This is a preprint of: “The extended 16-th Hilbert problem for discontinuous piecewise linear centers separated by a non-regular line”, Marina Esteban, Jaume Llibre, Clàudia Valls, Internat. J. Bifur. Chaos Appl. Sci. Engrg., vol. 31(15), 2150225, 2021. DOI: [10.1142/S0218127421502254] THE EXTENDED 16-TH HILBERT PROBLEM FOR DISCONTINUOUS PIECEWISE LINEAR CENTERS SEPARATED BY A NON-REGULAR LINE MARINA ESTEBAN, JAUME LLIBRE, AND CLAUDIA VALLS Abstract. We study the discontinuous piecewise differential systems formed by two linear centers separated by a non-regular straight line. We provide upper bounds for the maximum number of limit cycles that these discontinuous piecewise differential systems can exhibit and we show that these upper bounds are reached. 1. Introduction and statement of the main result One of the main interesting objects in the study of differential systems are limit cycles. A limit cycle is a periodic orbit of the differential system isolated in the set of all periodic orbits of the system. Limit cycles have played and are playing an important role for explaining phsyical phenomena, see for instance the limit cycle of van der Pol equation [18, 19], or the one of the Belousov-Zhavotinskii model [2, 21], etc. The extended 16th Hilbert problem, that is, to find an upper bound for the maxi- mum number of limit cycles that a given class of differential systems can exhibit, is in general an unsolved problem. Only for very few classes of differential system this problem has been solved. For the class of discontinuous piecewise differential sys- tems here studied, we can obtain the solution by using the first integrals of the two linear centers which form the discontinuous piecewise differential system separated by a non-regular line. The study of the piecewise linear differential systems goes back to Andronov, Vitt and Khaikin [1], and nowadays such systems still continue to receive the attention of many researchers. These differential systems are widely used to model processes appearing in electronics, mechanics, economy, etc., see for instance the books of di Bernardo et al. [3] and Simpson [20], the survey of Makarenkov and Lamb [17], as well as hundreds of references quoted in these last three works. The simplest class of discontinuous piecewise differential systems are the planar ones formed by two pieces separated by a straight line having a linear differential system in each piece. Several authors have tried to determine the maximum number of limit cycles for this class of discontinuous piecewise differential systems. Thus, in one of the first papers dedicated to this problem, Giannakopoulos and Pliete [8] 2010 Mathematics Subject Classification. Primary 34A05. Secondary 34C05, 37C10. Key words and phrases. discontinuous piecewise linear systems, linear centers, non-regular line. 1 2 MARINA ESTEBAN, JAUME LLIBRE, AND CLAUDIA VALLS in 2001, showed the existence of discontinuous piecewise linear differential systems with two limit cycles. Then, in 2010 Han and Zhang [9] found other discontinuous piecewise linear differential systems with two limit cycles and they conjectured that the maximum number of limit cycles for discontinuous piecewise linear differential systems with two pieces separated by a straight line is two. But in 2012 Huan and Yang [11] provided numerical evidence of the existence of three limit cycles in this class of discontinuous piecewise linear differential systems. In 2012, Llibre and Ponce [14] inspired by the numerical example of Huan and Yang, proved for the first time that there are discontinuous piecewise linear differential systems with two pieces separated by a straight line having three limit