This is a preprint of: “Limit cycles in planar piecewise linear Hamiltonian systems with three zones without equilibrium points”, Alexander Fernandes da Fonseca, Jaume Llibre, Luis Fernando Mello, Internat. J. Bifur. Chaos Appl. Sci. Engrg., vol. 30(11), 2050157 (8 pages), 2020. DOI: [10.1142/S0218127420501576] Limit cycles in planar piecewise linear Hamiltonian systems with three zones without equilibrium points Alexander Fernandes da Fonseca∗ Instituto de Matemática e Computação, Universidade Federal de Itajubá, Avenida BPS 1303, Pinheirinho, CEP 37.500–903, Itajubá, MG, Brazil. e–mail: alexff@unifei.edu.br Jaume Llibre Departament de Matemàtiques, Universitat Autònoma de Barcelona, Bellaterra, 08193–Barcelona, Catalonia, Spain. e–mail: jllibre@mat.uab.cat Luis Fernando Mello Instituto de Matemática e Computação, Universidade Federal de Itajubá, Avenida BPS 1303, Pinheirinho, CEP 37.500–903, Itajubá, MG, Brazil. e–mail: lfmelo@unifei.edu.br Abstract We study the existence of limit cycles in planar piecewise linear Hamilto- nian systems with three zones without equilibrium points. In this scenario, we have shown that such systems have at most one crossing limit cycle. Key–words: Piecewise smooth vector field, limit cycle, Hamiltonian vector field. 2010 Mathematics Subject Classification: Primary 34A36, 34C25, 34A26. ∗ Corresponding author. 1 1 Introduction and statement of the main re- sults The study of piecewise smooth vector fields, especially the planar case, has grown over the last thirty years mainly due to its great applicability as a mathematical model of a series of applied phenomena in relay systems, me- chanics, electrical circuits, among others. As a landmark for such studies, we cited the book of Andronov et al. [1], and more recently Filippov’s book [5] and the book of di Bernardo et al. [2]. Many of the studies developed so far consider piecewise smooth vector fields with two zones and few studies are found with more zones. A piecewise smooth vector field with three zones in the plane is composed of three C r vector fields X, Y and Z, r ≥ 1, defined on R2 and separated by a pair of disjoint connected unbounded smooth curves Σ1 and Σ2 . The separation curves Σi are obtained by considering Σi = h−1 i (0), where hi : R −→ R 2 are differentiable functions having 0 as a regular value, for i = 1, 2. Thus R2 = R1 ∪ R2 ∪ R3 ∪ Σ1 ∪ Σ2 , where the zones Ri , i = 1, 2, 3, are unbounded disjoint regions defined on the complement of the separation curves. So, a piecewise smooth vector field with three zones in the plane can be written as    X(x, y), h1 (x, y) ≤ 0, W (x, y) = Y (x, y), h1 (x, y) ≥ 0 and h2 (x, y) ≤ 0, (1)   Z(x, y), h2 (x, y) ≥ 0. System (1) will be denoted by W = (X, Y, Z, Σ1 , Σ2 ), or simply by W = (X, Y, Z) when the separation curves Σ1 and Σ2 are well understood. We will use the vector field X and the separation curve Σ1 in the next definitions. However, they can be easily adapted to the vector fields Y and Z and the separation curve Σ2 . The contact between the smooth vector field X and the separation curve Σ1 is characterized by the derivative of h1 in the direction of the vector field X, that is, by the expression Xh1 (p) = h∇h1 (p), X(p)i, where h·, ·i is the 2 usual inner product in R2 . Some important subsets of Σ1 are the following ones (the same for Σ2 ): (a) Crossing set: Σc1 = {p ∈ Σ1 | Xh1(p) · Y h1 (p) > 0}; (b) Sliding set: Σs1 = {p ∈ Σ1 | Xh1 (p) < 0, Y h1 (p) > 0}; (c) Escaping set: Σe1 = {p ∈ Σ1 | Xh1(p) > 0, Y h1 (p) < 0}. When p ∈ Σ1 and Xh1 (p) = 0 we say