This is a preprint of: “Limit cycles in continuous and discontinuous piecewise linear differential systems with two pieces separated by a straight line”, Jaume Llibre, Bul. Acad. Ştiinţe Repub. Mold. Mat., num. 2, 3–12, 2019. LIMIT CYCLES IN CONTINUOUS AND DISCONTINUOUS PIECEWISE LINEAR DIFFERENTIAL SYSTEMS WITH TWO PIECES SEPARATED BY A STRAIGHT LINE JAUME LLIBRE Abstract. This paper is a survey on the study of the maxi- mum number of limit cycles of planar continuous and discontinu- ous piecewise linear differential systems defined in two half–planes separated by a straight line L. We restrict our attention to the crossing limit cycles, i.e. to the limit cycles having exactly two points on the straight line L. We summarize the results known by now and describe the tools for obtaining them. 1. Introduction and results Every continuous or discontinuous piecewise linear differential sys- tem with two pieces separated by a straight line in the plane R2 after a linear change of variables can be written into the form    −    −  ẋ a11 a− 12 x b1 = + for x < 0; ẏ a−21 a − 22 y b− 2 and        ẋ a+ + 11 a12 x b+ 1 = + for x > 0. ẏ a21 a+ + 22 y b+ 2 The goal of this paper is to summarize what is known about the following question: How many limit cycles can such continuous or discontinuous piecewise linear differential systems have? For continu- ous piecewise systems both linear differential systems coincides on the straight line x = 0, and for the definition of discontinuous piecewise systems we follow the rules of Filippov [9]. 2010 Mathematics Subject Classification. 34C05, 34C07, 37G15. Key words and phrases. discontinuous differential system, continuous differential system, limit cycle, piecewise linear differential system. 1 2 J. LLIBRE The study of the piecewise linear differential systems goes back to Andronov, Vitt and Khaikin [1], and nowadays such systems still con- tinue 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 [3] and Simpson [37], the survey of Makarenkov and Lamb [35], as well as hundreds of references quoted in these last three works. The simplest possible continuous but nonsmooth piecewise linear differential systems are the ones having only two pieces separated by a straight line in the plane R2 . In 1990 Lum and Chua conjectured in [33, 34] that such continuous piecewise linear differential systems have at most one limit cycle. In 1998 this conjecture was proved by Freire et al. [10]. In 2013 a new and shorter proof was done by Llibre, Ordóñez and E. Ponce [26], and recently another proof has been provided by Carmona, Fernández-Sánchez and Novaes [6]. But in all these papers the authors forgot to analyze the case when the two linear differential systems which form the piecewise linear differential system have no equilibrium points, this case was studied in Llibre and Teixeira [28] in 2016 where it is proved that the continuous piecewise systems have no limit cycles, but the discontinuous piecewise system can have at most one limit cycle. The main objective of this paper is to study the problem of Lum and Chua just for the class of discontinuous piecewise linear differ- ential systems in the plane with two pieces separated by a straight line. Several authors have tried to determine the maximum number of limit cycles for this class of discontinuous piecewise linear differential systems. Thus in one of the first papers dedicated to this problem Giannakopoulos and Pliete [13] in 2001 shown the existence of dis- continuous piecewise linear differential systems with two limit cycles. Then in 2010 Han and Zhang [16] 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 [17] provided numerical evidence of the existence of three limit cycles in this class of discontinuous piece- wise linear differential systems. In 2012 Llibre and Ponce [27] 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. More precisely, they showed that the following discontinuous piecewise linear 3 differential system with two zones separated by the straight line x = 1  + A X if x ≥ 1, (1) Ẋ = A− X if x < 1, where X = (x, y)T with  19   4 20  −1 −    3 3  A+ =  50  and A =  −  19 377 26 1 − 50 750 15 has the three limit cycles shown in Figure 1. Figure 1. The first example with 3 limit cycles. Later on other authors obtained also three limit cycles for discontin- uous piecewise linear differential systems with two pieces separated by a straight line, see