This is a preprint of: “Limit cycles in Filippov systems having a circle as switching manifold”, Jaume Llibre, Marco Antonio Teixeira, Chaos, vol. 32, 053106, 2022. LIMIT CYCLES IN FILIPPOV SYSTEMS HAVING A CIRCLE AS SWITCHING MANIFOLD JAUME LLIBRE1 AND MARCO ANTONIO TEIXEIRA2 Abstract. It is known that planar discontinuous piecewise linear differential systems separated by a straight line have no limit cycles when both linear differential systems are centers. Here we study the limit cycles of the planar discontinuous piecewise linear differential systems separated by a circle, when both linear differential systems are centers. Our main results show that such discontinuous piecewise differ- ential systems can have 0, 1 or 2 limit cycles, but no more limit cycles than 2. 1. Introduction and statement of the main result 1.1. Historical facts. The problem of existence of limit cycles has been extensively treated in the literature since the early days of celestial mechanics. More recently, much work has been done on the rigorous mathematical foundation of nonsmooth dynamical systems problems, in particular, in the search of typical minimal sets that there are no counterparts in the smooth universe. It is worth to mention that, some existing smooth techniques are useful in solving many nonsmooth problems. Some of the orbits of planar differential systems are difficult to study, this is the case of the limit cycles. Recall that a limit cycle of a differential system (S) is a periodic solution of (S) which is isolated in the set of all periodic solutions of (S). Concerning the nonsmooth universe and in the 2–dimensional case one can find many results on the existence of limit cycles when the switching set is an imbedded curve in R2 , see [1–15, 18–29]. One of the main properties of smooth integrable systems in the plane R2 is that their periodic orbits usually appear in continuous one–parameter families, in contrast to the periodic orbits of piecewise nonsmooth integrable systems which typically are limit cycles, see [2–6, 18–28]. Andronov, Vitt and Khaikin [1] started the study of the discontinuous piecewise linear differential systems in the plane, mainly motivated for their applications to some mechanical problems. Recently the interest for this kind of differential systems increased due mainly to the fact that these differential systems model many processes appearing in mechanics, electronics, economy,... See for these applications the survey of Makarenkov and Lamb [27], and the books of Simpson [30] and of 2010 Mathematics Subject Classification. 34C29, 34C25, 47H11. Key words and phrases. limit cycles, discontinuous piecewise differential systems, linear differen- tial centers. 1 2 J. LLIBRE AND M.A. TEIXEIRA di Bernardo, Budd, Champneys and Kowalczyk [7], together with the hundreds of references quoted in these works. In the plane the discontinuous piecewise linear differential systems separated by a straight line are the easiest possible discontinuous piecewise linear differential systems. For these systems a crossing limit cycle is a limit cycle which has exactly two points on the line of discontinuity. But for these easy systems there are open questions as the following: Is three the maximum number of crossing limit cycles that a discontinuous piecewise linear differential systems separated by a straight line can have? See for instance [6, 10–12, 14, 20, 23, 23]. 1.2. Setting the problem. The main goal of this article is to discuss the existence of limit cycles of nonsmooth piecewise integrable systems in the plane R2 where the switching set is concentrated on a closed curve. Recently, the mentioned great interest in non-smooth dynamics leads us the need to explore new challenges in this subject. In the classical smooth case we find in the literature an endless number of papers involving the problem on the existence of limit cycles in the plane, see for instance the books [31, 32] and the hundreds of references therein. Moreover, problems involving the existence of limit cycles bifurcating from a smooth center are extensively studied in the last