cycles. Later on, other authors obtained also three limit cycles for discontinuous piecewise linear differential systems with two pieces separated by a straight line, see Braga and Mello [4] in 2013, Buzzi, Pessoa and Torregrosa [5] in 2013, Liping Li [13] in 2014, Freire, Ponce and Torres [7] in 2014, and Llibre, Novaes and Teixeira [15] in 2015. But proving that discontinuous piecewise linear differential systems separated by a straight line have at most three limit cycles is an open problem. We consider the discontinuous piecewise differential systems of the form  F1 (x, y) = (f1 (x, y), g1 (x, y)) if (x, y) ∈ S1 , (1) (ẋ, ẏ) = F(x, y) = F2 (x, y) = (f2 (x, y), g2 (x, y) if (x, y) ∈ S2 , where fi , gi are linear polynomials for i = 1, 2 and the regions S1 and S2 are defined by S1 = {(x, y) ∈ R2 : x > 0, y > αx}, S2 = {(x, y) ∈ R2 : x < 0, or x > 0, y < αx}, being α ∈ R. System (1) is bi-valued on the non-regular separation line (2) S = S1 ∩ S2 = {(0, y) : y ≥ 0} ∩ {(x, αx) : x > 0}. As usual, system (1) is denoted by F = (F1 , F2 , S) or simply by F = (X, Y ), when the separation line S is well understood. In order to establish a definition for the trajectories of F and to investigate its behaviour we need a criterion for the transition of the orbits between S1 and S2 across S. The gradient ∇S(p) at the point p of the discontinuity line is given by the vector (−1, 0) at the points p = (0, y) with y > 0, and by the vector (α, −1) at the points p = (x, αx) with x > 0. The contact between the vector field F1 (or F2 ) and the discontinuity line S, is characterized by the derivative of S in the direction of the vector field F1 (or F2 ), also known as the Lie derivative of S with respect to F1 (or F2 ), that is by the expression F1 S(p) = h∇S(p), F1 (p)i , and for i ≥ 2 we define Fi1 S(p) = ∇Fi−1 1 S(p), F1 (p) , where h., .i is the usual inner product in R2 . The basic results of the discontinuous piecewise differential systems in this context were stated by Filippov [6]. We can divide the discontinuity line S in the following three sets: (a) Crossing set S c : {p ∈ S : F1 S(p) · F2 S(p) > 0}; (b) Escaping set S e : {p ∈ S : F1 S(p) > 0 and F1 S(p) < 0}; (c) Sliding set S s : {p ∈ S : F1 S(p) < 0 and F1 S(p) > 0}. PIECEWISE LINEAR CENTERS SEPARATED BY A NON-REGULAR LINE 3 The escaping S e or sliding S s regions are respectively defined on points of S where both vector fields F1 and F2 simultaneously point outwards or inwards from S, while the interior of its complement in S defines the crossing region S c . The complementary of the union of these regions is the set formed by the tangency points between F1 or F2 with S. Following the Filippov’s convention [6] the discontinuous piecewise linear dif- ferential systems can have sliding or crossing limit cycles. A sliding limit cycle contains sliding segments on the line of discontinuity, whereas crossing limit cycles contain only crossing points. In this paper we work only with crossing limit cycles, or simply limit cycles. In [12] the authors proved that any piecewise differential system of the form (1) can be transformed into a piecewise differential system with α = 0 by means of an invertible linear transformation. Thus, it is not restrictive to consider α = 0. In this paper, we study the maximum number of limit cycles of systems (1) with α = 0 formed by two linear differential centers, namely  ẋ = −Ax − (A2 + Ω2 )y + B, (3) for (x, y) ∈ R1 , ẏ = x + Ay + C, and  ẋ = −ax − (a2 + ω 2 )y + b, (4) for (x, y) ∈ R2 , ẏ = x + ay + c, with Ω, ω > 0, A, B, C, a, b, c ∈ R, A, a 6= 0 and in the regions R1 and R2 are R1 = {(x, y) ∈ R2 : x ≥ 0, y ≥ 0}, R2 = {(x, y) ∈ R2 : x ≤ 0, or x ≥ 0, y ≤ 0}. Each system (3) and (4) have, respectively, the first integrals H1 (x, y) = (x + Ay)2 + 2(Cx − By) + y 2 Ω2 , (5) H2 (x, y) = (x + ay)2 + 2(cx − by) + y 2 ω 2 . Now, the discontinuity line is denoted by R = Ry ∪ Rx = {(0, y) : y ≥ 0} ∪ {(x, 0) : x ≥ 0}. We note that, differential systems (3) and (4) are the most general expressions of linear differential systems, see Lemma 1 of [16] for more details . Following the notation of [12], we denote by I2 a crossing limit cycle having two intersection points with either Rx , or Ry ; by II2 a crossing limit cycle having one intersection point with Rx and another point with Ry (see Figure 1(a)); and by II4 a crossing limit cycle having two intersection points with Rx and two intersection points with Ry (see Figure 1(b)). The study of the existence of limit cycles of type I2 is the study of limit cycles existing for two linear centers separated by a straight line, and it was proved in Theorem 3 of [16] that such piecewise differential systems have no limit cycles. Then, we restrict our analysis to limit cycles of II2 or II4 type. The main result of the present paper is to study the maximum number of limit cycles of types II2 and II4 that the discontinuous piecewise linear differential sys- tems (3)-(4) can exhibit. 4 MARINA ESTEBAN, JAUME LLIBRE, AND CLAUDIA VALLS 5 1.5 4 3 1.0 2 1 0.5 0 -1 0.0 -8 -6 -4 -2 0 0.0 0.5 1.0 1.5 2.0 (a) The pair of limit cycles of type (b) The limit cycle of type II4 for II2 for system (8)–(9). system (11)-(12). Figure 1 Theorem 1. Consider discontinuous piecewise differential systems separated by the non-regular line R and formed by two arbitrary linear differential centers (3)- (4). The maximum number of limit cycles of these discontinuous piecewise linear differential systems are: (a) two of type II2 and there exists systems with exactly two limit cycles of this type (see Figure 1(a)); (b) one of type II4 and there are systems with exactly one limit cycle of this type (see Figure 1(b)). The proof of Theorem 1 is given in the following section. Other paper with several non-regular lines of discontinuity is [10]. 2. Proof of Theorem 1 We separate the proof of both statements of Theorem 1. Proof of statement (a) of Theorem 1. We consider the discontinuous planar linear differential system (3)-(4). If there exists a crossing limit cycle of type II2 , then it must intersect the non-regular separation curve R in two points of the form (x, 0) and (0, y), both different from the origin. Since the functions H1 and H2 defined in (5) are first integrals of the systems (3) and (4) respectively, these points must satisfy the equations H1 (x, 0) − H1 (0, y) = 2Cx + x2 + 2By − A2 y 2 − y 2 Ω2 = 0, (6) H2 (x, 0) − H2 (0, y) = 2cx + x2 + 2by − a2 y 2 − y 2 ω 2 = 0. Assume that there exists a crossing limit cycle of type II2 passing trough the points (x1 , 0), (0, y1 ). Then, by solving equations in (6) with x = x1 and y = y1 with respect to C and c we get −x21 + y1 (−2B + y1 (A2 + Ω2 )) −x21 + y1 (−2b + y1 (a2 + ω 2 )) C= , c= . 2x1 2x1 PIECEWISE LINEAR CENTERS SEPARATED BY A NON-REGULAR LINE 5 Now, assume that there exists a second crossing limit cycle of type II2 passing through the points (x2 , 0) and (0, y2 ) with 0 < x1 < x2 and 0 < y1 < y2 . Case 1: Assume x2 y1 − x1 y2 6= 0. Using the obtained values of C and c we solve equations in (6) for x = x2 and y = y2 for the parameters B and b, i.e. we solve the system H1 (x2 , 0) − H1 (0, y2 ) = x22 + 2By2 − A2 y22 − y22 Ω2 x2 (−x21 + y1 (−2B + y1 (A2 + Ω2 ))) + = 0, x1 (7) H2 (x2 , 0) − H2 (0, y2 ) = x22 + 2by2 − a2 y22 − y22 ω 2 x2 (−x21 + y1 (−2b + y1 (a2 + ω 2 ))) + = 0, x1 and we get −x21 x2 + x2 y12 (A2 + Ω2 ) + x1 (x22 − y22 (A2 + Ω2 )) B= , 2(x2 y1 − x1 y2 ) −x2 x2 + x2 y12 (a2 + ω 2 ) + x1 (x22 − y22 (a2 + ω 2 )) b= 1 . 