Braga and Mello [4], 2013, Buzzi, Pessoa and Tor- regrosa [5], 2013, Liping Li [21], 2013, Freire, Ponce and Torres [12], 2014, and Llibre, Novaes and Teixeira [23], 2015. The linear differential system that we consider in every half–plane extended to the full plane is either a focus (F) (include in this class of foci the centers), or a node (N) or a saddle (S). Thus in the beginning we distinguish six classes or types of planar discontinuous piecewise linear differential systems: FF, FN, FS, NN, NS and SS. Limit cycles of discontinuous piecewise linear differential systems with two pieces separated by a straight line have been studied by many 4 J. LLIBRE authors. In addition to the papers quoted by now we mention also some more articles. In 2010 Shui, Zhang and Li [36] obtained some general results on the class of discontinuous piecewise linear differential systems here consid- ered, mainly about the existence of heteroclinic and homoclinic orbits. In 2012 Freire, Ponce and Torres [11] obtained a Liénard-like canon- ical form which is topologically equivalent to the original system if one restricts the attention to orbits with no points in the sliding set. Moreover they characterize some bifurcations leading to the birth of crossing periodic orbits and localize certain parameter regions where the number of crossing limit cycles is exactly two. In 2013 Artés, Llibre, Medrado and Teixeira [2] studied the case SS and showed that this case can produce a focus on the straight line of discontinuity, and that the maximum number of limit cycles that can bifurcate from this focus is two. The paper of Llibre, Teixeira and Torregrosa [30] was submitted at the beginning of 2012 and published in 2013. In it for the six types of systems FF, FN, FS, NN, NS, SS they provide examples with two limit cycles. In 2013 Huan and Yang [18] showed that systems SS can exhibit two limit cycles, and in 2014 they proved in [19] that systems NN can also exhibit two limit cycles. In 2015 Euzébio and Llibre [8] proved that the maximum number of crossing limit cycles, when one of the two linear differential systems has its equilibrium point on the straight line of discontinuity, is larger than two and smaller than four. This result was improved by Llibre, Novaes and Teixeria [24] in 2015 by proving that such a maximum is two. In 2015 Gouveia, Llibre and Novaes [14] studied the linear differential center (x0 , y 0 ) = (−y, x) perturbed in the class of all discontinuous piecewise linear differential systems with two zones separated by the straight line y = 0. Using the Bendixson transformation they provided sufficient conditions to ensure the existence of a crossing limit cycle coming from the infinity. In 2017 Castillo, Llibre and Verduzco [7] studied the creation or destruction of a crossing limit cycle when a sliding segment changes its stability, this phenomenon is known as a pseudo–Hopf bifurcation. In that paper, under generic conditions, the authors did an unfolding for 5 such bifurcation, and proved the existence and uniqueness of a crossing limit cycle for such class of piecewise systems. In 2019 Shimin Li and Llibre [22] provided the phase portraits in the Poincaré disc of those piecewise systems having a unique finite singular point which is a node or a focus. Moreover they also gave sufficient and necessary conditions for the existence and uniqueness of limit cycles in that class of systems. We recall that there are three classes of linear nodes: nodes with different eigenvalues N, nodes with equal eigenvalues whose linear part does not diagonalize N’, and nodes with equal eigenvalues whose linear part diagonalizes, called star nodes. Clearly if we have a star node this prevents the existence of crossing periodic orbits. Moreover now we shall distinguish between focus F and centers C. Then we can consider 15 classes or types of planar discontinuous piecewise linear differential systems: FF, FC, FN, FN’, FS, CC, CN, CN’, CS, NN, NN’, NS, N’N’, N’S and SS. Summarizing the results of these articles we have that the maximum number of known limit cycles which one of these systems can exhibit is given in the following table. F C N N’ S F 3 2 3 3 3 C – 0 1 1 1 N – – 2 2 2 N’ – – – 2 2 S – – – – 2 But the main open question remains: Is 3 the maximum number of crossing limit cycles which a discontinuous piecewise linear differential systems with a straight line of separation can have? 2. Computation of the crossing periodic orbits We want to recall the four different more usual techniques for com- puting analytically the periodic solutions of discontinuous piecewise lin- ear differential systems based on the first integrals, or on the Poincaré map, or on the Poincaré map plus the Newton-Kantorovich Theorem, or on the averaging theory or equivalently the Melnikov integral. 