that p is a tangential singularity of X. A tangential singularity is called a fold point of X if Xh1 (p) = 0 but X 2 h1 (p) 6= 0, where X 2 h1 (p) = h∇Xh1 (p), X(p)i. Moreover, p ∈ Σ1 is a visible (invisible, respectively) fold point of X if Xh1 (p) = 0 and X 2 h1 (p) > 0 (X 2 h1 (p) < 0, respectively). In order to define a trajectory of a piecewise smooth vector field with three zones in the plane passing through a crossing point p ∈ Σ1 , it is enough to concatenate the trajectories of X and Y by p. However, in the sliding and escaping sets we need to define an auxiliary vector field and use the Filippov’s convention (see [5]). We say that an equilibrium point p of X is real if h1 (p) < 0 and it is virtual if h1 (p) > 0. A piecewise smooth vector field with three zones in the plane W = (X, Y, Z, Σ1 , Σ2 ) is called continuous if X(p) = Y (p), ∀p ∈ Σ1 and Y (q) = Z(q), ∀q ∈ Σ2 . Otherwise, it is called discontinuous. Without being exhaustive we present some results relative to the study of piecewise smooth systems with three zones in the plane. A beautiful and simple example can be found in the introductory chapter of the book [4] where the authors studied the problem of a particle under the gravity law rolling in an inclined plane followed by a horizontal displacement and next moving in another inclined plane. This problem is modeled by a piecewise linear system with three zones in the plane with a continuous of equilibria. 3 In fact, as far as we know, the studies found in the pertinent literature deal with piecewise smooth systems with three zones in the plane with at least one equilibrium point which can be real or virtual. This is the case, for instance, in the studies carried out in [3], [6], [7], [8], [9], [10]. We emphasize that in these articles the authors studied the existence and the number of limit cycles under the hypothesis of the continuity of the vector fields in the separation curves which are straight lines. In this article we are mainly interested in the study of the existence and the number of crossing limit cycles of a piecewise smooth vector field with three zones in the plane W = (X, Y, Z, Σ1 , Σ2 ) under the following assumptions: H1. The separation curves Σ1 and Σ2 are straight lines. H2. The vector fields X, Y and Z are linear. H3. The vector fields X, Y and Z are Hamiltonian. H4. The vector fields X, Y and Z have no equilibrium points, neither real nor virtual. We emphasize that when the vector fields X, Y and Z are Hamiltonian the solution curves of the respective differential equations are contained in the level sets of the Hamiltonian functions. This greatly simplifies the analysis of the transitions maps in and between the separation curves by seeking for points in Σi that are on the same level curves of these Hamiltonian functions. In this case, we avoid working with transition times. We can state the main results of this article. Theorem 1. Continuous planar piecewise linear Hamiltonian systems with three zones separated by two straight lines and without equilibrium points, neither real nor virtual, have no crossing limit cycles. 4 Theorem 2. Discontinuous planar piecewise linear Hamiltonian systems with three zones separated by two straight lines and without equilibrium points, neither real nor virtual, have at most one crossing limit cycle. The article is organized as follows. In section 2 are presented the proofs of Theorems 1 and 2. Examples of discontinuous planar piecewise linear Hamiltonian systems with three zones separated by two straight lines and without equilibrium points, neither real nor virtual, are analyzed in section 3. In particular, we give an example of a discontinuous planar piecewise linear Hamiltonian system with three zones separated by two straight lines and without equilibrium points, neither real nor virtual, with one crossing limit cycle. 