decades, see the book [8] and the references quoted there. We recall that a center of a planar differential system is a singular point p of the system for which there is a neighborhood U such that all the orbits of U \ {p} are periodic. It is interesting to note that if we consider discontinuous piecewise linear differ- ential systems formed by two centers separated by a straight line, then such systems have no crossing limit cycles, for a proof see Theorem 4 of [20]. In this paper we deal with discontinuous piecewise linear differential systems (C) formed by two centers separated by a circle, that without loss of generality we can assume that is the circle S1 = {(x, y) ∈ R2 : x2 + y 2 = 1}. A crossing periodic orbit for a system (C) is a periodic orbit which has exactly two points on the circle S1 of discontinuity, and of course a crossing limit cycle for a system (S) in (C) is a crossing periodic orbit isolated in the set of all crossing periodic orbits of system (S). To provide a sharp upper bound on the number of limit cycles for a class of planar differential systems is in general a very difficult problem, consider for instance the 16-th Hilbert problem. Thus, in particular, for the class of quadratic polynomial differential systems we have systems with four limit cycles, but we do not know if there is a uniform bound for the maximum number of limit cycles for all the systems of this class. See for more details [17, 18]. We shall provide an upper bound for the number of limit cycles of the planar discontinuous piecewise linear differential systems separated by a circle, when both linear differential systems are centers. We stress that the respective proof is non- trivial although the expressions of the systems are very friendly. It should be noted LIMIT CYCLES IN FILIPPOV SYSTEMS 3 that, as far as the authors know, there are few developed techniques or tools within the theory of discontinuous piecewise linear differential systems. So new techniques in this regard are, of course, notoriously welcomed. 1.3. Statement of the main results. The objective of this paper is to study the number of crossing limit cycles that systems in the class (C) can have, and to present an exotic center. Our main result is the following. Figure 1: Three periodic solutions of the center (0, −1) of the discontinuous piecewise linear differential system formed by the centers (1) and (2). Figure 2: The unique limit cycle of the discontinuous piecewise linear differential system formed by the centers (3) and (4). Theorem 1. Let (C) be the class of planar discontinuous piecewise linear differ- ential systems formed by two centers separated by the circle S1 . Then the following statement hold. (a) There are systems in (C) with infinitely many crossing periodic orbits, in fact with a continuum of crossing periodic orbits forming a center, see Fig- ure 1. 4 J. LLIBRE AND M.A. TEIXEIRA (b) There are systems in (C) without crossing periodic orbits. (c) There are systems in (C) having exactly one crossing limit cycle, see Figure 2. (d) There are systems in (C) having exactly two crossing limit cycles, see Figure 3. (e) Every system in (C) has at most two crossing limit cycles. Figure 3: The two limit cycles of the discontinuous piecewise linear differential system formed by the centers (6) and (7). Theorem 1 is proved in section 2. The results of Theorem 1 show once again that the shape of the discontinuity line plays a main role on the number of crossing limit cycles that discontinuous planar piecewise linear differential systems can exhibit, see also [4, 5, 26, 28]. Figure 4: An exotic center. LIMIT CYCLES IN FILIPPOV SYSTEMS 5 We exhibit in section 3 a planar discontinuous piecewise linear differential sys- tems formed by two centers separated by the circle S1 possessing an unusual center (exotic center). That means that its phase portrait contains the origin as a canoni- cal smooth center and there are on the switching set two 2–fold singularities of the center type, see Figure 4. 