This is a preprint of: “Limit cycles of discontinuous piecewise differential systems formed by linear centers in R2 and separated by two circles”, Maria Elisa Anacleto, Jaume Llibre, Clàudia Valls, Claudio Vidal, Nonlinear Anal. Real World Appl., vol. 60, 103281, 2021. DOI: [10.1016/j.nonrwa.2020.103281] LIMIT CYCLES OF DISCONTINUOUS PIECEWISE DIFFERENTIAL SYSTEMS FORMED BY LINEAR CENTERS IN R2 AND SEPARATED BY TWO CIRCLES MARIA ELISA ANACLETO1 , JAUME LLIBRE2 , CLAUDIA VALLS3 AND CLAUDIO VIDAL1 Abstract. We show that discontinuous planar piecewise differential systems formed by linear centers and separated by two concentric circles can have at most three limit cycles. Usually is a difficult problem to provide the exact upper bound that a class of differential systems can exhibit. Here we also provide examples of such systems with zero, one, two, or three limit cycles. 1. Introduction and statement of the main result In the qualitative theory of the differential systems in R2 one of the main difficult objects to study are the limit cycles. Recall that a limit cycle is an isolated periodic solution in the set of all periodic solutions of the differential systems, see for instance the second part of the famous 16th Hilbert problem [5, 7, 9]. The study of piecewise linear discontinuous differential systems started with Andronov, Vitt and Khaikin in [1]. Due to the fact that these systems model many real phenomena and different modern devices, they have became a topic of great interest these last twenty years. For more details see for instance the books [2, 19] and the references therein. In recent years many authors have been widely interested in solving the second part of the 16-th Hilbert’s problem for continuous and discontinuous piecewise linear differential systems in R2 , that is to determine an upper bound for the maximum number of limit cycles for these class of differential systems. The easiest continuous piecewise linear differential systems are formed by two linear differential systems separated by a straight line. It is known that such systems have at most one limit cycle, see [4,13,17,18]. But if both linear differential systems are linear centers, then it is known that the continuous piecewise linear differential systems have no limit cycles, see [11, 15]. Also the easiest discontinuous piecewise linear differential systems are formed by two linear differential systems separated by a straight line. It is known that this class of differential systems can have three limit cycles, but at this moment it is unknown if three is the maximum number of limit cycles that this class can 2010 Mathematics Subject Classification. 34C29, 34C25, 47H11. Key words and phrases. Limit cycles, linear centers, continuous piecewise linear differential sys- tems, discontinuous piecewise differential systems, first integrals. 1 2 M.E. ANACLETO, J. LLIBRE, C. VALLS AND C. VIDAL exhibit, see for instante [8, 10, 14]. Other works on the limit cycles of discontinuous piecewise linear differential systems can be found in [6, 12]. A natural further step is to know which is the maximum number of limit cycles of continuous or discontinuous piecewise linear differential systems separated by other objects rather than a straight line and in particular if these linear systems are centers. This is the goal of this paper. More precisely, in this paper we deal with discontinuous piecewise differential systems formed by two linear centers separated by two concentric circles. Without loss of generality we can assume that these circles are S1 = {(x, y) ∈ R2 : x2 + y 2 = 1}, and S2 = {(x, y) ∈ R2 : x2 + y 2 = s2 }, with s > 1. We use the Filippov conventions for defining the discontinuous piece- wise differential system on S1 and on S2 , see [3]. We denote by (C) the class of planar discontinuous piecewise differential systems formed by three linear centers and separated by two circles S1 and S2 . We note that if we try to obtain continuous piecewise differential systems formed by three linear centers separated by the circles S1 and S2 , then we obtain a global linear center in the plane, and consequently we do not have limit cycles. Therefore the interesting piecewise differential systems formed by three linear centers sepa- rated by the circles S1 and S2 are the discontinuous ones, the ones that we shall study. Here a crossing periodic orbit for a system in (C) is a periodic orbit which has exactly four points of discontinuity, that is, two points in the circle S1 and the other two points in the circle S2 . A crossing limit cycle for a system in (C) is a crossing periodic orbit isolated in the set of all crossing periodic orbits of the system. We remark that we are not interested in studying the crossing limit cycles for the systems in (C) having only two intersection points with either the circle S1 or the circle S2 , because in [16] it was proved that such systems can have at most two crossing limit cycles intersecting a circle in two points, and that there are examples exhibiting two such crossing limit cycles. The objective of this paper is to study the number of crossing limit cycles that systems in the class (C) can have intersecting in two points each of the circles S1 and S2 . In what follows when we talk about a crossing limit cycle, it will be a crossing limit cycle that intersects in two points each of the circles S1 and S2 . Our main result is the following. Theorem 1. The following statements hold. (a) Every system in (C) has at most three crossing limit cycles. (b) There are systems in (C) having three crossing limit cycles. (c) There are systems in (C) having exactly two crossing limit cycles. (d) There are systems in (C) having exactly one crossing limit cycle. (e) There are system in (C) without crossing limit cycles. The proof of Theorem 1 is given in the next section. LIMIT CYCLES FOR GENERALIZED PWLS WITH ONLY CENTERS 3 2. Proof of Theorem 1 To prove Theorem 1 we fist recall the following lemma, proved in [15] which provides a normal form for an arbitrary linear differential system having a center. Lemma 2. A linear differential system having a center can be written as 4b2 + ω 2 (1) ẋ = −bx − y + d, ẏ = ax + by + c, 4a with a > 0 and ω > 0. We prove each statement of Theorem 1 separately. Proof of statement (a) of Theorem 1. In the bounded region R1 limited by the cir- cle S1 we consider the arbitrary linear differential center (1), which has the first integral H1 (x, y) = 4(ax + by)2 + 8a(cx − dy) + ω 2 y 2 . In the bounded region R2 limited by the circles S1 and S2 we consider the arbitrary linear differential center 4B 2 + Ω2 (2) ẋ = −Bx − y + D, ẏ = Ax + By + C, 4A with A > 0 and Ω > 0. This system has the first integral H2 (x, y) = 4(Ax + By)2 + 8A(Cx − Dy) + Ω2 y 2 . Finally in the region R3 outside the circle S2 we consider the arbitrary linear dif- ferential center 4β 2 + w2 (3) ẋ = −βx − y + δ, ẏ = αx + βy + γ, 4α with α > 0 and w > 0, that has first integral H3 (x, y) = 4(αx + βy)2 + 8α(γx − δy) + w2 y 2 . Take the scaling of the times σ = at in R1 , τ = At in R2 and ν = αt in R3 . These three scalings change the velocity in which the orbits of system (1), (2) and (3) are traveled, but they