2 Proofs of Theorems 1 and 2 Without loss of generality, from the assumption H1, we can take h1 , h2 : R2 −→ R given by h1 (x, y) = x + 1 and h2 (x, y) = x − 1. Thus, the separation lines can be written as Σ1 = h−1 1 (0) = {(x, y) ∈ R : x = −1}, 2 Σ2 = h−1 2 2 (0) = {(x, y) ∈ R : x = 1}. These two straight lines decompose the plane into three regions: RL = {(x, y) ∈ R2 : x < −1}, RC = {(x, y) ∈ R2 : −1 < x < 1}, RR = {(x, y) ∈ R2 : x > 1}. Thus, R2 = RL ∪ RC ∪ RR ∪ Σ1 ∪ Σ2 . Taking into account the assumptions H2, H3 and H4, we can write a piecewise linear Hamiltonian vector field with three zones and without equilibrium points in the plane as   X (x, y), x ≤ −1,  L W (x, y) = XC (x, y), −1 ≤ x ≤ 1, (2)   XR (x, y), x ≥ 1, 5 where  XL (x, y) = −λL bL x + bL y + γL , −λ2L bL x + λL bL y + δL ,  XC (x, y) = −λC bC x + bC y + γC , −λ2C bC x + λC bC y + δC ,  XR (x, y) = −λR bR x + bR y + γR , −λ2R bR x + λR bR y + δR , and δk 6= λk γk , k ∈ {L, C, R}. In fact, in order to satisfy conditions H2 and H4 these vector fields can be written as Xk (x, y) = (ak x + bk y + γk , λk ak x + λk bk y + δk ) , (3) where δk 6= λk γk , k ∈ {L, C, R}. Then in order to satisfy condition H3 the divergence of (3) must be zero, that is ak = −λk bk . If bk = 0 for some k ∈ {L, C, R}, then the vector field Xk has the form Xk (x, y) = (γk , δk ), with δk 6= λk γk . So the trajectories are straight lines implying that W has no periodic orbits. Henceforth, we consider bk 6= 0, for k ∈ {L, C, R}. As the vector fields XL , XC and XR in (2) are linear and Hamiltonian, there are quadratic functions Hk : R2 −→ R (Hamiltonian functions) such that   ∂Hk ∂Hk Xk (x, y) = − (x, y), (x, y) , k ∈ {L, C, R}. ∂y ∂x By elementary calculations, we found λ2k bk 2 bk Hk (x, y) = − x + λk bk xy − y 2 + δk x − γk y, k ∈ {L, C, R}. (4) 2 2 From the assumption of the continuity of W expressed in Theorem 1, we obtain XL (−1, y) = XC (−1, y) and XC (1, y) = XR (1, y), ∀y ∈ R. These equalities imply that λL = λC = λR = λ, bL = bC = bR = b, γL = γC = γR = γ, δL = δC = δR = δ. 6 Therefore, from (2), a continuous piecewise linear Hamiltonian vector field with three zones and without equilibrium points in the plane can be written as  W (x, y) = −λbx + by + γ, −λ2 bx + λby + δ , δ 6= λγ, b 6= 0. (5) By hypothesis this vector field has no equilibrium points. So it has no periodic orbits and, in particular, it has no limit cycles. In short, Theorem 1 is proved. Now we will begin the proof of Theorem 2. From (2) a piecewise linear Hamiltonian vector field with three zones and without equilibrium points in the plane has a crossing periodic orbit if there are crossing points (−1, y0), (−1, y3 ) ∈ Σ1 and crossing points (1, y1 ), (1, y2) ∈ Σ2 such that HL (−1, y0 ) = HL (−1, y3 ), HC (−1, y0 ) = HC (1, y1), (6) HR (1, y1 ) = HR (1, y2), HC (1, y2 ) = HC (−1, y3 ). In fact, in this situation, if we take αL = HL−1 ({HL (−1, y0 )}) ∩ RL , αC,1 = HC−1 ({HC (−1, y0 )}) ∩ RC , αC,2 = HC−1 ({HC (−1, y3 )}) ∩ RC , −1 αR = HR ({HR (1, y1)}) ∩ RR , then α = αR ∪ αC,1 ∪ αL ∪ αC,2 is a crossing periodic orbit of W . See Figure 1. 