2. Proof of Theorem 1 Proof of statement (a) of Theorem 1. We consider in the bounded region limited by the circle S1 the linear differential center 1 (1) ẋ = 1 − y, ẏ =x. 4 In all this paper the dot denotes derivative with respect to the time t, except if we say explicitly another thing. This system has the first integral 1 2 H1 (x, y) = x + y 2 − 2y. 4 In the unbounded region of R2 with boundary the circle S1 we have the linear differential center 1 (2) ẋ = − − y, ẏ = x. 2 This system has the first integral H2 (x, y) = x2 + y 2 + y. For all α ∈ [0, 2π) \ {±π/2} the two ellipses H1 (x, y) = H1 (cos α, sin α) and H2 (x, y) = H2 (cos α, sin α), intersect exactly with the circle S1 in the two points (± cos α, sin α). Therefore the discontinuous piecewise linear differential system formed by the centers (1) and (2) has a continuum of crossing periodic solutions, which intersect the circle at the two points (± cos α, sin α) for all α ∈ [0, 2π) \ {±π/2}. See three crossing periodic solutions of this continuum of crossing periodic solutions corresponding to the values of α equal to 0, 1.0572619009906146.. and 5.114273122257285.. in Figure 1.  Proof of statement (b) of Theorem 1. This statement has an easy proof it is suffi- cient to consider the discontinuous piecewise linear differential system having the linear center ẋ = −y, ẏ = x in the bounded region limited by the circle S1 , and an arbitrary center different from the previous one in the unbounded region limited by S1 . Clearly a such discontinuous piecewise linear differential system has no crossing periodic orbits.  Proof of statement (c) of Theorem 1. In the unbounded region limited by the circle S1 we consider the linear differential center 1 x y y 1 (3) ẋ = − − , ẏ = x + − . 3 2 2 2 2 6 J. LLIBRE AND M.A. TEIXEIRA This system has the first integral  y 2 x y H1 (x, y) = 4 x + −8 + + y2 . 2 2 3 In the bounded region of R2 with boundary the circle S1 we have the linear differential center 2 (4) ẋ = − y, ẏ = 1 + x. 5 This system has the first integral  2 2 2 H2 (x, y) = (x + 1) + −y . 5 This discontinuous piecewise differential system formed by the linear differential systems (3) and (4) has exactly one crossing limit cycle, because the unique real solutions (α, β, γ, δ) of the system H1 (α, β) = H1 (γ, δ), H2 (α, β) = H2 (γ, δ), (5) α2 + β 2 = 1, γ 2 + δ 2 = 1, where (α, β, γ, δ) satisfying (α, β) 6= (γ, δ) are √  √ √  √ ! 10 232 − 7 29 −928 − 175 29 10 232 + 7 29 175 29 − 928 , , , , 2697 2697 2697 2697 and the one interchanging (α, β) by (γ, δ), which defines the same crossing limit cycle. See this crossing limit cycle in Figure 2.  Proof of statement (d) of Theorem 1. In the unbounded region limited by the circle S1 we consider the linear differential center 1 √  5 √  (6) ẋ = −x − y − 9 + 5 3 , ẏ = 2x + y + 1+ 3 . 8 8 This system has the first integral  √   √  H1 (x, y) = 2x2 + x 2y + 2 + y y + 2 + 1 . In the bounded region of R2 with boundary the circle S1 we have the linear differential center 5y 1 1 1 (7) ẋ = x − + √ + , ẏ = x − y − √ . 4 2 8 2 This system has the first integral  √   √  H2 (x, y) = 4x2 − 4x 2y + 2 + y 5y − 4 2 − 1 . This discontinuous piecewise differential system formed by the linear differential systems (6) and (7) has exactly two crossing limit cycles, because the unique real solutions  (α, β, γ, δ) of the  system  (10) are the four solutions  (1, 0, 0, 1), (0, 1, 1, 0), 1 1 1 1 1 1 1 1 −√ , −√ , √ , −√ and √ , − √ , − √ , − √ . Hence, the pairs (1, 0), 2 2 2 2 2 2 2 2 LIMIT CYCLES IN FILIPPOV SYSTEMS 7     1 1 1 1 (0, 1) and − √ , − √ , √ , − √ are the intersection points of the two cross- 2 2 2 2 ing limit cycles with the circle S1 . See these two crossing limit cycles in Figure 3.  