do not change the orbits, and consequently they do not change the crossing limit cycles that the discontinuous piecewise linear differential systems may have. Note that after these scalings of the time we can always assume, without loss of generality, that a = A = α = 1, with the dot in the systems denoting derivative with respect to the new times σ, τ and ν. The proof of statement (a) will be done by contradiction. We thus assume that the discontinuous piecewise linear differential system formed by the three linear centers (1), (2) and (3) have four crossing periodic orbits that intersect the circle S1 in the points (xi , yi ) for i = 2, 3 and the circle S2 in the points (xi , yi ) for i = 1, 4, 4 M.E. ANACLETO, J. LLIBRE, C. VALLS AND C. VIDAL so these points must satisfy the system f1 = H2 (x1 , y1 ) − H2 (x2 , y2 ) = 0, f2 = H1 (x2 , y2 ) − H1 (x3 , y3 ) = 0, f3 = H2 (x3 , y3 ) − H2 (x4 , y4 ) = 0, (4) f4 = H3 (x4 , y4 ) − H3 (x1 , y1 ) = 0, f5 = x21 + y12 − s2 = 0, f6 = x24 + y42 − s2 = 0, f7 = x22 + y22 − 1 = 0, f8 = x23 + y32 − 1 = 0, or equivalently (5) f1 =4(By1 + x1 )2 − 4(By2 + x2 )2 + 8C(x1 − x2 ) − 8D(y1 + y2 ) + Ω2 (y12 − y22 ) = 0, f2 =4(by2 + x2 )2 − 4(by3 + x3 )2 + 8c(x2 − x3 ) − 8d(y2 + y3 ) + ω 2 (y22 − y32 ) = 0, f3 =4(By3 + x3 )2 − 4(By4 + x4 )2 + 8C(x3 − x4 ) − 8D(y3 + y4 ) + Ω2 (y32 − y42 ) = 0, f4 =4(βy4 + x4 )2 − 4(βy1 + x1 )2 + 8γ(x4 − x1 ) − 8δ(y4 + y1 ) + w2 (y42 − y12 ) = 0, f5 =0, f6 = 0, f7 = 0, f8 = 0. Assume that the four solutions of this system are (p1 , p2 , p3 , p4 ), (q1 , q2 , q3 , q4 ), (t1 , t2 , t3 , t4 ) and (w1 , w2 , w3 , w4 ) with pi , qi , t1 , wi ∈ R2 , pi 6= pj , qi 6= qj , ti 6= tj , and wi 6= wj for i 6= j and i, j = 1, 2, 3, 4. We rewrite these points in the following way (6) pi = (xi , yi ), qi = (ki , li ), ti = (mi , ni ), wi = (hi , ji ), for i = 1, 2, 3, 4. Substituting the solutions (p1 , p2 , p3 , p4 ) and (q1 , q2 , q3 , q4 ), with pi , qi given in (6) inside (5) we can determine the parameters d, δ, c and γ in function of the coordinates of the points pi , qi as follows (7) (4b2 + ω 2 )(y22 − y32 ) + 8x2 (by2 + c) − 8x3 (by3 + c) + 4x22 − 4x23 d= , 8(y2 − y3 )   4β 2 + w2 y12 − y42 + 4x21 + 8x1 (γ + βy1 ) − 4x24 − 8x4 (γ + βy4 ) δ= , 8(y1 − y4 ) 1 c= 4 l2 b2 y32 + 2by3 (x3 − k2 ) + 2bk2 y2 8(l2 − l3 )(x2 − x3 ) − 8(k2 − k3 )(y2 − y3 )   −(by2 + x2 )2 + x23 + b2 l22 (y2 − y3 ) − b2 l3 y32 + y3 (bl3 + k3 )2 − 2bl3 x3 − k22  +l3 (−by2 (bl3 + 2k3 )) + (by2 + x2 )2 − x23 + 4y2 (k2 − k3 )(k2 + k3 )  +ω 2 (l2 − l3 )(y2 − y3 )(l2 + l3 − y2 − y3 ) , 1 γ= (8β(k1 l1 (y1 − y4 ) + k4 l4 (y4 − y1 ) 8(l1 − l4 )(x1 − x4 ) − 8(k1 − k4 )(y1 − y4 ) − (l1 − l4 )(x1 y1 − x4 y4 )) + 4(k1 − k4 )(k1 + k4 )(y1 − y4 )  + (l1 − l4 ) w2 (y1 − y4 )(l1 + l4 − y1 − y4 ) − 4x21 + 4x24  +4β 2 (l1 − l4 )(y1 − y4 )(l1 + l4 − y1 − y4 ) . LIMIT CYCLES FOR GENERALIZED PWLS WITH ONLY CENTERS 5 We assume that the points (t1 , t2 , t3 , t4 ), (w1 , w2 , w3 , w4 ) with ti , wi as in (6) satisfies system (5), then we can obtain the remaining parameters, namely, ω 2 and w2 . In fact, from equation f2 we obtain ω 2 = S/T , where S = 4(−(k2 − k3 )(y3 (−b2 n22 − 2bm2 n2 + (bn3 + m3 )2 + 2bx3 (n2 − n3 ) − m22 ) + b2 n22 y2 + b2 y32 (n2 − n3 ) − b2 n2 y22 − b2 n23 y2 + b2 n3 y22 + 2bm2 n2 y2 − 2bm3 n3 y2 − 2bn2 x2 y2 + 2bn3 x2 y2 + k2 ((n2 − n3 )(x2 − x3 ) − (m2 − m3 )(y2 − y3 )) + k3 ((n2 − n3 )(x2 − x3 ) − (m2 − m3 )(y2 − y3 )) + y2 (m2 − m3 )(m2 + m3 ) − n2 x22 + n2 x23 + n3 x22 − n3 x23 ) + l2 (m2 (b2 (y32 − y22 ) + 2b(k2 y2 − k2 y3 + n2 x2 − n2 x3 − x2 y2 + x3 y3 ) − x22 + x23 ) + m3 (b2 (y2 − y3 )(y2 + y3 ) + 2b(−k2 y2 + k2 y3 − n3 x2 + n3 x3 + x2 y2 − x3 y3 ) + (x2 − x3 )(x2 + x3 )) + b(n2 − n3 )(x2 − x3 )(b(n2 + n3 ) − 2k2 ) + m22 (x2 − x3 ) + m23 (x3 − x2 )) + l3 (m2 (b2 (y2 − y3 )(y2 + y3 ) + 2b(−k3 y2 + k3 y3 + n2 (x3 − x2 ) + x2 y2 − x3 y3 ) + (x2 − x3 )(x2 + x3 )) + m3 (b2 (y32 − y22 ) + 2b(k3 y2 − k3 y3 + n3 (x2 − x3 ) − x2 y2 + x3 y3 ) − x22 + x23 ) + b(n2 − n3 )(x3 − x2 )(b(n2 + n3 ) − 2k3 ) + m22 (x3 − x2 ) + m23 (x2 − x3 )) + l22 b2 ((m2 − m3 )(y2 − y3 ) + (n3 − n2 )(x2 − x3 )) + l32 b2 ((n2 − n3 )(x2 − x3 ) − (m2 − m3 )(y2 − y3 ))), and T = (k2 − k3 )(n2 − n3 )(y2 − y3 )(n2 + n3 − y2 − y3 ) + l22 ((n2 − n3 )(x2 − x3 ) − (m2 − m3 )(y2 − y3 )) + l2 ((m2 − m3 )(y2 − y3 )(y2 + y3 ) + (n3 − n2 )(n2 + n3 )(x2 − x3 )) + l32 ((m2 − m3 )(y2 − y3 ) + (n3 − n2 )(x2 − x3 )) + l3 ((n2 − n3 )(n2 + n3 )(x2 − x3 ) − (m2 − m3 )(y2 − y3 )(y2 + y3 )). Analogously from equation f4 we obtain w2 = R/U , where R =4(−(k1 − k4 )(k1 ((n1 − n4 )(x1 − x4 ) − (m1 − m4 )(y1 − y4 )) − k4 (m1 − m4 )(y1 − y4 ) + k4 (n1 − n4 )(x1 − x4 ) + 2β(m1 n1 (y1 − y4 ) + m4 n4 (y4 − y1 ) − (n1 − n4 )(x1 y1 − x4 y4 )) + (m1 − m4 )(m1 + m4 )(y1 − y4 ) − (n1 − n4 )(x1 − x4 )(x1 + x4 ) + β 2 (n1 − n4 )(y1 − y4 ) × (n1 + n4 − y1 − y4 )) + l1 (2β(k1 (m1 − m4 )(y1 − y4 ) − k1 (n1 − n4 )(x1 − x4 ) + m1 (n1 (x1 − x4 ) − x1 y1 + x4 y4 ) + m4 (n4 (x4 − x1 ) + x1 y1 − x4 y4 )) + β 2 ((n1 − n4 ) × (n1 + n4 )(x1 − x4 ) − (m1 − m4 )(y1 − y4 )(y1 + y4 )) + (m1 − m4 )(x1 − x4 ) × (m1 + m4 − x1 − x4 )) + l4 (2β(−k4 (m1 − m4 )(y1 − y4 ) + k4 (n1 − n4 )(x1 − x4 ) + m1 (n1 (x4 − x1 ) + x1 y1 − x4 y4 ) + m4 (n4 (x1 − x4 ) − x1 y1 + x4 y4 )) + β 2 ((m1 − m4 )(y1 − y4 )(y1 + y4 ) + (n4 − n1 )(n1 + n4 )(x1 − x4 )) + (m1 − m4 )(x1 − x4 )(−m1 − m4 + x1 + x4 )) + β 2 l12 ((m1 − m4 )(y1 − y4 ) + (n4 − n1 )(x1 − x4 )) + β 2 l42 ((n1 − n4 )(x1 − x4 ) − (m1 − m4 )(y1 − y4 ))), and U =(k1 − k4 )(n1 − n4 )(y1 − y4 )(n1 + n4 − y1 − y4 ) + l12 ((n1 − n4 )(x1 − x4 ) − (m1 − m4 )(y1 − y4 )) + l1 ((m1 − m4 )(y1 − y4 )(y1 + y4 ) + (n4 − n1 )(n1 + n4 )(x1 − x4 )) + l42 ((m1 − m4 )(y1 − y4 ) + (n4 − n1 )(x1 − x4 )) + l4 ((n1 − n4 )(n1 + n4 )(x1 − x4 ) − (m1 − m4 )(y1 − y4 )(y1 + y4 )). 6 M.E. ANACLETO, J. LLIBRE, C. VALLS AND C. VIDAL Substituting pi , qi , ti , wi as in (6) in equations f2 and f4 , we obtain the expressions for b and β. In order to simplify these expressions, we use the relations x2i = s2i − yi2 , ki2 = s2i − li2 , m2i = s2i − n2i , h2i = s2i − ji2 with si = 1 for i = 2, 3 and si = r for i = 1, 4. Doing this we get  b = h23 + j32 − 1 (−(k2 − k3 )(−k2 (m2 − m3 )(y2 − y3 ) + k2 (n2 − n3 )(x2 − x3 ) − k3 (m2 − m3 )(y2 − y3 ) + k3 (n2 − n3 )(x2 − x3 ) + (m2 − m3 )(m2 + m3 )(y2 − y3 ) − (n2 − n3 )(x2 + x3 )(x2 − x3 )) + l2 (m2 − m3 )(x2 − x3 )(m2 + m3 − x2 − x3 ) − l3 (m2 − m3 )(x2 − x3 )(m2 + m3 − x2 − x3 )),  β = h21 + j12 − r2 (−(k1 − k4 )(−k1 (m1 − m4 )(y1 − y4 ) + k1 (n1 − n4 )(x1 − x4 ) − k4 (m1 − m4 )(y1 − y4 ) + k4 (n1 − n4 )(x1 − x4 ) + (m1 − m4 )(m1 + m4 )(y1 − y4 ) − (n1 − n4 )(x1 + x4 )(x1 − x4 )) + l1 (m1 − m4 )(x1 − x4 )(m1 + m4 − x1 − x4 ) − l4 (m1 − m4 )(x1 − x4 )(m1 + m4 − x1 − x4 )). Since h23 + j32 − 1 = 0 and h21 + j12 − r2 = 0 (because they are points on the circles of radius 1 and r, respectively), we obtain that b = 0 and β = 0. Therefore we get that the piecewise linear differential center is formed by the linear differential centers ẋ = −y, ẏ = x, in the regions R1 and R3 . This is a contradiction because in the region R1 inside the circle of radius 1 we have the first integral H1 (x, y) = x2 + y 2 . Then using this first integral it is not possible to generate crossing limit cycles. Therefore there is no discontinuous piecewise linear differential systems (1), (2) and (3) having more than three crossing limit cycles.  