6 J. LLIBRE 2.1. Computation of crossing periodic orbits using first inte- grals. . Consider a differential system (2) ẋ = f (x), x ∈ Ω ⊂ Rn , where Ω is an open subset of Rn , and f (x) = (f1 (x), . . . , fn (x)) is a C k function defined in Ω with k ≥ 1. A first integral of the differential system (2) is a continuous function H(x) defined in the domain of definition Ω of the differential system, which is not constant in any neighborhood, but such that H(x) is constant on each orbit of the differential system (2). If H(x) is C 1 then it is a first integral if and only if it satisfies dH ∂H ∂H = f1 (x) + . . . + fn (x) = 0, dt ∂x1 ∂xn for all the points x ∈ Ω. Using the first integrals of the linear differential systems, Llibre and Zhang [31] have proved that the numbers of the previous table, for the discontinuous piecewise linear differential systems in R2 formed by two pieces separated by a straight line when at least one of the two linear differential systems is a center, are exactly the maximum number of crossing limit cycles that such systems can exhibit. This result also was proved previously without using the first integrals by Llibre, Novaes and Teixeira in [24]. See also the paper of Llibre and Teixeira [29] for additional examples of computing periodic orbits using first integrals. 2.2. Computation of crossing periodic orbits using Poincaré maps. The use of the Poincaré maps for computing periodic orbits is the most often used method. In fact the methods for computing peri- odic solutions using the averaging theory or equivalently the Melnikov integral are also based on the Poincaré map. The first proof that there are discontinuous piecewise linear differen- tial systems with two zones separated by a straight line having 3 limit cycles was given for the system (2). That proof uses first the Poincaré map, and then the Newton–Kantorovich Theorem. Let Br (x0 ) be the points x ∈ Rn such that |x − x0 | < r, i.e. the open ball with center x0 and radius r. We denote by Br (x0 ) the closure of Br (x0 ). For a proof of the next theorem see [20]. 7 Theorem 1 (Newton–Kantorovich Theorem). Given a function f : C ⊂ Rn → Rn and a convex C0 ⊂ C, assume that f is C 1 in C0 and that the following assumptions hold: (a) |Df (z) − Df (z 0 )| ≤ γ|z − z 0 | for all z, z 0 ∈ C0 , (b) |Df (z0 )−1 f (z0 )| ≤ α, (c) |Df (z0 )−1 | ≤ β, for some z0 ∈ C0 . Consider √ 1± 1 − 2h h = αβγ, r1,2 = α. h If h ≤ 1/2 and Br1 (z0 ) ⊂ C0 , then the sequence {zk } defined by zk+1 = zk − Df (zk )−1 f (zk ) for k = 0, 1, ... is contained in Br1 (z0 ) and converges to the unique zero of f (z) con- tained in C0 ∩ Br2 (z0 ). Now we shall present the scheme of the proof that the piecewise linear differential system (2) has the three limit cycles of Figure 1. The solution (x+ (t), y + (t)) of system Ẋ = A+ X which pass through the point (1, Y ) when the time t = 0 is x+ (t) = e19t/50 (cos t − Y sin t) , y + (t) = e19t/50 (Y cos t + sin t) , The solution (x− (t), y − (t)) of system Ẋ = A− X which passes through the point (1, Y ) when the time t = 0 is 1 −t/5 x− (t) = e (15 cos t − 100Y sin t + 23 sin t) , 15 1 −t/5 y − (t) = e (750Y cos t − 1150Y sin t + 377 sin t) . 750 Assume that through the point (1, Y ) passes a periodic solution (x+ (t), y + (t)) ∪ (x− (t), y − (t)). Then if t+ > 0 is the smallest time such that x+ (−t+ ) = 1, and t− > 0 is the smallest time such that x− (t− ) = 1, we have that y + (−t+ ) = y − (t− ). 8 J. LLIBRE Hence a periodic solution of our system is characterized by a solution (t+ , t− , Y ) of the system f1 (t+ , t− , Y ) = x+ (−t+ ) − 1 = 0, (3) f2 (t+ , t− , Y ) = x− (t− ) − 1 = 0, f3 (t+ , t− , Y ) = y + (−t+ ) − y − (t− ) = 0. So we need to find the solutions (t+ , t− , Y ) of system (3). We shall − prove that there are 3 isolated solutions (t+ k , tk , Yk ) for k = 1, 2, 3 of the previous system near t+ 1 = 1.48663280365501727068752016595.., t− 1 = 3.45108332296097156801770033378.., Y1 = 1.68119451051893996990946394580.., t+ 2 = 0.85668322292111353096467693501.., t− 2 = 3.78234111866076263903523393333.., Y2 = 0.96579985584668225513544927484.., t+ 3 = 0.39178388598443280650466201997.., t− 3 = 4.66507269554569223774066305515.., Y3 = 0.61885416518252501376067323768... So these 3 solutions will provide the 3 limit cycles of our discontinu- ous piecewise linear differential system (2). For proving the existence of these 3 isolated solutions of system (3) we shall use the Newton– Kantorovich Theorem. For the complete details of the proof see [27]. 