7 Σ1 Σ2 αC,2 (1, y2 ) (−1, y3 ) αL αR (−1, y0 ) (1, y1 ) αC,1 RL RC RR Figure 1: A crossing periodic orbit of W : α = αR ∪ αC,1 ∪ αL ∪ αC,2 . From the above four equations in (6) and (4), we obtain 1 − (y0 − y3 ) (2bL λL + bL y0 + bL y3 + 2γL ) = 0, (7) 2 1 1 −λC bC y0 − bC y0 2 − 2δC − γC y0 − λC bC y1 + bC y1 2 + γC y1 = 0, (8) 2 2 1 − (y1 − y2 ) (−2bR λR + bR y1 + bR y2 + 2γR ) = 0, (9) 2 1 1 λC bC y2 − bC y2 2 + 2δC − γC y2 + λC bC y3 + bC y3 2 + γC y3 = 0. (10) 2 2 As y0 6= y3 and y1 6= y2 , equation (7) can be solved for y0 as well as equation (9) can be solved for y2 . Substituting the obtained expressions of y0 and y2 into equations (8) and (10), respectively, we get the equations of two hyperbolas in the plane y1 y3 given by (y1 − A)2 (y3 − B)2 (y1 − C)2 (y3 − D)2 − = 1, − = 1, (11) K K K K 8 where γC A = λC − , bC γL γC B = −2 − 2λL + λC + , bL bC γR γC C = −2 + 2λR − λC + , bR bC   γC D = − λC + , bC δC − λC γC K=4 . bC We have the following three possibilities: Case 1. The hyperbolas do not intersect. So, the discontinuous piece- wise linear Hamiltonian vector field with three zones and without equilibrium points in the plane given in equation (2) has no periodic solutions. See Ex- ample 1. Case 2. The hyperbolas coincide. In this case, the discontinuous piece- wise linear Hamiltonian vector field with three zones and without equilibrium points in the plane given in equation (2) has infinitely many periodic solu- tions. See Example 2. Case 3. The hyperbolas intersect at two points. Thus, the discontin- uous piecewise linear Hamiltonian vector field with three zones and without equilibrium points in the plane given in equation (2) has only one limit cycle. It is easy to check that the two hyperbolas (11) at most can intersect at two points. See Example 3. In short, under the hypotheses of Theorem 2, a discontinuous piecewise linear Hamiltonian vector field with three zones and without equilibrium points in the plane has at most one limit cycle. Theorem 2 is proved. 9 3 Some examples In Figures 2, 3 and 4, related to Examples 1, 2, 3, we will use the following notations: the separation lines Σ1 and Σ2 are illustrated in dashed lines while the sliding sets are depicted in black solid lines. Black dots are used for fold points. Example 1. Consider W = (XL , XC , XR ) defined by XL (x, y) = (−x + y + 1, −x + y + 3) , (12) XC (x, y) = (−x + y + 1, −x + y + 2) , (13) XR (x, y) = (−2x + y + 1, −4x + 2y + 1) . (14) In this case we have bL = 1, λL = 1, γL = 1, δL = 3, bC = 1, λC = 1, γC = 1, δC = 2, bR = 1, λR = 2, γR = 1, δR = 1. The Hamiltonian functions (4) are, respectively, given by x2 y2 HL (x, y) = − + xy − + 3x − y, 2 2 x2 y2 HC (x, y) = − + xy − + 2x − y, 2 2 y2 HR (x, y) = −2x2 + 2xy − + x − y. 2 Then, the equations of the two hyperbolas (11) are y12 (y3 + 2)2 (y1 − 2)2 (y3 + 2)2 − = 1, − = 1, 4 4 4 4 It is easy to show that the two hyperbolas do not intersect. The point (−1, −2) ∈ Σ1 is an invisible fold point of XL and a visible fold point of XC . The point (1, 0) ∈ Σ2 is an invisible fold point of XC while the point (1, 1) ∈ Σ2 is an invisible fold point of XR . They are the endpoints of the sliding set in Σ2 . The phase portrait of W is depicted in Figure 2. 10 10 5 0 -5 -2 0 2 4 6 Figure 2: Phase portrait of W defined by (12), (13) and (14). Example 2. Considering the parameters bL = 1, λL = 1, γL = 1, δL = 3, bC = 1, λC = 1, γC = 1, δC = 2, bR = 1, λR = 1, γR = 1, δR = −1, W = (XL , XC , XR ) can be written as XL (x, y) = (−x + y + 1, −x + y + 3) , (15) XC (x, y) = (−x + y + 1, −x + y + 2) , (16) XR (x, y) = (−x + y + 1, −x + y − 1) . (17) 11 4 2 0 -2 -4 -3 -2 -1 0 1 2 3 4 Figure 3: Phase portrait of W defined by (15), (16) and (17). So the Hamiltonian functions (4) have the expressions x2 y2 HL (x, y) = − + xy − + 3x − y, 2 2 x2 y2 HC (x, y) = − + xy − + 2x − y, 2 2 x2 y2 HR (x, y) = − + xy − − x − y. 2 2 The equations of the two hyperbolas (11) are equal and given by y12 (y3 + 2)2 − = 1. 