Proof of statement (b) of Theorem 1. We construct a system in (C) having three crossing limit cycles. In the bounded region R1 limited by the circle S1 we consider the linear differential center (8) ẋ = 4x − 11.9153y + 2.59286, ẏ = 8x − 4y − 7.70389, with the first integral H1 (x, y) = − 7.703888863058826x − 2.59286097618567y + 4x2 − 4xy + 5.957636997088071y 2 . In the bounded region R2 limited by the circles S1 and S2 with s = 2 we consider the linear differential center (9) ẋ = −2.48171x − 3.50967y + 5.2645, ẏ = 8x + 2.48171y − 12.2078, with the first integral H2 (x, y) = − 12.207843699316152x − 5.264498126903808y + 4x2 + 2.481708216718382xy + 1.7548327089679396y 2 . Finally in the region R3 outside the circle S2 we consider the linear differential center (10) ẋ = −8x + 4y + 12.2222, ẏ = 8x − 4y − 12.2222, LIMIT CYCLES FOR GENERALIZED PWLS WITH ONLY CENTERS 7 with the first integral H3 (x, y) = − 12.222157553827293x − 3.9748620974752176y + 4x2 − 4xy + 6.143143553839666y 2 . The discontinuous piecewise differential system formed by the previous three linear differential centers has three crossing limit cycles (p1 , p2 , p3 , p4 ), (q1 , q2 , q3 , q4 ) and (t1 , t2 , t3 , t4 ) with pi , qi , ti pi = (xi , yi ) = (si cos ρi , si sin ρi ), qi = (ki , li ) = (si cos νi , si sin νi ), ti = (mi , ni ) = (si cos ui , si sin ui ), where ρi , νi , ui ∈ [0, 2π), i = 1, 2, 3, 4, si = 1 for i = 2, 3 and si = 2 for i = 1, 4. Moreover, ρ1 = ρ2 = π/4, ρ3 = ρ4 = 1/10, ν1 = ν2 = π/2, ν3 = ν4 = −π/4, u1 = 2, u2 = 2.2778851991902505, u3 = 4 and u4 = 4.760510529471427. See the three crossing limit cycles of this system in Figure 1.  4 3 2 1 -2 2 4 -1 -2 Figure 1: The three crossing limit cycles of the discontinuous piecewise differential systems formed by the linear centers (8), (9) and (10). Proof of statement (c) of Theorem 1. We construct a system in (C) having exactly two crossing limit cycles. In the bounded region R1 limited by the circle S1 we consider the linear differential center 2 (11) ẋ = − y, ẏ = x − 1, 5 with the first integral   2 2y H1 (x, y) = 4x + 8 −x − + 4y 2 . 5 In the bounded region R2 limited by the circles S1 and S2 with s = 2 we consider the linear differential center 11x 18292y 19x 11y (12) ẋ = + − 4, ẏ = − − − 10, 19 6859 2 19 with the first integral  2 19x 11y H2 (x, y) = 4 − − − 76(4y − 10x) + 100y 2 . 2 19 8 M.E. ANACLETO, J. LLIBRE, C. VALLS AND C. VIDAL And in the region R3 outside the circle S2 we consider the linear differential center 1 x (13) ẋ = 16y + , ẏ = 1 − , 5 4 with the first integral x2  y H3 (x, y) = −2 x− + 16y 2 . 4 5 This discontinuous piecewise differential system formed by the previous linear centers (11), (12) and (13) has two crossing limit cycles, which are the unique two real solutions (pj1 , pj2 , pj3 , pj4 ) with j = 1, 2 satisfying (4). Since we want to use Groebner bases, we use the rational parametrization of the circle for obtain polynomial equations, so we write the points pi = (xi , yi ) as   R(1 − t2i ) 2Rti (14) (xi , yi ) = , , 1 + t2i 1 + t2i with ti 6= tj , i 6= j for i = 1, 2, 3, 4. Doing this system (4) reduces to the polynomial system (15) f1 =116603t41 t44 − 566048t41 t34 − 169146t41 t24 − 311904t41 t4 + 1214043t41 + 251256t31 t44 + 502512t31 t24 + 251256t31 + 59434t21 t44 − 1132096t21 t34 − 685836t21 t24 − 623808t21 t4 + 2254314t21 + 187720t1 t44 + 375440t1 t24 + 187720t1 − 432117t44 − 566048t34 − 1266586t24 − 311904t4 + 665323, f2 =2t1 t2 + 5t1 + 5t2 − 2, f3 = − 116603t42 t43 + 566048t42 t33 + 169146t42 t23 + 311904t42 t3 − 1214043t42 − 251256t32 t43 − 502512t32 t23 − 251256t32 − 59434t22 t43 + 1132096t22 t33 + 685836t22 t23 + 623808t22 t3 − 2254314t22 − 187720t2 t43 − 375440t2 t23 − 187720t2 + 432117t43 + 566048t33 + 1266586t23 + 311904t3 − 665323, f4 =2t33 t34 + 305t33 t24 + 2t33 t4 − 10t33 + 305t23 t34 − 2t23 t24 − 10t23 t4 − 2t23 + 2t3 t34 − 10t3 t24 + 2t3 t4 − 325t3 − 10t34 − 2t24 − 325t4 − 2, f5 =0, f6 = 0, f7 = 0, f8 = 0. Doing the Groebner basis using Mathematica we can eliminate the variables t2 , t4 using the equations (f2 , f3 ) and (f1 , f4 ), respectively. Moreover we get two poly- nomials: p1 of degree 8 and p2 of degree 24 with respect to variables t1 and t3 . Solving the system obtained for these two polynomials we obtain the following six real solutions (t11 , t13 ) = (3.113683889, −3.345831873), (t21 , t23 ) = (−1.208512958, 3.444180431), (t31 , t33 ) = (−0.07918973421, −2.003557002), (t41 , t43 ) = (0.7258741121, −0.9412890517), (t51 , t53 ) = (−0.2525471397, 0.8580640103), (t61 , t63 ) = (0.4948650262, 0.5345302181). LIMIT CYCLES FOR GENERALIZED PWLS WITH ONLY CENTERS 9 Therefore (15) has the next six solutions (t11 , t12 , t13 , t14 ) = (3.113684, −1.208513, −3.345832, 3.444180), (t21 , t22 , t23 , t24 ) = (−1.208513, 3.113684, 3.444180, −3.345832), (t31 , t32 , t33 , t34 ) = (0.7258741, −0.2525471, −0.941289, 0.8580640), (t41 , t42 , t43 , t44 ) = (−0.2525472, 0.7258742, 0.8580642, −0.941289), (t51 , t52 , t53 , t54 ) = (−0.07919058, 0.4948659, −2.003559, 0.5345301), (t61 , t62 , t63 , t64 ) = (0.4948641, −0.07918876, 0.5345306, −2.003555). The solutions (tj1 , tj2 , tj3 , tj4 ) for j = 2, 4, 6, define the same solutions as for j = 1, 3, 4. Therefore the solutions of system (4) in the variables (xi , yi ) are (p11 , p12 , p13 , p14 ) =(0.880088, −0.474811, 0.309849, 0.950786, 0.303784, 1.97679, 0.120863, −1.99634), (p21 , p22 , p23 , p24 ) =(−0.187158, −0.98233, −0.812997, 0.582268, −1.68902, 1.07109, − 1.67199, −1.09748), (p31 , p32 , p33 , p34 ) =(0.987536, −0.157392, 0.606566, 0.795033, −1.20227, −1.59829, 1.11109, 1.66297). The third solution does not define a crossing limit cycle. See the two crossing limit cycles in Figure 2.  3 2 1 -5 5 10 15 20 -1 -2 -3 Figure 2: The two crossing limit cycles of the discontinuous piecewise differential systems formed by the linear centers (11), (12) and (13). Proof of statement (d) of Theorem 1. We construct a system in (C) having exactly one crossing limit cycle. In the bounded region R1 limited by the circle S1 we consider the linear differential center x 2849y 34 213x y (16) ẋ = − + + , ẏ = − + + 1, 3 1917 25 77 3 10 M.E. ANACLETO, J. LLIBRE, C. VALLS AND C. VIDAL with the first integral  2   y 213x 1704 34y H1 (x, y) = 4 − − x− + 16y 2 . 3 77 77 25 In the bounded region R2 limited by the circles S1 and S2 with s = 2 we consider the linear differential center 11x 18292y 19x 11y (17) ẋ = + − 4, ẏ = − − − 10, 19 6859 2 19 with the first integral  2 19x 11y H2 (x, y) = 4 − − − 76(4y − 10x) + 100y 2 . 2 19 Finally in the region R3 outside the circle S2 we consider the linear differential center 23x 9117y 2 11x 23y 131 (18) ẋ = − + , ẏ = − − , 22 5324 5 9 22 88 with the first integral  2   11x 23y 88 131x 2y H3 (x, y) = 4 − + − − + 4y 2 . 