A first integral of this system is   √ 1046 1 H3 (x, y) = 4x2 + x −4y + 32 10 − + y(13y + 144). 9 9 24 R. BENTERKI AND J. LLIBRE Figure 28. Two crossing limit cycles of the discontinuous piecewise linear differential system (48)–(49). Figure 29. Two crossing limit cycles of the discontinuous piecewise linear differential system (50)–(51). Figure 30. Two crossing limit cycles of the discontinuous piecewise linear differential system (52)–(53). The second linear center is 37y √ 1123 (41) ẋ = −x − + 4, ẏ = x + y − 8 10 − , 36 36 25 Figure 31. Two crossing limit cycles of the discontinuous piecewise linear differential system (54)–(55). Figure 32. Two crossing limit cycles of the discontinuous piecewise linear differential system (56)–(57). Figure 33. Two crossing limit cycles of the discontinuous piecewise linear differential system (58)–(59). in the region c2 (x, y) < 0. Its first integral is   √ 2246 1 H4 (x, y) = 4x2 + x 8y − 64 10 − + y(37y − 288). 9 9 26 R. BENTERKI AND J. LLIBRE Figure 34. Two crossing limit cycles of the discontinuous piecewise linear differential system (60)–(61). Figure 35. Two crossing limit cycles of the discontinuous piecewise linear differential system (62)–(63). Figure 36 Figure 37 The real solutions of the system of equations (28) when l = 3 and k =√4, for √the discontinuous √ linear differential system (40)–(41) are (4, −2 3, 4, 2 3) √ piecewise and (5, −2 10, 6, 3 10), producing the two crossing limit cycles of Figure 24. 27 Figure 38 Figure 39 For the second configuration in the class C2 we consider the differential linear center x 85y 1  √  y 194 (42) ẋ = − + 10 + 154 , ẏ = x − − , 9 324 45 9 405 in the region c2 (x, y) > 0. Its first integral is 1   √   H3 (x, y) = −72 5x + 154 + 10 y + 4x(405x − 388) + 425y 2 . 405 The second linear center is √ 17y 2 154 4 (43) ẋ = −2x − − − 4, ẏ = x + 2y − , 4 5 25 in the region c2 (x, y) < 0. Its first integral is 16  √  32x H4 (x, y) = 4x2 + 5x + 154 + 10 y − + 17y 2 . 5 25 The real solutions of the system of equations (28) when l = 3 and k = p 4, for the discontinuous p piecewise linearp differential system p (42)–(43) are (1/5, 2 14/5/5, 4/5, 2 11/5/5) and (1/5, −2 14/5/5, 4/5, −2 11/5/5), given place to the two crossing limit cycles of Figure 25. Now we prove the statement for the third configuration. In the region c2 (x, y) > 0 we consider the linear differential center 1 √  ẋ = 20x − 52y + 27 10 − 115 , (44) 200 √  1 ẏ = x + −4y − 3 70 + 10 , 40 which has the first integral 1   √  1  √  H3 (x, y) = 4x2 − x 4y + 3 70 + 10 + y 26y − 27 10 + 115 . 5 25 In the region c2 (x, y) < 0 we consider the linear differential center √ −640x − 328y + 243 10 + 3680 ẋ = , 3200 (45) y 3 273 ẏ = x + + √ − , 5 2 10 64 28 R. BENTERKI AND J. LLIBRE with its first integral r ! 2 8y 2 273 1  √  H4 (x, y) = 4x + x +6 − + y 164y − 243 10 − 3680 . 5 5 8 400 The real solutions of the system of equations (28) when l = 3 and k√= 4, for√the discontinuous√piecewise linear differential system (44)–(45) are (5, −2 10, 6, 3 10) and (3, 0, 5, 2 10), producing the two crossing limit cycles of Figure 26. We prove the statement for the forth configuration. In the region c2 (x, y) > 0 we consider the linear differential center x 13y 1 √ √  √  ẋ = − + 691 10 7 − 2 + 288 3 7 − 13 , 6 144 864 (46) r y 7 √  1 √  ẏ = x − − 4 8−3 7 + 3103 − 1382 7 , 6 5 216 which has the first integral r !  