We shall use the next result which is well known, see for instance [24, 26]. Lemma 2. Doing a rescaling of the independent variable and a linear change of variables every linear differential system in R2 having a center can be written as 4b2 + ω 2 (8) ẋ = −bx − y + d, ẏ = ax + by + c, 4a with a > 0 and ω > 0. Proof of statement (e) of Theorem 1. In the bounded region limited by the circle S1 we consider the arbitrary linear differential center (8), which has the first integral H1 (x, y) = 4(ax + by)2 + 8a(cx − dy) + y 2 ω 2 . In the unbounded region limited by the circle S1 we consider the arbitrary linear differential center 4B 2 + Ω2 (9) ẋ = −Bx − y + dD, 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) + y 2 Ω2 . If this discontinuous piecewise differential system formed by the linear differential systems (8) and (9) has a crossing limit cycle which intersect the circle x2 + y 2 = 1 in the pair of points (α, β) and (γ, δ), then these points must satisfy the system H1 (α, β) = H1 (γ, δ), H2 (α, β) = H2 (γ, δ), α2 + β 2 = 1, γ 2 + δ 2 = 1, with (α, β) 6= (γ, δ), or equivalently the system H1 (α, β) = h1 , H2 (α, β) = h2 , H1 (γ, δ) = h1 , (10) H2 (γ, δ) = h2 , α2 + β 2 = 1, γ 2 + δ 2 = 1. The two first equations of system (10) has at most four real solutions for (α, β), because these solutions are the intersection of two ellipses. These four solutions are also the four solutions of the third and fourth equations of system (10). Since these four solutions on the circle x2 + y 2 = 1 must be the intersection points of the crossing limit cycles with the circle, we can have at most two crossing limit cycles.  This completes the proof of Theorem 1. 8 J. LLIBRE AND M.A. TEIXEIRA 3. Exotic Center y2 Let f (x, y) = x2 + y 2 − 1, H0 (x, y) = x2 + and 2 (−x sin θ + y cos θ)2 H1 (x, y; θ) = (x cos θ + y sin θ)2 + . 2 Notice that the Hamiltonian vector field X0 originated by H0 has four tangential contacts with Σ = f −1 (0) at p1 = (0, 1), p2 = (1, 0), p3 = (0, −1) and p4 = (−1, 0). Since the level curves of H1 are the same level curves of H0 after a rotation of angle θ, we obtain that the Hamiltonian vector field Yθ generated by H1 has also four contact points at q1 (θ) = (sin θ, cos θ), q2 (θ) = (cos θ, − sin θ) q3 (θ) = (− sin θ, − cos θ), and q4 (θ) = (− cos θ, sin θ). Of course the orbits of X0 (resp. Yθ ) are symmetric with respect to the lines containing the segments p1 p2 and p3 p4 (resp. q1 q2 and q3 q4 ). If θ = π/2 or θ = 3π/2, we have that all the tangency points of the Hamilton- ian systems X0 and Yθ coincide generating four fold–fold points. Since the phase portrait of this discontinuous piecewise linear differential system is symmetric with respect to the x-axis and the y-axis, we have that p1 and p3 are parabolic fold– fold points surrounded by crossing periodic orbits presenting a behavior similar to a singularity of type center. Also, the singularities p2 and p4 are connected by four orbits giving rise to pseudo–separatrices for the discontinuous piecewise linear system formed by X0 and Yθ . If θ = π/4, 3π/4, 5π/4, 7π/4, then it follows from symmetry that the orbits of Yθ give rise to two connections between the fold–regular points p1 and p2 . If θ is different from the values mentioned in items 1 and 2 the two previous paragraphs, then we have that the system presents a polycycle having two parabolic fold–fold singularities. Acknowledgments The first author is supported by the Ministerio de Economı̀a, Industria y compet- itividad, Agencia Estatal de Investigación grant MTM2016-77278-P (FEDER), the Agència de Gestió d’Ajusts Universitaris i de Recerca grant 2017SGR1617, and the H2020 European Research Council grant MSCA-RISE-2017-777911. The second author is supported by ???????? References [1] A. Andronov, A. Vitt and S. Khaikin, Theory of oscillations, Pergamon Press, Oxford, 1966. [2] J.C. Artés, J. Llibre, J.C. Medrado and M.A. Teixeira, Piecewise linear differential systems with two real saddles, Math. Comp. Sim. 95 (2013), 13–22. [3] 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 Dyn. 73 (2013), 1283–1288. [4] D.C. Braga and L.F. Mello, More than three limit cycles in discontinuous piecewise linear differential systems with two zones in the plane, Int. J. 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Soc., Providence, RI, 1992. 1 Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bel- laterra, Barcelona, Catalonia, Spain E-mail address: jllibre@mat.uab.cat 2 Departamento de Matemática, Universidade Estadual de Campinas, Caixa Postal 6065, 13083–970, Campinas, SP, Brazil E-mail address: teixeira@ime.unicamp.br