2(x2 y1 − x1 y2 ) Now we substitute the obtained values of the parameters C, c, B, b in equations (6) and solving them with respect to x and y, obtaining the solutions (x, y) = (0, 0), (x, y) = (x1 , y1 ), (x, y) = (x2 , y2 ),   (x1 − x2 )(x2 y1 + x1 y2 ) (y1 − y2 )(x2 y1 + x1 y2 ) (x, y) = , . x1 y2 − x2 y1 x2 y1 − x1 y2 Due to the conditions 0 < x1 < x2 and 0 < y1 < y2 , the last solution has one negative component, so the maximum number of crossing limit cycles of type II2 for system (3)-(4) is two. Case 2: Assume x1 y2 = x2 y1 . We solve equations (7) now for the parameters A and a, and we get p p (x1 − y1 Ω)(x1 + y1 Ω) (x1 − y1 ω)(x1 + y1 ω) A= , a= . y1 y1 Substituting the values of C, c, A, a in (6) and taking into account that x1 y2 = x2 y1 , we obtain that y1 y= x, x1 and so, in this case there exists a continuum of periodic orbits, and consequently no limit cycles. In summary, the maximum number of limit cycles of type II2 that the discon- tinuous piecewise linear differential systems (3)-(4) can have is two. Now we give a discontinuous piecewise linear differential system (3)-(4) having exactly two limit cycles of type II2 . In region R1 we consider the linear differential center 3 43 (8) ẋ = − − 2x − 8y, ẏ = + x + 2y, 2 4 6 MARINA ESTEBAN, JAUME LLIBRE, AND CLAUDIA VALLS with the first integral   2 43 3 H1 (x, y) = 4y + 2 x + y + (x + 2y)2 ; 4 2 and in region R2 we consider the linear differential center 7 (9) ẋ = −x − 2y, ẏ = +x+y 4 with the first integral 7 H2 (x, y) = y 2 + x + (x + y)2 . 2 In this case, the two solutions of equations (6) are     1 3 (x1 , y1 ) = ,1 , (x2 , y2 ) = 1, , 2 2 and the corresponding limit cycles are shown in Figure 1(a).  Proof of statement (b) of Theorem 1. We consider again the discontinuous piece- wise linear differential systems (3)-(4). If there exists a limit cycle of type II4 , then it has four intersection points on the discontinuity line R of the form (x1 , 0), (x2 , 0), (0, y1 ) and (0, y2 ), satisfying 0 < x1 < x2 , 0 < y1 < y2 and the equations H1 (x1 , 0) − H1 (0, y1 ) = 2Cx1 + 2By1 + x21 − (A2 + Ω2 )y12 = 0, H1 (x2 , 0) − H1 (0, y2 ) = 2Cx2 + 2By2 + x22 − (A2 + Ω2 )y22 = 0, (10) H2 (x1 , 0) − H2 (x2 , 0) = (x1 − x2 )(2c + x1 + x2 ) = 0,  H2 (0, y1 ) − H2 (0, y2 ) = (y1 − y2 ) 2b − (y1 + y2 )(a2 + ω 2 ) = 0. Since x1 < x2 and y1 < y2 , we can obtain from the third and fourth equations that 2b x2 + x1 = −2c, y2 + y1 = . a2 + ω 2 Using these last expressions, we can write the first and second equations of (10) in terms of x1 and y1 , and we get P (x1 , y1 ) = 2Cx1 + 2By1 + x21 − (A2 + Ω2 )y12 = 0, 1 Q(x1 , y1 ) = 2 (2c + x1 )(2c − 2C + x1 )(a2 + ω 2 )2 − (2b (a + ω 2 )2 − y1 (a2 + ω 2 ))(−2B(a2 + ω 2 ) + (2b − y1 (a2 + ω 2 ))  (A2 + Ω2 )) = 0. PIECEWISE LINEAR CENTERS SEPARATED BY A NON-REGULAR LINE 7 Now, we compute the resultant of the polynomials P and Q with respect to the variable y1 , and we obtain the following polynomial in the variable x1 : R(x1 ) =16(A2 + Ω2 )(A2 b2 − (a2 + ω 2 )(bB + c(c − C)(a2 + ω 2 )) + b2 Ω2 )(A4 b2 + 2B 2 (a2 + ω 2 )2 − (a2 + ω 2 )(3bB + c(c − C)(a2 + ω 2 ))Ω2 + b2 Ω4 + A2 (−(a2 + ω 2 )(3bB + c(c − C)(a2 + ω 2 )) + 2b2 Ω2 )) + 32c(a2 + ω 2 )2 (A2 + Ω2 )(−A4 b2 − B 2 (a2 + ω 2 )2 + (a2 + ω 2 )(2bB + (c − C)2 (a2 + ω 2 ))Ω2 − b2 Ω4 + A2 ((a2 + ω 2 )(2bB + (c − C)2 (a2 + ω 2 )) − 2b2 Ω2 ))x1 + (a2 + ω 2 )2 (A2 + Ω2 )(−A4 b2 − B 2 (a2 + ω 2 )2 + (a2 + ω 2 )(2bB + (c − C)2 (a2 + ω 2 ))Ω2 − b2 Ω4 + A2 ((a2 + ω 2 )(2bB + (c − C)2 (a2 + ω 2 )) − 2b2 Ω2 ))x21 . The polynomial R(x1 ) is quadratic in the variable x1 , whose solutions are x1 and −x1 − 2c. This fact implies that equations (10) have only a pair of solutions, that is (x1 , x2 ) = (x1 , −x1 − 2c) (and analogously for (y1 , y2 )). Hence the maximum number of limit cycles of type II4 that the discontinuous piecewise linear differential system (3)-(4) can have is one. Now we give a discontinuous piecewise differential systems with the discontinuity line R formed by two linear differential centers having exactly one limit cycle of type II4 . In region R1 we consider the linear differential center 17 191 (11) ẋ = − x − 2y, ẏ = − +x+y 8 128 with the first integral   191 17 H1 (x, y) = −2 x + y + y 2 + (x + y)2 , 128 8 and in region R2 we consider the linear differential center 187 (12) ẋ = + x − 2y, ẏ = −1 + x − y, 128 with the first integral 187 H2 (x, y) = −2(x + y) + y 2 + (x − y)2 . 128 For this case, the solutions of equations (10) satisfying 0 < x1 < x2 and 0 < y1 < y2 are x1 = 0.267515, x2 = 1.73249, y1 = 0.187568, y2 = 1.27337, and the corresponding limit cycle is shown in Figure 1(b).  3. Conclusions Every discontinuous piecewise differential system separated by a non-regular line of the form (2) and formed by two arbitrary linear differential centers can be written as the discontinuous piecewise differential system (2)–(3). For this class of discon- tinuous piecewise differential systems we provide an upper bound for the maximum 8 MARINA ESTEBAN, JAUME LLIBRE, AND CLAUDIA VALLS number of limit cycles that they can exhibit, i.e. we have solved the extended 16th Hilbert problem to this class of discontinuous piecewise linear systems. More precisely, this class of discontinuous piecewise differential systems can have two kinds of limit cycles, denoted by II2 and II4, see Figures 1(a) and 1(b), respec- tively. The maximum number of limit cycles of type II2 is two and of type II4 is one. Moreover, we have shown that these upper bounds are reached, see Theorem 1. Acknowlegments The first author is partially supported by the Ministerio de Economı́a y Compet- itividad in the frame of project PGC2018-096265-B-I00, and by the Consejerı́a de Economı́a y Conocimiento de la Junta de Andalucı́a, under grant P12-FQM-1658. The second author is supported by the Ministerio de Ciencia, Innovación y Uni- versidades, Agencia Estatal de Investigación grants MTM2016-77278-P (FEDER) and PID2019-104658GB-I00 (FEDER), the Agència de Gestió d’Ajuts Universi- taris i de Recerca grant 2017SGR1617, and the H2020 European Research Coun- cil grant MSCA-RISE-2017-777911. The third author is partially supported by FCT/Portugal through UID/MAT /04459/2019. References [1] A. Andronov, A. Vitt and S. Khaikin, Theory of Oscillations, Pergamon Press, Oxford, 1966 (Russian edition ≈ 1930). [2] B. P. Belousov, A periodic reaction and its mechanism, Collection of Short Papers on Radiation Medicine for 1958 (1959).(Moscow: Med. Publ.) [in Russian]. [3] M. di Bernardo, C. J. 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W. Simpson, Bifurcations in Piecewise-Smooth Continuous Systems, World Sci. Ser. Nonlinear Sci. Ser. A, vol. 69, World Scientific, Singapore, 2010. [21] A.M. Zhabotinsky, Periodical oxidation of malonic acid in solution (a study of the Belousov reaction kinetics), Biofizika 9 (1964), 306–311. Dept. Matemática Aplicada II and Instituto de Matemáticas (IMUS),, Escuela Técnica Superior de Ingenierı́a de la Universidad de Sevilla,, Camino de los Descubrimientos s/n, 41092 Sevilla (SPAIN) Email address: marinaep@us.es Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain Email address: jllibre@mat.uab.cat Centro de Análise Matemática, Geometria e Sistemas Dinâmicos, Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisboa, Por- tugal Email address: cvalls@math.tecnico.ulisboa.pt