2.3. Computation of crossing periodic orbits using averaging theory or equivalently the Melnikov integral. The equivalence between the methods based on the averaging theory and the Melnikov integral for finding periodic orbits was proved by Han, Romanovski and Zhang [15]. The next theorem is the first-order averaging theory developed, for computing periodic orbits of discontinuous piecewise differential sys- tems separated by the curve h(t, x) = 0, by Llibre, Novaes and Teixeira [25], and it can be used for studying the periodic orbits of discontinuous piecewise linear differential systems separated by a straight line. Theorem 2. Consider the following discontinuous differential system (4) x0 (t) = εF (t, x) + ε2 R(t, x, ε), with F (t, x) = F1 (t, x) + sign(h(t, x))F2 (t, x), R(t, x, ε) = R1 (t, x, ε) + sign(h(t, x))R2 (t, x, ε), 9 where F1 , F2 : R × D → Rn , R1 , R2 : R × D × (−ε0 , ε0 ) → Rn and h : R × D → R are continuous functions, T –periodic in the variable t, D is an open subset of Rn , h is a C 1 function having 0 as a regular value, and Σ0 = {0} × D \ h−1 (0). Define the averaged function f : D → Rn as Z T f (x) = F (t, x)dt. 0 We assume (i) F1 , F2 , R1 , R2 and h are locally L–Lipschitz with respect to x; (ii) for a ∈ Σ0 with f (a) = 0, there exist a neighborhood V of a such that f (z) 6= 0 for all z ∈ V \{a} and dB (f, V, a) 6= 0, i.e. the Brouwer degree of f at a is not zero. (iii) If ∂h/∂t(t0, z0 ) = 0 for some (t0 , z0 ) ∈ M, then h∇x h, F1 i2 − h∇x h, F2 i2 (t0 , z0 ) > 0. Then for |ε| > 0 sufficiently small, there exists a T –periodic solution x(t, ε) of system (4) such that x(0, ε) → a as ε → 0. We note that if the function f (z) is C 1 and the Jacobian of f at a is not zero, then dB (f, V, a) 6= 0, see for instance Lloyd [32]. Acknowledgements The author is partially supported by the Ministerio de Ciencia, Inno- vación y Universidades, Agencia Estatal de Investigación grants MTM2016- 77278-P (FEDER), the Agència de Gestió d’Ajuts Universitaris i de Re- cerca grant 2017SGR1617, and the H2020 European Research Council grant MSCA-RISE-2017-777911. References [1] A. Andronov, A. Vitt and S. Khaikin, Theory of Oscillations, Pergamon Press, Oxford, 1966 (Russian edition ≈ 1930). [2] J. C. Artés, J. Llibre, J. C. Medrado and M. A. Teixeira, Piecewise linear differential systems with two real saddles, Math. Comput. Simul. 95 (2013), 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., vol. 163, Springer-Verlag, London, 2008. [4] D. C. Braga and L. F. Mello, Limit cycles in a family of discontinuous piecewise linear differential systems with two zones in the plane, Nonlinear Dynam. 73 (2013), 1283–1288. 10 J. LLIBRE [5] C. Buzzi, C. Pessoa and J. Torregrosa, Piecewise linear perturbations of a linear center, Discrete Contin. Dyn. Syst. 33 (2013), 3915–3936. [6] V. Carmona, F. Fernández-Sánchez and D. D. Novaes, The Lum–Chua conjecture revisited, preprint (2019). [7] J. Castillo, J. Llibre and F. Verduzco, The pseudo–Hopf bifurcation for planar discontinuous piecewise linear differential systems, Nonlinear Dynam. 90 (2017), 1829–1840. [8] R. D. Euzébio and J. Llibre, On the number of limit cycles in discontinuous piecewise linear differential systems with two pieces separated by a straight line, J. Math. Anal. Appl. 424 (2015), 475–486. [9] A. F. Filippov, Differential equations with discontinuous right–hand sides, translated from Russian. Mathematics and its Applications (Soviet Series), vol. 18, Kluwer Academic Publishers Group, Dordrecht, 1988. [10] E. Freire, E. Ponce, F. Rodrigo and F. Torres, Bifurcation sets of continuous piecewise linear systems with two zones, Int. J. Bifurcation and Chaos 8 (1998), 2073–2097. [11] E. Freire, E. Ponce and F. Torres, Canonical discontinuous planar piece- wise linear systems, SIAM J. Applied Dynamical Systems 11 (2012), 181–211. [12] E. Freire, E. Ponce and F. Torres, A general mechanism to generate three limit cycles in planar Filippov systems with two zones, Nonlinear Dy- namics 78 (2014), 251–263. [13] F. Giannakopoulos and K. Pliete, Planar systems of piecewise linear differential equations with a line of discontinuity, Nonlinearity 14 (2001), 1611– 1632. [14] M. R. A. Gouveia, J. Llibre and D. D. Novaes, On limit cycles bifur- cating from the infinity in discontinuous piecewise linear differential systems, Appl. Math. Comput. 