9 22 9 88 5 This discontinuous piecewise differential system formed by the linear centers (16), (17) and (18) has exactly one crossing limit cycle, coming from the unique real solution (p1 , p2 , p3 , p4 ) with pi = (xi , yi ), pi 6= pj for i 6= j and i, j = 1, 2, 3, 4 that satisfies system (4). Again since we want to use Groebner bases we use the rational parametrization (14) and system (4) reduces to the polynomial system (19) f1 =116603t41 t44 − 566048t41 t34 − 169146t41 t24 − 311904t41 t4 + 1214043t41 + 251256t31 t44 + 502512t31 t24 + 251256t31 + 59434t21 t44 − 1132096t21 t34 − 685836t21 t24 − 623808t21 t4 + 2254314t21 + 187720t1 t44 + 375440t1 t24 + 187720t1 − 432117t44 − 566048t34 − 1266586t24 − 311904t4 + 665323, f2 =6248781t31 t32 − 8413925t31 t22 + 3788631t31 t2 − 3690225t31 − 8413925t21 t32 − 8708931t21 t22 − 3690225t21 t2 − 6248781t21 + 3788631t1 t32 − 3690225t1 t22 + 1328481t1 t2 + 1033475t1 − 3690225t32 − 6248781t22 + 1033475t2 − 3788631, f3 = − 116603t42 t43 + 566048t42 t33 + 169146t42 t23 + 311904t42 t3 − 1214043t42 − 251256t32 t43 − 502512t32 t23 − 251256t32 − 59434t22 t43 + 1132096t22 t33 + 685836t22 t23 + 623808t22 t3 − 2254314t22 − 187720t2 t43 − 375440t2 t23 − 187720t2 + 432117t43 + 566048t33 + 1266586t23 + 311904t3 − 665323, f4 =810216t33 t34 − 243515t33 t24 − 1193544t33 t4 − 713295t33 − 243515t23 t34 − 2813976t23 t24 − 713295t23 t4 − 810216t23 − 1193544t3 t34 − 713295t3 t24 − 3197304t3 t4 − 1183075t3 − 713295t34 − 810216t24 − 1183075t4 + 1193544, f5 =0, f6 = 0, f7 = 0, f8 = 0. Doing the Groebner basis using Mathematica we can eliminate the variables t2 , t4 using the equations (f2 , f3 ) and (f1 , f4 ), respectively. Moreover, we get two poly- nomials p3 and p4 both of degree 24 with respect to variables t1 and t3 . Solving the LIMIT CYCLES FOR GENERALIZED PWLS WITH ONLY CENTERS 11 system formed by these two polynomials we obtain its two unique real solutions (t11 , t13 ) = (−0.2761369370, −5.638072382), (t21 , t22 ) = (−1.456914010, −0.9512182090). Therefore system (19) has the two solutions (t11 , t12 , t13 , t14 ) = (−1.4576899, −0.27613694, −0.95176617, −5.6380724), (t21 , t22 , t23 , t24 ) = (−0.27613694, −1.4576899, −5.6380724, −0.95176617), From these two solutions we obtain the solutions of system (4) in the variables (xi , yi ), i.e. (p11 , p12 , p13 , p14 ) =(−0.359971, −0.932963, 0.858302, −0.513146, 0.0987913, −1.99756, − 1.878, −0.687824), (p21 , p22 , p23 , p24 ) =(0.858302, −0.513146, −0.359971, −0.932963, −1.878, −0.687824, 0.0987913, −1.99756). But the both solution define the same crossing limit cycle. See this crossing limit cycle of the piecewise differential system in Figure 3.  3 2 1 -2 -1 1 2 3 -1 -2 Figure 3: The crossing limit cycle of the discontinuous piecewise differential system formed by the linear centers (16), (17) and (18). Proof of statement (e) of Theorem 1. To prove statement (e) it is sufficient con- sider the discontinuous piecewise linear differential system having the linear center ẋ = −y, ẏ = x in the region R2 limited by the circles S1 and S2 with first integral H(x, y) = x2 + y 2 , and two arbitrary different centers in the region R1 limited by the circle S1 and in the region R3 outside the circle S2 . Then it is impossible to connect orbits of the region R1 with orbits of the region R3 , so they do not exist crossing limit cycle for this discontinuous piecewise differential system.  3. Appendix The polynomials p1 , p2 , p3 and p4 of the proofs of statements (c) and (d), p1 =a1 t41 t43 + a2 t41 t33 + a3 t41 t23 + a4 t41 t3 + a5 t41 + a6 t31 t43 + a7 t31 t23 + a8 t31 + a9 t21 t43 + a10 t21 t33 + a11 t21 t23 + a12 t21 t3 + a13 t21 + a14 t1 t43 + a15 t1 t23 + a16 t1 + a17 t43 + a18 t33 + a19 t23 + a20 t3 + a21 12 M.E. ANACLETO, J. LLIBRE, C. VALLS AND C. VIDAL where the coefficients ai are a1 = 54607 a2 = −16415392 a3 = −11558994 a4 = −9045216 a5 = 31880367 a6 = −10198280 a7 = −20396560 a8 = −10198280 a9 = −4653694 a10 = −32830784 a11 = −32643804 a12 = −18090432 a13 = 58997826 a14 = −20969016 a15 = −41938032 a16 = −20969016 a17 = −7078753 a18 = −16415392 a19 = −25825714 a20 = −9045216 a21 = 24747007 p2 =b1 t12 12 11 12 10 12 9 12 8 12 7 12 6 12 5 12 3 t1 + b2 t3 t1 + b3 t3 t1 + b4 t3 t1 + b5 t3 t1 + b6 t3 t1 + b7 t3 t1 + b8 t3 t1 + b9 t43 t12 3 12 2 12 12 12 12 11 11 11 1 + b10 t3 t1 + b11 t3 t1 + b12 t3 t1 + b13 t1 + b14 t3 t1 + b15 t3 t1 + b16 t10 11 9 11 8 11 7 11 6 11 5 11 4 11 3 t1 + b17 t3 t1 + b18 t3 t1 + b19 t3 t1 + b20 t3 t1 + b21 t3 t1 + b22 t3 t1 + b23 t33 t11 2 11 11 11 12 10 11 10 10 10 1 + b24 t3 t1 + b25 t3 t1 + b26 t1 + b27 t3 t1 + b28 t3 t1 + b29 t3 t1 + b30 t93 t10 8 10 7 10 6 10 5 10 4 10 3 10 1 + b31 t3 t1 + b32 t3 t1 + b33 t3 t1 + b34 t3 t1 + b35 t3 t1 + b36 t3 t1 + b37 t23 t10 10 10 12 9 11 9 10 9 9 9 1 + b38 t3 t1 + b39 t1 + b40 t3 t1 + b41 t3 t1 + b42 t3 t1 + b43 t3 t1 + b44 t83 t91 + b45 t73 t91 + b46 t63 t91 + b47 t53 t91 + b48 t43 t91 + b49 t33 t91 + b50 t23 t91 + b51 t3 t91 + b52 t91 + b53 t12 8 11 8 10 8 9 8 8 8 7 8 6 8 3 t1 + b54 t3 t1 + b55 t3 t1 + b56 t3 t1 + b57 t3 t1 + b58 t3 t1 + b59 t3 t1 + b60 t53 t81 + b61 t43 t81 + b62 t33 t81 + b63 t23 t81 + b64 t3 t81 + b65 t81 + b66 t12 7 11 7 3 t1 + b67 t3 t1 + b68 t10 7 9 7 8 7 7 7 6 7 5 7 4 7 3 7 3 t1 + b69 t3 t1 + b70 t3 t1 + b71 t3 t1 + b72 t3 t1 + b73 t3 t1 + b74 t3 t1 + b75 t3 t1 + b76 t23 t71 + b77 t3 t71 + b78 t71 + b79 t12 6 11 6 10 6 9 6 8 6 3 t1 + b80 t3 t1 + b81 t3 t1 + b82 t3 t1 + b83 t3 t1 + b84 t73 t61 + b85 t63 t61 + b86 t53 t61 + b87 t43 t61 + b88 t33 t61 + b89 t23 t61 + b90 t3 t61 + b91 t61 + b92 t12 5 11 5 10 5 9 5 8 5 7 5 6 5 5 5 3 t1 + b93 t3 t1 + b94 t3 t1 + b95 t3 t1 + b96 t3 t1 + b97 t3 t1 + b98 t3 t1 + b99 t3 t1 + b100 t43 t51 + b101 t33 t51 + b102 t23 t51 + b103 t3 t51 + b104 t51 + b105 t12 4 11 4 3 t1 + b106 t3 t1 + b107 t10 4 9 4 8 4 7 4 6 4 5 4 4 4 3 t1 + b108 t3 t1 + b109 t3 t1 + b110 t3 t1 + b111 t3 t1 + b112 t3 t1 + b113 t3 t1 + b114 t33 t41 + b115 t23 t41 + b116 t3 t41 + b117 t41 + b118 t12 3 11 3 10 3 3 t1 + b119 t3 t1 + b120 t3 t1 + b121 t93 t31 + b122 t83 t31 + b123 t73 t31 + b124 t63 t31 + b125 t53 t31 + b126 t43 t31 + b127 t33 t31 + b128 t23 t31 + b129 t3 t31 + b130 t31 + b131 t132 2 11 2 10 2 9 2 3 t1 + b133 t3 t1 + b134 t3 t1 + b135 t3 t1 + b136 t83 t21 + b137 t73 t21 + b138 t63 t21 + b139 t53 t21 + b140 t43 t21 + b141 t33 t21 + b142 t23 t21 + b143 t3 t21 + b144 t21 + b145 t12 11 10 9 8 3 t1 + b146 t3 t1 + b147 t3 t1 + b148 t3 t1 + b149 t3 t1 + b150 t73 t1 + b151 t63 t1 + b152 t53 t1 + b153 t43 t1 + b154 t33 t1 + b155 t23 t1 + b156 t3 t1 + b157 t1 + b158 t12 11 10 9 8 7 6 3 + b159 t3 + b159 t3 + b160 t3 + b161 t3 + b162 t3 + b163 t3 + b164 t53 + b165 t43 + b166 t33 + b167 t23 + b168 t3 + b169 , LIMIT CYCLES FOR GENERALIZED PWLS WITH ONLY CENTERS 13 where the coefficients bi are b1 = 303991592446001788228938487 b2 = 1408045233262894200409887264 b3 = −610348310045438609985979526 b4 = 150794822399721357288599264 b5 = 3175292247340716691173055833 b6 = −320740208014163780149933504 b7 = −1704899763947849707377573908 b8 = −405573064211186754732584256 b9 = 62276987629682561558117113 b10 = −175671366937609682473351264 b11 = −909165874719067126843100166 b12 = 8410996908736764862587488 b13 = 67790690035723395857234967 b14 = 769919768775604394983693704 b15 = 608614820298158930816555520 b16 = 1006338413132290284840738864 b17 = 1431321555103103515287098880 b18 = 1106920085504028821353449720 b19 = 1051042237498193081988510720 b20 = 1785102108481388009724969120 b21 = 274182185151143817728302080 b22 = 598987412480135128015248120 b23 = 77449776729327796556689920 