y 2 1 √ 7 √  y2 H3 (x, y) = 4 x − + 3103 − 1382 7 − 864 8−3 7 x+ 6 27 5 4 1 √ √ √  + 3744 − 864 7 − 691 10 7 − 2 y. 108 In the region c2 (x, y) < 0 we consider the linear differential center √ √  √  9x 977y 74387 10 2 + 3 7 − 11340 157 + 29 7 ẋ = − + √  , 70 22050 44100 25 + 8 7 (47) q √  297548 85075 − √ − 5832 35 8 − 3 7 9y 7 ẏ = x − + , 70 9450 with its first integral  q  297548 √   2 4 85075 − √ − 5832 35 8 − 3 7 x 9y 7 y2 H4 (x, y) = 4 x − + + 70 √  √ 4725 √  9 2 11340 157 + 29 7 − 74387 10 2 + 3 7 y + √  . 11025 25 + 8 7 The real √ solutions√ of the system √ of equations (28) when l = 3 and k = 4, are (5, −2 10, 10, 3 70) and (3, 0, 8, 2 70), therefore the discontinuous piecewise lin- ear differential system (46)–(47) has the two crossing limit cycles of Figure 27. For the fifth configuration. In the region c2 (x, y) > 0 we consider the linear differential center x 85y 1 √  ẋ = − + 10 + 154 , 9 324 45 (48) y 194 ẏ = x − − , 9 405 which has the first integral  y 2 8   √   H3 (x, y) = 4 x − − 194x + 9 10 + 154 y + y 2 . 9 405 29 In the region c2 (x, y) < 0 we consider the linear differential center 1 √  ẋ = −20x − 29y − 4 10 + 154 , 100 (49) y 298 ẏ = x + − , 5 625 with its first integral  y 2 8   √   H4 (x, y) = 4 x + + 25 10 + 154 y − 298x + y 2 . 5 625 The real p solutions of p the system of equationsp (28) when l =p3 and k = 4, are (1/5, 2 14/5/5, 4/5, 2 11/5/5) and (1/5, −2 14/5/5, 4/5, −2 11/5/5), then the discontinuous piecewise linear differential system (48)–(49) has the two crossing limit cycles of Figure 28. For the sixth configuration we consider the linear differential center 1 √  ẋ = 300x − 181y + 496 10 − 1500 , 900 (50) y 1 ẏ = x − − , 3 2 in the region c2 (x, y) > 0, this system has the first integral 4 1  √  H3 (x, y) = 4x2 − x(2y + 3) + y 181y − 992 10 + 3000 . 3 225 The second linear differential center is 1 √  ẋ = 140x − 265y + 608 10 − 700 , 980 (51) y 1 ẏ = x − − , 7 2 in the region c2 (x, y) < 0, this system has the following first integral 8  √  53y 2 H4 (x, y) = − 35x + 152 10 − 175 y + 4(x − 1)x + . 245 49 The real√solutions of the system of equations (28) when l = 3 and k = 4, are (3, 0, 5, 2 10) and (0, 0, 1, 0), so the discontinuous piecewise linear differential sys- tem (50)–(51) has the two crossing limit cycles of Figure 29. The second part of this proof analyze all the possible configurations of two cross- ing limit cycles of the class C5 . For the first configuration we consider the linear differential center x y 1 √  y √ 9 (52) ẋ = − + 7 − 2 , ẏ = x − + 7 − , 4 8 3 4 2 in the region c5 (x, y) > 0, with its first integral  √  1  √  H5 (x, y) = 4x2 + x −2y + 8 7 − 36 + y 3y − 16 7 + 32 . 6 The second linear differential center is x y 1 √  y √ 9 (53) ẋ = − + 7 − 2 , ẏ = x − + 7 − , 4 8 3 4 2 30 R. BENTERKI AND J. LLIBRE in the region c5 (x, y) < 0, with its first integral x 17y 1  √  H6 (x, y) = − − + 11 + 2 7 . 6 450 6 For this case the real solutions of the system of equations (35) when l = 5 and k = 6, for the discontinuous √ piecewise linear differential system (52)–(53) are (0, 0, 8, 24) and (3, 6, 6, 6 7). So the discontinuous piecewise linear differential system (52)– (53) has two crossing limit cycles, see them in Figure 30. For the second possible configuration we consider the first linear differential center x 85y 7 y 197 (54) ẋ = − − − , ẏ = x + + , 6 144 30 6 360 in the region c5 (x, y) > 0, with its first integral 1 1 H5 (x, y) = 4x2 + x(60y + 197) + y(425y + 336). 45 180 In the region c5 (x, y) < 0 we consider the linear differential center 17y 7 117 (55) ẋ = x − + , ẏ = x − y + , 16 5 200 with its first integral   2 117 1 H6 (x, y) = 4x + x − 8y + y(85y − 224). 