271 (2015), 365–374. [15] M. Han, V. G. Romanovski and X. Zhang, Equivalence of the Melnikov function method and the averaging method, Qual. Theory Dynam. Sys. 15 (2016), 471–479. [16] M. Han and W. Zhang, On Hopf bifurcation in non–smooth planar systems, J. of Differential Equations 248 (2010), 2399–2416. [17] S. M. Huan and X. S. Yang, On the number of limit cycles in general planar piecewise systems, Discrete Cont. Dyn. Syst., Series A 32 (2012), 2147–2164. [18] S. M. Huan and X. S. Yang, Existence of limit cycles in general planar piecewise linear systems of saddle–saddle dynamics, Nonlinear Anal. 92 (2013), 82–95. [19] S. M. Huan and X. S. Yang, On the number of limit cycles in general planar piecewise linear systems of node–node types, J. Math. Anal. Appl. 411 (2014), 340–353. [20] E. Isaacson and H. B. Keller, Analysis of numerical methods, John Wiley and Sons, New York, 1966. [21] L. Li, Three crossing limit cycles in planar piecewise linear systems with saddle-focus type, Electron. J. Qual. Theory Differ. Equ. 2014, No. 70, pp. 14. [22] S. Li and J. Llibre, Phase portraits of piecewise linear continuous differential systems with two zones separated by a straight line, J. Differential Equations 266 (2019), 8094–8109. 11 [23] J. Llibre, D. D. Novaes and M. A. Teixeira, Limit cycles bifurcating from the periodic orbits of a discontinuous piecewise linear differential center with two zones, Int. J. Bifurcation and Chaos 25 (2015), 1550144, pp. 11. [24] J. Llibre, D. D. Novaes and M. A. Teixeira, Maximum number of limit cycles for certain piecewise linear dynamical systems, Nonlinear Dyn. 82 (2015), 1159–1175. [25] J. Llibre, D. D. Novaes and M. A. Teixeira, On the birth of limit cycles for non–smooth dynamical systems, Bull. Sci. Math. 139 (2015), 229–244. [26] J. Llibre, M. Ordóñez and E. Ponce, On the existence and uniqueness of limit cycles in planar piecewise linear systems without symmetry, Nonlinear Anal. Series B: Real World Appl. 14 (2013), 2002–2012. [27] J. Llibre and E. Ponce, Three nested limit cycles in discontinuous piecewise linear differential systems with two zones, Dyn. Cont. Disc. Impul. Syst., Series B 19 (2012), 325–335. [28] J. Llibre and M. A. Teixeira, Piecewise linear differential systems without equilibria produce limit cycles?, Nonlinear Dyn. 88 (2017), 157–164. [29] J. Llibre and M. A. Teixeira, Periodic orbits of continuous and discontin- uous piecewise linear differential systems via first integrals, São Paulo J. Math. Sci. 12 (2018), 121–135. [30] J. Llibre, M. A. Teixeira and J. Torregrosa, Lower bounds for the maximum number of limit cycles of discontinuous piecewise linear differential systems with a straight line of separation, Int. J. Bifurcation and Chaos 23 (2013), 1350066, pp. 10. [31] J. Llibre and X. Zhang, Limit cycles for discontinuous planar piecewise linear differential systems separated by one straight line and having a center, J. Math. Anal. Appl. 467 (2018), 537–549. [32] N. G. Lloyd, Degree theory, Cambridge Tracts in Mathematics, No. 73. Cam- bridge University Press, Cambridge–New York–Melbourne, 1978. [33] R. Lum and L. O. Chua, Global properties of continuous piec ewise-linear vector fields. Part I: Simplest case in R2 , Internat. J. Circuit Theory Appl. 19 (1991), 251–307. [34] R. Lum and L. O. Chua, Global properties of continuous piecewise-linear vector fields. Part II: Simplest symmetric case in R2 , Internat. J. Circuit The- ory Appl. 20 (1992), 9–46. [35] O. Makarenkov and J. S. W. Lamb, Dynamics and bifurcations of non- smooth systems: a survey, Phys. D 241 (2012), 1826–1844. [36] S. Shui, X. Zhang and J. Li, The qualitative analysis of a class of planar Filippov systems, Nonlinear Anal. 73 (2010), 1277–1288. [37] D. J. W. Simpson, Bifurcations in Piecewise-Smooth Continuous Systems, World Sci. Ser. Nonlinear Sci. Ser. A, vol. 69, World Scientific, Singapore, 2010. 1 Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain Email address: jllibre@mat.uab.cat