b24 = −141824611475603542032144336 b25 = 31603094271432476346355200 b26 = 173788643378306408181172104 b27 = 1588372403382159061063149654 b28 = 8092975591745815633602116544 b29 = −3248693041709167069185540444 b30 = 211406878520421069494121024 b31 = 19938833981071197564188426058 b32 = −2121423179067511352527553664 b33 = −10009180546277357760027249288 b34 = −2156318684700125763185882496 b35 = 1048296922418733508883452938 b36 = −905812191897963002200866624 b37 = −4698504379735191229617889884 b38 = 62646607676949750357457728 b39 = 470711522393073093178291734 b40 = 3275446737045079891833146776 b41 = 2798389897743457301579527680 b42 = 3430073858498420097704336016 b43 = 6384456051429072119840709120 b44 = 3437215685906732064733269480 b45 = 4256413077567038680571489280 b46 = 7086432988050350508983168480 b47 = 655959555828558576937681920 b48 = 1805773207367510400930991080 b49 = 88554595955003083626094080 b50 = −1396578676010723603677244784 b51 = 102941964007868368998720000 b52 = 601492540218724645512852376 b53 = 1863185952562665535144523113 b54 = 18368537004625498191672760800 b55 = −10118633311500150875666933562 b56 = −3645531639364227160097798880 b57 = 47742911997329850459275889735 b58 = −7908940207973105675684819520 b59 = −29927104493544736358479363180 b60 = −5586703187951216642169113280 b61 = 1322636002702229535320877735 b62 = −2216882120199644964482927520 b63 = −10708672793085961168931909562 b64 = 104935955560450798251925920 b65 = 823337441741604282540503113 b66 = 4610181176067349319811463504 b67 = 4934488104553140293973765120 b68 = 673222061320160933366852064 b69 = 10443232592303538229139911680 b70 = −1826353774138329805781401680 b71 = 5118958520150359554781009920 b72 = 6459378942855282670710355520 b73 = −1347774333705502370068101120 b74 = −1787454240730479998434739280 b75 = −951934381413631957788410880 b76 = −5963483656776835980091631136 b77 = 6053984691832031894553600 b78 = 172744186200068108390253904 b79 = −1621708105402125393637763148 b80 = 20878158753803570951808551040 b81 = −17588058999912760831475813128 b82 = −12899199805848832415420900480 b83 = 58653267579916511026612363020 b84 = −15440556721247650813313611520 b85 = −50934444886457453503292294000 b86 = −7665732406157195645450753280 b87 = −1867587215720614217259928180 b88 = −2791367357827462082945909120 b89 = −12614307807012810344397195528 b90 = 51148822457498709060684160 b91 = 374525371112801525600025652 b92 2312913197949713418835292976 b93 = 4148805890329529288800711680 b94 = −6389500990001080093169282784 b95 = 7559997156913701546733317120 b96 = −10609352208188163894569321520 b97 = 728772418724086020836321280 b98 = 43899111766812611810796480 b99 = −4884992862282772005446830080 b100 = −6208792316160131100545359920 b101 = −2460343804730642330472913920 b102 = −9161706662669674635942223584 b103 = −257769790307956562122368000 b104 = −1002077214505360541021321424 14 M.E. ANACLETO, J. LLIBRE, C. VALLS AND C. VIDAL b105 = −3971430621041392996112889447 b106 = 12374289928344354025078994400 b107 = −12344000458594114924782416922 b108 = −16330806431518499543110022880 b109 = 46437110133303585391251181335 b110 = −14124913545548867479214291520 b111 = −42477362711277341585174494380 b112 = −3946994294230016748826649280 b113 = −33305509685054214583510665 b114 = −1059408107241328466289647520 b115 = −4873852326922304202721184922 b116 = 97755453094933815991727520 b117 = 378892493096299322127122553 b118 = 200730767420231523311578024 b119 = 1644632956482672247915461120 b120 = −5190506023107083703857888016 b121 = 2171512967117559308974241280 b122 = −5226483846546228429709648680 b123 = −2128889600225717310326983680 b124 = 3195078713173340252023634720 b125 = −4473735425890521681580139520 b126 = −528169211751130452219818280 b127 = −2097914965047015435859668480 b128 = −4390595229150387720190227216 b129 = −279949150017098125665292800 b130 = −832100248207003813406591576 b131 = −1962737440728361463430634346 b132 = 3571645054628274995872119744 b133 = −1135972992384322472318023644 b134 = −8698405461151187684087549376 b135 = 23946171151847913614553237258 b136 = −4881243111994631139488859264 b137 = −16439941847018161873601156488 b138 = 1104550916440195157556415104 b139 = 2661394009267641937515912138 b140 = 862790136065306374490947776 b141 = 1979063068004556038527706916 b142 = 199041204126846535197342528 b143 = 760913305201756890114011734 b144 = −59789326956919740039496520 b145 = 240540093260984882325542400 b146 = −865085042503842495052102320 b147 = 101613906181427386827225600 b148 = 442302423721227403531571400 b149 = −944074338868289374965273600 b150 = 3264005580186614462109154400 b151 = −1318294267990666805410790400 b152 = 1902466505411470364808179400 b153 = −600610360955680350244569600 b154 = −251184784074299007561286320 b155 = −87464244753745424300736000 b156 = −137243848567305068804488520 b157 = −286675820524507063672675993 b158 = 391907233944915164121144864 b159 = 531122523280024395014816794 b160 = −1560655139996593684095106336 b161 = 3760122027727895557551059433 b162 = −88629050773656370280979904 b163 = −5171315387161144294026388308 b164 = 1221770371975293894648126144 b165 = −932340071814670759258887287 b166 = 617572180718464402590098336 b167 = 588807353130457320567680154 b168 = 101734228697173615509243488 b169 = 241273820025099567960612487 p3 =c1 t12 12 12 11 10 12 9 12 8 12 7 12 6 12 5 12 1 t3 + c2 t1 t3 + c3 t3 t1 + c4 t3 t1 + c5 t3 t1 + c6 t3 t1 + c7 t3 t1 + c8 t3 t1 + c9 t43 t12 3 12 2 12 12 12 12 11 11 11 1 + c10 t3 t1 + c11 t3 t1 + c12 t3 t1 + c13 t1 + c14 t3 t1 + c15 t1 t3 + c16 t10 11 9 11 8 11 7 11 6 11 5 11 4 11 3 t1 + c17 t3 t1 + c18 t3 t1 + c19 t3 t1 + c20 t3 t1 + c21 t3 t1 + c22 t3 t1 + c23 t33 t11 2 11 11 11 12 10 11 10 10 10 1 + c24 t3 t1 + c25 t3 t1 + c26 t1 + c27 t3 t1 + c28 t3 t1 + c29 t1 t3 + c30 t93 t10 8 10 7 10 6 10 5 10 4 10 3 10 1 + c31 t3 t1 + c32 t3 t1 + c33 t3 t1 + c34 t3 t1 + c35 t3 t1 + c36 t3 t1 + c37 t23 t10 10 10 12 9 11 9 10 9 9 9 1 + c38 t3 t1 + c39 t1 + c40 t3 t1 + c41 t3 t1 + c42 t3 t1 + c43 t1 t3 + c44 t83 t91 + c45 t73 t91 + c46 t63 t91 + c47 t53 t91 + c48 t43 t91 + c49 t33 t91 + c50 t23 t91 + c51 t3 t91 + c52 t91 + c53 t12 8 11 8 10 8 9 8 8 8 7 8 6 8 3 t1 + c54 t3 t1 + c55 t3 t1 + c56 t3 t1 + c57 t3 t1 + c58 t3 t1 + c59 t3 t1 + c60 t53 t81 + c61 t43 t81 + c62 t33 t81 + c63 t23 t81 + c64 t3 t81 + c65 t81 + c66 t12 7 11 7 3 t1 + c67 t3 t1 LIMIT CYCLES FOR GENERALIZED PWLS WITH ONLY CENTERS 15 + c68 t10 7 9 7 8 7 7 7 6 7 5 7 4 7 3 7 3 t1 + c69 t3 t1 + c70 t3 t1 + c71 t3 t1 + c72 t3 t1 + c73 t3 t1 + c74 t3 t1 + c75 t3 t1 + c76 t23 t71 + c77 t3 t71 + c78 t71 + c79 t12 6 11 6 10 6 9 6 8 6 3 t1 + c80 t3 t1 + c81 t3 t1 + c82 t3 t1 + c83 t3 t1 + c84 t73 t61 + c85 t63 t61 + c86 t53 t61 + c87 t43 t61 + c88 t33 t61 + c89 t23 t61 + c90 t3 t61 + c91 t61 + c92 t12 5 11 5 10 5 9 5 8 5 7 5 6 5 5 5 3 t1 + c93 t3 t1 + c94 t3 t1 + c95 t3 t1 + c96 t3 t1 + c97 t3 t1 + c98 t3 t1 + c99 t3 t1 + c100 t43 t51 + c101 t33 t51 + c102 t23 t51 + c103 t3 t51 + c104 t51 + c105 t12 4 11 4 3 t1 + c106 t3 t1 + c107 t10 4 9 4 8 4 7 4 6 4 5 4 4 4 3 t1 + c108 t3 t1 + c109 t3 t1 + c110 t3 t1 + c111 t3 t1 + c112 t3 t1 + c113 t1 t3 + c114 t33 t41 + c115 t23 t41 + c116 t3 t41 + c117 t41 + c118 t12 3 11 3 10 3 3 t1 + c119 t3 t1 + c120 t3 t1 + c121 t93 t31 + c122 t83 t31 + c123 t73 t31 + c124 t63 t31 + c125 t53 t31 + c126 t43 t31 + c127 t31 t33 + c128 t23 t31 + c129 t3 t31 + c130 t31 + c131 t12 2 11 2 10 2 9 2 3 t1 + c132 t3 t1 + c133 t3 t1 + c134 t3 t1 + c135 t83 t21 + c136 t73 t21 + c137 t63 t21 + c138 t53 t21 + c139 t43 t21 + c140 t33 t21 + c141 t23 t21 + c142 t3 t21 + c143 t21 + c144 t12 11 10 9 8 3 t1 + c145 t3 t1 + c146 t3 t1 + c147 t3 t1 + c148 t3 t1 + c149 t73 t1 + c150 t63 t1 + c151 t53 t1 + c152 t43 t1 + c153 t33 t1 + c154 t23 t1 + c155 t1 t3 + c156 t1 + c157 t12 11 10 9 8 7 6 3 + c158 t3 + c159 t3 + c160 t3 + c161 t3 + c162 t3 + c163 t3 + c164 t53 + c165 t43 + c166 t33 + c167 t23 + c168 t3 + c169 , where the coefficients ci are c1 = 492827014259845915299608227319536637731419 c2 = −2438080869483823805524125927423327101760864 c3 = −583967369729745691407332725982065481194222 c4 = −14030753213599793266190916869868749229728928 c5 = −2112398880888992440649575226736629098193739 c6 = −20704537678061895220347710778183777494702272 c7 = 16314059677623217238964322331384947587760444 c8 = −2952515995403178561164076745475852546919488 c9 = 31162658776998996833900763726327531493653461 c10 = 2995100914173664161345254044563304380156448 c11 = 8694194157488590194646535078660583863610258 c12 = 252497957065961189248293575799739160341728 c13 = 1082715163783039276166468232443383736985339 c14 = 3492602981916068139510311697321894896336160 c15 = −1086772216910382971847827935580184122246400 c16 = 19104079234490622993495661251498388385943360 c17 = −6479921110772705738680288056214355877292800 c18 = 47513651424426686589606400322923368008940000 c19 = −8645153532208056857462556794504613480307200 c20 = 71743869842303927706128932848762131389449600 c21 = −490954940952138178301360594813097836428800 c22 = 61443705627491408447388292352607012026700000 c23 = 4467727356180986120302936646931141870124800 c24 = 23145629144821074004398951377060529369111360 c25 = 1706677658787390207974200567873797981292800 c26 = 1543618187781461527518541104468774212528160 c27 = −2323816561894020815291862412380876186228962 c28 = −15221232929930773458456472178578407675326784 c29 = −34262923829620036787064124222157609493388588 c30 = −84967373635139092702075928473474253572696768 c31 = −87563593152089934929362358545649791350450174 c32 = −129428210040115971570675908069145369017890432 c33 = −3289163113523431857186036315376701628705256 c34 = −31343870431795582756934128868875937872224128 c35 = 113595081364525278063153886360041182910873026 c36 = 6148329632500583640777991728611485795069888 c37 = 31478112592247655368997669319792092455630292 c38 = −1689390982001220570825035447768473853226432 c39 = 7427447651187831461330285031989406759342558 c40 = 13383012203227907917824621303121452747105648 c41 = −1421559284855548360829961693156575825875200 c42 = 74294378073956355629576991136310304510349088 c43 = −13568022171936691683662440860954055825350400 c44 = 181447380215363356673809177074365638872101520 c45 = −10409412470586033575120823305984180370649600 c46 = 263823724112907293601299014675982880399028160 c47 = 24084297326297726275359823584854414717401600 c48 = 210938020177403305017237463430401479109381520 c49 = 32232539800884528345582162109628026374726400 c50 = 65020571727123825284877363267494066440973088 c51 = 9885292891081911817933994386586911286150400 c52 = −2629738682007095093117892728101319378238352 c53 = −8585634321747039597044812015468889237484059 c54 = −38138639261621573185878772093209972833203360 c55 = −99459097371239746361552224007534640828174994 c56 = −208662958402028599241398852262553107190714720 c57 = −250110400612387426498394945973529777602342645 c58 = −325004178824310003530726853593163716251753280 c59 = −58308874505884115504607589998842526587829820 c60 = −99112247874103864818271271760567769256669120 c61 = 244442364943242843598877630775076244637925355 c62 = −6383670908668405183963835675976378354920480 c63 = 77409215265931032806019933596610819345940206 c64 = −10500086996941854444601098016745126992493280 c65 = 26917647511261211074128350933511426131836741 c66 = 19094043267862933710085655825380950597913792 c67 = 2772610408987514284138792210269974004083200 c68 = 107352615889938001273408807761836539829959552 c69 = −435715139505203306977082946120434998233600 c70 = 255794639028007854636979038401682749241718080 c71 = 25051281408735678347817878192373970590233600 c72 = 356024253763926654472363257956615418049552640 c73 = 83647205771681762950064507358973292712934400 c74 = 257807992454289979345818756939808912045238080 c75 = 76534654714198106208073087304993063641817600 c76 = 48522959886319009715933789759605571346375552 c77 = 21147055899744739196942333294784150521433600 c78 = −20796845209977102231177595688815082658982208 16 M.E. ANACLETO, J. LLIBRE, C. VALLS AND C. VIDAL c79 = −9849593356850675629057265912349452548668636 c80 = −49076228308296251339217265678542819154922880 c81 = −118477440137686412135260416080073153484440936 c82 = −265467476778339568055632356837203309190789760 c83 = −296055099828644208169001933424649463930059620 c84 = −419598797741386963273170324312105825561834240 c85 = −46194302401357875971815524917860928049905840 c86 = −141858835155420258253205093148770087801544960 c87 = 335802656231088162646087412919754618878724380 c88 = −27027475103068849017876051450881071906131840 c89 = 116934505413169408583611402486349874317120664 c90 = −20041261590291352793028539025577567630554240 c91 = 46395112003948398478782053411467226382777764 c92 = 13074535060747681538889606780253951477429088 c93 = 6376784916240002535742343118761989697036800 c94 = 74601482244135768710184485014226207152977728 c95 = 17109988613179601769992721432844435088473600 c96 = 171662240237361494428105915501401410658345120 c97 = 54418430666371151454317072290336175287526400 c98 = 224471342258156540611335495031295585356184960 c99 = 104429153609590278533044694929123115854105600 c100 = 136207429899252139163881906944016285022825120 c101 = 82159045091585497104397343124805497714342400 c102 = −8910406882712120111403793589855870289838272 c103 = 21415118451426770791419342171936111756326400 c104 = −30781787577781125920761242770005724939274912 c105 = −5082172265702464585247355603705187123392459 c106 = −34122512559156973906757888417258486428403360 c107 = −71107324421539628081805956301982089608425394 c108 = −184021892951246111748373441843811156181114720 c109 = −177778722876474358884925292729102608494168645 c110 = −293178985563079241017830172614709547461353280 c111 = 5630357503192470717671281385280068729202180 c112 = −100671692955629954960193684385262770495069120 c113 = 263758420017792989370778900707430135626099355 c114 = −26494512564605919430001480565518515188520480 c115 = 92812247919220694808917872229478948389689806 c116 = −17851229058440404249724176045928558402093280 c117 = 39460490256819789181883049067509939501928341 c118 = 4321739489821064172804252764526646351858592 c119 = 4183362550647183634987602511988674689555200 c120 = 25116824553329108660182684756403623031756352 c121 = 13654639510991575419668368541305731256710400 c122 = 54948306158767587465505621794879169295242080 c123 = 35271134885231417852010000752136784723289600 c124 = 65160775836306878710834209738570954892576640 c125 = 56708653914086864208841965094809501621158400 c126 = 26310914036538686895342134341455570473962080 c127 = 