25 20 The real solutions of the system of equations (35) when l = 5 and k = 6,√for the dis- √ continuous piecewise √ system (54)–(55) are (−1/5, 2/(5 5), −4/5, 4/(5 5)) √ linear differential and (−1/5, −2/(5 5), −4/5, −4/(5 5)), producing the two crossing limit cycles of Figure 31. For the third possible configuration for the class C5 we consider in the region c5 (x, y) > 0 the linear differential center x 29y 1 y √ 12 293 (56) ẋ = − − , ẏ = x − + 4 6 + √ − , 5 100 5 5 5 20 with its first integral 2  √ √  1 H5 (x, y) = 4x2 + x −4y + 80 6 + 48 5 − 293 + y(29y + 40). 5 25 In the region c5 (x, y) < 0 we consider the linear differential center x 17y 1 1  √ √  (57) ẋ = − − + , ẏ = x + 8y − 160 6 − 96 5 − 883 , 8 64 8 64 with its first integral   √ √ 883 17y 2 H6 (x, y) = 4x2 + x y − 20 6 − 12 5 − + − y. 8 16 In this case √ the √real solutions of√the system √ of equations (35) when l = 5 and k = 6, are (1, − 2, 1, 2) and (4, −4 5, 5, 5 6). So the discontinuous piecewise linear differential system (56)–(57) has the two crossing limit cycles of Figure 32. 31 For the fourth possible configuration of the class C5 we consider in the region c5 (x, y) < 0 the linear differential center √ √ 1 1440y + 704 6 + 972 3 + 4811 (58) ẋ = (−10x − 26y − 7), ẏ = x + , 100 14400 with its first integral √ √ 7200x2 + x(1440y + 704 6 + 972 3 + 4811) + 144y(13y + 7) H5 (x, y) = . 1800 In the region c5 (x, y) > 0 we consider the linear differential center √ p x 37y 7 y 9 3 22 2/3 2495 (59) ẋ = − − − , ẏ = x + + + + , 3 9 30 3 40 45 2592 with its first integral √ p 2 8y 9 3 176 2/3 2495 4 H6 (x, y) = 4x + x( + + + ) + y(185y + 21). 3 5 45 324 45 In this case the realpsolutions of the system p of equations (35) whenp l = 5 and k √= 6, are (−7/10, (7/10) 3/10, 7/10, (7/10) 3/10) and (−1/3, (1/3) 2/3, −1/4, − 3/8). So the discontinuous piecewise linear differential system (58)–(59) has the two cross- ing limit cycles of Figure 33. For the fifth possible configuration of the class C5 we consider in the region c5 (x, y) < 0 the linear differential center x 113y 1 y 19 5 4897 (60) ẋ = − − + , ẏ = x + − √ − √ + , 6 18 4 6 5 15 3 2 3600 with its first integral 1  √ √  2 H5 (x, y) = 4x2 − x −600y + 912 15 + 3000 2 − 4897 + y(113y − 9). 450 9 In the region c5 (x, y) > 0 we consider the linear differential center r 3 19 x 401y 3 y 1 5 20629 (61) ẋ = − − , ẏ = x − +√ + + , 10 100 20 10 2 25 20000 with its first integral r 3 √ 152 2 4y 5 20629 1 H6 (x, y) = 4x + x(− + 4 2 + + ) + y(401y + 30). 5 25 2500 25 The real solutions √ of the system p of equations (35) p when l = 5 and p k = 6, are (−1/2, 1/(2 2), −2/5, (3/2) 5/2) and (3/2, (3/2) 5/2, 3/2, (−3/2) 5/2). So the discontinuous piecewise linear differential system (60)–(61) has the two crossing limit cycles of Figure 34. For the sixth and last possible configuration of the class C5 we consider in the region c5 (x, y) > 0 the linear differential center x 10y 1 y 1 √ (62) ẋ = − − + , ẏ = x + + ( 6 − 8), 3 9 3 3 12 32 R. BENTERKI AND J. LLIBRE with its first integral 2 √ 8 H5 (x, y) = 4x2 + x(4y + 6 − 8) + y(5y − 3). 3 9 In the region c5 (x, y) < 0 we consider the linear differential center x 5y 1 y 1 457 (63) ẋ = − − + , ẏ = x + + √ − , 6 18 6 3 2 6 1536 with its first integral r 2 8y 2 457 1 H6 (x, y) = 4x + x( + 2 − )+ y(73y − 384). 3 3 192 144 The real solutions of the √ system of equations (35) when l = 5 and k = 6, are (3, 6, 3, −6) and (0, 0, 1, 2). So the discontinuous piecewise linear differential sys- tem (62)–(63) has the two crossing limit cycles of Figure 35. This completes the proof of statement (c) of Theorem 3.  