4 4 The point (−1, −2) ∈ Σ1 is an invisible fold point of XL and a visible fold point of XC . The point (1, 0) ∈ Σ2 is an invisible fold point of both XC and XR . The phase portrait of W is illustrated in Figure 3. 12 Example 3. From equation (2), consider W = (XL , XC , XR ) defined by XL (x, y) = (−x + y + 1, −x + y + 3) , (18) XC (x, y) = (−x + y − 1, −x + y + 2) , (19) XR (x, y) = (−2x + y + 1, −4x + 2y − 2) . (20) In this case we have bL = 1, λL = 1, γL = 1, δL = 3, bC = 1, λC = 1, γC = −1, δC = 2, bR = 1, λR = 2, γR = 1, δR = −2. Then the Hamiltonian functions (4) are, respectively, given by x2 y2 HL (x, y) = − + xy − + 3x − y, 2 2 x2 y2 HC (x, y) = − + xy − + 2x + y, 2 2 y2 HR (x, y) = −2x2 + 2xy − − 2x − y. 2 Equations (7), (8), (9) and (10) can be written as 1 (y0 − y3 )(y0 + y3 + 4) = 0, 2 1  −y02 + y12 − 4y1 − 8 = 0, 2 1 − (y1 − y2 )(y1 + y2 − 2) = 0, 2 1  −y22 + 4y2 + y32 + 8 = 0, 2 which imply that the equations of the two hyperbolas (11) have the form (y1 − 2)2 (y3 + 4)2 y12 y2 − = 1, − 3 = 1. 12 12 12 12 As y0 6= y3 and y1 6= y2 , the real symmetric solutions of the above system are √ √ √ √ y0 = −2 − 5, y1 = 1 − 2 5, y2 = 1 + 2 5, y3 = −2 + 5, 13 √ √ √ √ y0 = −2 + 5, y1 = 1 + 2 5, y2 = 1 − 2 5, y3 = −2 − 5. Note that both identities provide the same four points on the separation lines Σ1 and Σ2 . The points (−1, y0), (−1, y3 ) ∈ Σ1 , and (1, y1), (1, y2) ∈ Σ2 are crossing points because √ (XL (−1, y0 ) · (1, 0)) (XC (−1, y0) · (1, 0)) = 2 5 + 5 > 0, √ (XL (−1, y3 ) · (1, 0)) (XC (−1, y3) · (1, 0)) = 5 − 2 5 > 0, √  (XC (1, y1 ) · (1, 0)) (XR (1, y1) · (1, 0)) = 2 5 + 10 > 0, √ (XC (1, y2 ) · (1, 0)) (XR (1, y2) · (1, 0)) = 20 − 2 5 > 0. So the level sets of Hk containing the points √ √ √ √ (−1, −2 − 5), (−1, −2 + 5) ∈ Σ1 and (1, 1 − 2 5), (1, 1 + 2 5) ∈ Σ2 form a crossing limit cycle of W = (XL , XC , XR ) defined by (18), (19) and (20), see Figure 4. This crossing limit cycle is attractor since the associated Poincaré map can be written as q p P (y0) = − y0 2 + 4 y0 2 + 8y0 + 28 + 8y0 + 20, √ and its derivative evaluated at y0 = −2 − 5 gives √ 1  √  P ′ (−2 − 5) = 109 − 48 5 , 19 √ which satisfies 0 < P ′ (−2 − 5) < 1. The point (−1, −2) ∈ Σ1 is an invisible fold point of XL and the point (−1, 0) ∈ Σ1 is a visible fold point of XC . They are the endpoints of the sliding set in Σ1 . The point (1, 1) ∈ Σ2 is an invisible fold point of XR while the point (1, 2) ∈ Σ2 is an invisible fold point of XC , and they are the endpoints of the sliding set in Σ2 . 14 10 5 0 -5 -4 -2 0 2 4 6 Figure 4: Phase portrait of W defined by (18), (19) and (20) with the attracting crossing limit cycle. Conflict of interest The authors declare that they have no conflict of interest. Acknowledgements. The first and third authors are partially supported by Fundação de Amparo à Pesquisa do Estado de Minas Gerais – FAPEMIG [grant number APQ–01158–17]. The second author is partially supported by the Ministerio de Ciencia, Innovación y Universidades, Agencia Estatal de Investigación grants MTM2016-77278-P (FEDER), the Agència de Gestió d’Ajuts Universitaris i de Recerca grant 2017SGR1617, and the H2020 Eu- ropean Research Council grant MSCA-RISE-2017-777911. References [1] A. Andronov, A. Vitt, S. 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