41305648463209858365389389538488200651513600 c128 = −22720960628492070743883780548143283596467648 c129 = 10396852474010020223876659166498427186489600 c130 = −18024319923983421906518894954030091793197408 c131 = −1125002969761079454640643743261266930181922 c132 = −12093217807160562080241462843376330958782784 c133 = −21265579686952485525535597460059392318450348 c134 = −65640840281491212382211081078066041067608768 c135 = −53834367995688054524441882263226165499440574 c136 = −104664469067029119807763510989132303122978432 c137 = 16079375481294603024166020357060126186891544 c138 = −33211615754855752016170992494961780532576128 c139 = 105734685200422480615509581760981255520282626 c140 = −10212326132139845778048583857976930772738112 c141 = 34774765944655369305599473438282827293288532 c142 = −7617078681372297909152188388464117654314432 c143 = 16022341156649107222885282232531878871069598 c144 = 546974052057527699345306472081444121573808 c145 = 913975111339860772366185361073050700230400 c146 = 3254472818263915389281563228566178584332448 c147 = 3197036819470756287680716967080561118060800 c148 = 6589009804973044595065681967128584515453520 c149 = 7668244565973921463169073165654146916339200 c150 = 6556544600134649349878176201217168576365760 c151 = 11351453808928484172200593344992165926156800 c152 = −140804311839602487680472717135922086306480 c153 = 7916445641118925243785908255978882074387200 c154 = −6585722983165385395068723479751240032115552 c155 = 1950174690033467019439856470633352646739200 c156 = −3769885109957790462639499772041999422138192 c157 = −68684423999952508585735482541688566934421 c158 = −1691446355125215979754414262605771425664864 c159 = −2527061143852823572625761565219155128485262 c160 = −9360086128095368883956155412510591229536928 c161 = −6630305236928942947664202400225894960245339 c162 = −14803774783080914882705006384326456494894272 c163 = 4055345117805025211365989277907354279147644 c164 = −3677864516934594686681181621301413907687488 c165 = 16590212427743568372365743735873656177201861 c166 = −1208389043607450702526581383314927720515552 c167 = 4600219955329106189093067501246646716799218 c168 = −1248913155348573767928782374936212903850272 c169 = 2438796771237462754231229636119130017439499 p4 =d1 t12 12 11 12 10 12 9 12 8 12 7 12 6 12 5 12 4 12 3 12 3 t1 + d2 t3 t1 + d3 t3 t1 + d4 t3 t1 + d5 t3 t1 + d6 t3 t1 + d7 t3 t1 + d8 t3 t1 + d9 t3 t1 + d10 t3 t1 + d11 t23 t12 12 12 12 11 11 11 10 11 9 11 8 11 7 11 1 + d12 t3 t1 + d13 t1 + d14 t3 t1 + d15 t3 t1 + d16 t3 t1 + d17 t3 t1 + d18 t3 t1 + d19 t3 t1 + d20 t63 t11 5 11 4 11 3 11 2 11 11 11 12 10 11 10 1 + d21 t3 t1 + d22 t3 t1 + d23 t3 t1 + d24 t3 t1 + d25 t3 t1 + d26 t1 + d27 t3 t1 + d28 t3 t1 + d29 t10 10 9 10 8 10 7 10 6 10 5 10 4 10 3 10 2 10 3 t1 + d30 t3 t1 + d31 t3 t1 + d32 t3 t1 + d33 t3 t1 + d34 t3 t1 + d35 t3 t1 + d36 t3 t1 + d37 t3 t1 + d38 t3 t10 10 12 9 11 9 10 9 9 9 8 9 7 9 6 9 5 9 1 + d39 t1 + d40 t3 t1 + d41 t3 t1 + d42 t3 t1 + d43 t3 t1 + d44 t3 t1 + d45 t3 t1 + d46 t3 t1 + d47 t3 t1 + d48 t43 t91 + d49 t33 t91 + d50 t23 t91 + d51 t3 t91 + d52 t91 + d53 t12 8 11 8 10 8 9 8 8 8 3 t1 + d54 t3 t1 + d55 t3 t1 + d56 t3 t1 + d57 t3 t1 + d58 t73 t81 + d59 t63 t81 + d60 t53 t81 + d61 t43 t81 + d62 t33 t81 + d63 t23 t81 + d64 t3 t81 + d65 t81 + d66 t12 7 11 7 3 t1 + d67 t3 t1 + d68 t10 7 9 7 8 7 7 7 6 7 5 7 4 7 3 7 2 7 7 3 t1 + d69 t3 t1 + d70 t3 t1 + d71 t3 t1 + d72 t3 t1 + d73 t3 t1 + d74 t3 t1 + d75 t3 t1 + d76 t3 t1 + d77 t3 t1 + d78 t71 + d79 t12 6 11 6 10 6 9 6 8 6 7 6 6 6 5 6 4 6 3 t1 + d80 t3 t1 + d81 t3 t1 + d82 t3 t1 + d83 t3 t1 + d84 t3 t1 + d85 t3 t1 + d86 t3 t1 + d87 t3 t1 + d88 t33 t61 + d89 t23 t61 + d90 t3 t61 + d91 t61 + d92 t12 5 11 5 10 5 9 5 8 5 7 5 3 t1 + d93 t3 t1 + d94 t3 t1 + d95 t3 t1 + d96 t3 t1 + d97 t3 t1 LIMIT CYCLES FOR GENERALIZED PWLS WITH ONLY CENTERS 17 + d98 t63 t51 + d99 t53 t51 + d100 t43 t51 + d101 t33 t51 + d102 t23 t51 + d103 t3 t51 + d104 t51 + d105 t12 4 3 t1 + d106 t11 4 10 4 9 4 8 4 7 4 6 4 5 4 4 4 3 t1 + d107 t3 t1 + d108 t3 t1 + d109 t3 t1 + d110 t3 t1 + d111 t3 t1 + d112 t3 t1 + d113 t3 t1 + d114 t33 t41 + d115 t23 t41 + d116 t3 t41 + d117 t41 + d118 t12 3 11 3 10 3 9 3 3 t1 + d119 t3 t1 + d120 t3 t1 + d121 t3 t1 + d122 t83 t31 + d123 t73 t31 + d124 t63 t31 + d125 t53 t31 + d126 t43 t31 + d127 t33 t31 + d128 t23 t31 + d129 t3 t31 + d130 t31 + d131 t12 2 11 2 10 2 9 2 8 2 7 2 6 2 3 t1 + d132 t3 t1 + d133 t3 t1 + d134 t3 t1 + d135 t3 t1 + d136 t3 t1 + d137 t3 t1 + d138 t53 t21 + d139 t43 t21 + d140 t33 t21 + d141 t23 t21 + d142 t3 t21 + d143 t21 + d144 t12 11 3 t1 + d145 t3 t1 + d146 t10 9 8 7 6 5 4 3 3 t1 + d147 t3 t1 + d148 t3 t1 + d149 t3 t1 + d150 t3 t1 + d151 t3 t1 + d152 t3 t1 + d153 t3 t1 + d154 t23 t1 + d155 t3 t1 + d156 t1 + d157 t12 11 10 9 8 7 3 + d158 t3 + d159 t3 + d160 t3 + d161 t3 + d162 t3 + d163 t63 + d164 t53 + d165 t43 + d166 t33 + d167 t23 + d168 t3 + d169 , where the coefficients di are d1 = 9658413369501255163276163798755600773 d2 = 35317295585910647321645419281661863456 d3 = 15882166700089440555049128988946135766 d4 = −105388392832058172214709783809655818464 d5 = −350325349443127055311977332321980548393 d6 = −428381522780610943179781647065840260416 d7 = −15809160534680777659466556344143159372 d8 = 147223860351680840789833153696442108096 d9 = −6615219861428775459196868998657919693 d10 = −130457607152598424884258075670839593696 d11 = −66028157665812914294761894112469629034 d12 = −18827813456539996093060500369404280288 d13 = −4438557146599625422697512301083785327 d14 = −18862342571408613262814141417644415424 d15 = −27098581623582545267174529043905218880 d16 = −164348132527117286631607204863806542344 d17 = −108963336738218450613804747575390174400 d18 = −302829116835585602363956236127401975960 d19 = −206842844152946337723563587261549468800 d20 = −273865734699193772482856811909138308880 d21 = −231398698055413101271840354030223548800 d22 = −214120289749485949651766944724108262960 d23 = −142629302486948265492336291357833294400 d24 = −109393235536598237729637222417243283944 d25 = −36208693469845596597429306057674258880 d26 = −11795353606429131565563916921033480824 d27 = 57729284157066353866155204954743858226 d28 = 234656987160644447922639514794924595296 d29 = 121820635321241028646077092132716546224 d30 = −559506549687102693511196488061990037984 d31 = −2118567476935918029212416222051124573838 d32 = −2477495357417913260398615281171412696896 d33 = −203625537106801792577927445360122094672 d34 = 947931986191336347426324901057572154176 d35 = −102183554730617865953018442800748720138 d36 = −754420171555262600826706436279072729376 d37 = −395268568494608310922992670905533033376 d38 = −106507047854476707871067577816742947168 d39 = −28645092258199231344060934367609146074 d40 = −73970283591967734228551806698844431876 d41 = −134189170000435600917229454708240589120 d42 = −676324943209845484012609069384477401456 d43 = −513753920087134271811361522644923025600 d44 = −1166568377609732222391110648329955513540 d45 = −912386100447926698984152962989285251200 d46 = −943207740013608747433032966943709157120 d47 = −966450443162048947134195117857859491200 d48 = −766862279563437264907714826444795926540 d49 = −579812415326676879975230163786352385600 d50 = −432571835027556162389038838648966019856 d51 = −146183322525855960931055940981095709120 d52 = −44703577485873172307303593503556706476 d53 = 191315042643624893059513334646493610352 d54 = 679843111749526518876465654522124399680 d55 = 740250951786465306163790653404374475432 d56 = −1051495204182701241337520378923832357760 d57 = −4567298913320431082807614362880426101740 d58 = −5601443897393791581320665217099419427840 d59 = 49378570712444040552799773994988051760 d60 = 2974683728903811939898123617369472019840 d61 = 210689382944802130009482394572683194760 d62 = −1530144929589544094017213291483429106240 d63 = −711268829965776415000812749116172887768 d64 = −176991913799198956655158497920363796480 d65 = −42828243453612253016847113385061063948 d66 = −71403228819803075647235037594292987164 d67 = −276340325428861805181413653270972231680 d68 = −826830748066137855145509801288955145784 d69 = −958698623616697982429997693878013158400 d70 = −1028876778950424268929788905061289143060 d71 = −1448686586234176482148961046779916876800 d72 = −230538758872980450035127216361479577680 d73 = −1300323422737240100069148907901297996800 d74 = −357915239475244484172345572739976525060 d75 = −707679954094433657504337184620036838400 d76 = −433986950721373872114326335761653083384 d77 = −173684819403533862335565282891614791680 d78 = −33161210415020348545593838016529151564 d79 = 350871651626256743176411109116152088858 d80 = 1089714202226686546548282214306454394880 d81 = 1859003338875912157376044120131439040308 d82 = −725364496690152082851904149676874420480 d83 = −4615250140408917891828870452214289122690 d84 = −6330834902933215056529571936973326750720 d85 = 1225792967970035594750060525069677451320 d86 = 5112490037263997374490230331701725099520 d87 = 1247826542758169314920002701108756600310 d88 = −1370741153366541356303913048202579205120 d89 = −381782134472456115772808601958238312492 d90 = −68397626439287293434385493446302240000 d91 = 4254886772984477457428118126258612658 d92 = 12133046104963011972623401053615565724 d93 = −291844218340594690554353431215560197120 d94 = −244846173890652927950500933729293270856 d95 = −885120217451118186476551063313792665600 d96 = 431235518431186969261268616512628765460 d97 = −978105138303039807026880710070743731200 d98 = 1627071619724896597197103977159274604880 d99 = −488292671827481256785879223370613171200 d100 = 944651435641691915646022513671241027460 18 M.E. ANACLETO, J. LLIBRE, C. VALLS AND C. VIDAL d101 = −123529706847342385625553413706364825600 d102 = 30059231916484628334052636883333050744 d103 = −20066174212377439944357268308262917120 d104 = 24264677572886400700741149075829026124 d105 = 298947609281456947471060011296941304352 d106 = 1015571184479274887945888415901154623680 d107 = 1870011388733431021645249868715687489432 d108 = 113151275426306775770268615262777482240 d109 = −3004180338976207791453575875424774221740 d110 = −3841787582529920346766663371050306307840 d111 = 747067120340221896655180543948731311760 d112 = 4553872354179739072488325653565854579840 d113 = 958780374827108496447812202232516424760 d114 = −670791068842013309414772587386715106240 d115 = −58684982615440335953157510489007593768 d116 = 33911229657016281872715783221501147520 d117 = 18362457607105685410733138741227700052 d118 = 23269440792038131824508691360632637556 d119 = −155931232278433642897511076231639961280 d120 = 25419288014496715869287060255471171536 d121 = −404343560709587259839531397696287086400 d122 = 592419701955607140632641978191485710740 d123 = −225480698930525274925672195451130492800 d124 = 1269518902180883447935607718727406758720 d125 = 215576642185627504458109575682347427200 d126 = 761647174705759245502456606068942163740 d127 = 269877299716650008822665528091137793600 d128 = 96936655371041921329274815608046861936 d129 = 77232287031650846380904078422306998720 d130 = 14538528113017567174562318969864280156 d131 = 99404931705960888059481036045796059546 d132 = 519991046988837856069467748686791865696 d133 = 727569019673169628475399569098135618144 d134 = 361617615635429269013891901312864074016 d135 = −1647514051751930093065880066470931466038 d136 = −1279658395429461206312885276116288152896 d137 = −671248897741124756602582618684246756272 d138 = 1812692743336137088705829198395213818176 d139 = −218528322121937423421642457624993652338 d140 = −360583181081856471158675468729082937376 d141 = −119562357104602583186787956541658633456 d142 = −15383958031757819775071058032257116768 d143 = −11198162492150780674426888209440712754 d144 = −508747876203552493948483304158608880 d145 = −33336750989847701874516372622716625600 d146 = −16687604062468319930738350572016104280 d147 = −73131383526251234595421253190974728000 d148 = 65798843887621757503692849348880689800 d149 = 9481109433318411212836843567432144000 d150 = 174253433616727022366824654279569224400 d151 = 138597636382504507037194654979298544000 d152 = 81574667942253395974990381907171624800 d153 = 122910165309287351426775629605984872000 d154 = −16030056360066960445720948593712796280 d155 = 33588270896504788323322698762376574400 d156 = −5328990759479858994369392838053381880 d157 = 16524980377249508600570472543143773653 d158 = 113630644561456796396389988083229969056 d159 = 132889555745628761287666923265136799046 d160 = 120478890846627130432752811044710309536 d161 = −332924394618770796648409687579530053193 d162 = −216375653076266651973962259045461604416 d163 = −260914433744903730131510657201421953772 d164 = 197334823356238957446102006522742364096 d165 = −109137199266638537567166444593893234493 d166 = −154350900419504599973050998669751065696 d167 = −9990020729345629479621282675926173754 d168 = −28380134726799150205020914227035374688 d169 = 1756153198289277660124414662768185553 4. Acknowledgements The second author is supported by the Ministerio de Ciencia, Innovación y Uni- versidades, Agencia Estatal de Investigación grants MTM2016-77278-P (FEDER), the Agència de Gestió d’Ajuts Universitaris i de Recerca grant 2017SGR1617, and the H2020 European Research Council grant MSCA-RISE-2017-777911. The third author is partially supported by FCT/Portugal through UID/MAT/04459/2013. The present paper is part of the thesis of the first author. References [1] A. Andronov, A. Vitt and S. Khaikin, Theory of Oscillations, Pergamon Press, Oxford, 1966. [2] M. Di Bernardo, C.J. Budd, A.R. Champneys and P. Kowalczyk, Piecewise–Smooth Dynamical Systems: Theory and applications, Appl. Math. Sci. Series 163, Springer–Verlag, London, 2008. [3] A.F. Filippov, Differential Equations with Discontinuous Right–Hand Sides, Translated from the Russian, Mathematics and its Applications (Soviet Series) 18, Kluwer Academic Pub- lishers Group, Dordrecht, 1988. [4] E. Freire, E. Ponce, F. 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Simpson, Bifurcations in Piecewise–Smooth Continuous Systems, World Scientific Series on Nonlinear Science A, vol 69, World scientific, Singapure, 2010. 1 Departamento de Matemática, Universidad del Bio-Bio, Concepción, Avda. Collao 1202, Chile Email address: manacleto@ubiobio.cl, clvidal@ubiobio.cl 2 Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bel- laterra, Barcelona, Catalonia, Spain Email address: jllibre@mat.uab.cat 3 Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais 1049-001, Lisboa, Portugal Email address: cvalls@math.ist.utl.pt