Figure 40. Three limit cycles of the discontinuous piecewise linear differential system (64)–(65). Figure 41. Three limit cycles of the discontinuous piecewise linear differential system (66)–(67). 33 Proof of statement (d) of Theorem 3. First we prove the statement for the class C2 . For this class we consider the linear differential center (64) x 1 √ √ √ 1 √ √ √ ẋ = + (10 − 28 6 − 5 10 − 20 15)y + (−12 + 2 6 + 6 10 + 15), 5 180 36 √ y 6 ẏ = x − − − 2, 5 5 in the region c2 (x, y) > 0. This differential system has the first integral √ √ √ √ √ √ −8x(( 15 − 2 6)y + 3( 10 − 4)) − 4y( 10y + y − 10 15 + 15 6) H7 (x, y) = − p √ 5 39 − 12 10 p √ √ √ 12 39 − 12 10(x − 4)x + y(2 15y − 3 6y − 6) √ √ . 6 6 − 3 15 The other linear differential center is (65) 1 √ √ √ 1 √ √ √ ẋ = x + (2 − 28 6 − 10 − 20 15)y + (−60 + 2 6 + 30 10 + 15), 36 36 √ ẏ = x − y − 6 − 2, in the region c2 (x, y) < 0. It has the first integral √ √ √ √ √ √ −8x(( 15 − 2 6)y + 3( 10 − 4)) − 4y( 10y + y − 10 15 + 15 6) H8 (x, y) = − p √ 39 − 12 10 p √ √ √ 12 39 − 12 10(x − 4)x + y(2 15y − 3 6y − 6) + √ √ . 6 6 − 3 15 This discontinuous piecewise differential system formed by the linear differential centers (64)–(65) has exactly three crossing limit cycles,√ because the √ real solutions √ of the√system√(28) when l = 7 and k = 8, are (2, 0, 3, 6), (3, − 6, 4, 2 6) and (4, −2 6, 5, 2 15). See Figure 40. Second we prove the statement for the class C5 . For this class we consider the linear differential center x 37y 1 ẋ = − + , 6 36 24 (66) √ √ y 4697 + 1056 5 + 378 30 ẏ = x − + , 6 7200 in the region c5 (x, y) > 0. This differential system has the first integral √ √  !  y 2 4697 + 1056 5 + 378 30 x y H7 (x, y) = 4 x − +8 − + 4y 2 . 6 7200 24 The other linear differential center is x 5y 1 ẋ = − − − , 6 18 24 (67) √ √ y 2605 − 528 5 − 189 30 ẏ = x + + , 6 3600 34 R. BENTERKI AND J. LLIBRE in the region c5 (x, y) < 0. It has the first integral 1   √ √   H8 (x, y) = 1800x2 + x 600y − 189 30 − 528 5 + 2605 + 50y(10y + 3) . 450 This discontinuous piecewise differential system formed by the linear differential centers (66)–(67) has exactly three crossing limit cycles, because √ the real√ solu- tions of the√system (35) when l p = 7 and k = 8, are (−1/4, 3/8, −1/4, − 3/8), (−4/5, 4/(5 5), −7/10, (−7/10) 3/10) and (−0.9063569280622145544995606620.., −0.2773556931549898416488535525.., −0.9742505235096835971802596877.., 0.1563345074809235486186539264..). See Figure 41.  References [1] A. Andronov, A. Vitt and S. Khaikin, Theory of Oscillations, Pergamon Press, Oxford, 1966. [2] J.C. Artes, J. Llibre, J.C. Medrado and M.A. Teixeira, Piecewise linear with two real saddles, Math. Comput. Simulation 95 (2014), 13–22. [3] M. Di Bernardo, C.J. Budd, A.R. Champneys and P. Kowalczyk, Piecewise-Smooth Dynamical Systems: Theory and Applications, Appl. Math. Sci. 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Simpson, Bifurcations in piecewise-Smooth Continuous Systems, World Scientific series on Nonlinear Science A, vol. 69, World scientific, singapure, 2010. 35 1 Département de Mathématiques, Université Bachir El Ibrahimi, Bordj Bou Arréridj, Bordj Bou Arréridj 34265, El Anasser, Algeria Email address: r.benterki@univ-bba.dz 2 Departament de Matematiques, Universitat Autònoma de Barcelona, 08193 Bel- laterra, Barcelona, Catalonia, Spain Email address: jllibre@mat.uab.cat