This is a preprint of: “Limit cycles of planar piecewise differential systems with linear Hamiltonian saddles”, Jaume Llibre, Clàudia Valls, Symmetry (Basel), vol. 13, 1128:1–10, 2021. DOI: [10.3390/sym13071128] PIECEWISE DIFFERENTIAL SYSTEMS WITH ONLY LINEAR HAMILTONIAN SADDLES CAN CREATE LIMIT CYCLES? JAUME LLIBRE1 AND CLAUDIA VALLS2 Abstract. We study the continuous and discontinuous planar piecewise differen- tial systems formed only by linear Hamiltonian saddles and separated by one or two parallel straight lines. When these piecewise linear differential systems are either con- tinuous or discontinuous and are separated by one straight-line, or are continuous and are separated by two parallel straight lines, we show that they have no limit cycles. On the other hand, when these piecewise linear differential systems are discontinuous and are separated by two parallel straight lines, we show that they can have at most one limit cycle. Moreover we show that this upper bound is reached by providing an example of such a system with one limit cycle. 1. Introduction and statement of the main results The study of limit cycles (i.e., periodic orbits of a differential system in R2 isolated in the set of all periodic orbits of that system) goes back essentially to Poincaré [22] at the end of the nineteenth century and their existence became important in the applications because many phenomena are related with their existence, see for instancethe Van der Pol oscillator [25, 26], or the Belousov–Zhabotinskii chemical reaction [3, 27]. The study of continuous piecewise linear differential systems separated by one or two parallel straight lines appears in a natural way in control theory (see for instance the books [2, 9, 11, 12, 17, 21]). The easiest continuous piecewise linear differential systems are the ones formed by two linear differential systems and separated by a straight line and for such systems it is known that they can have at most one limit cycle (see for instance [7, 14, 18, 19] and the references therein). In the present paper we first show that if both linear differential systems are Hamil- tonian saddles, then the continuous piecewise linear differential system has not limit cycles. Theorem 1. A continuous piecewise linear differential system separated by one straight line formed by two Hamiltonian linear saddles has no limit cycles. The proof of Theorem 1 is given in section 3. Theorem 1 can be extended to contin- uous piecewise linear differential systems separated by two parallel straight lines and formed by three linear Hamiltonian saddles. 2000 Mathematics Subject Classification. 34C05, 34C23, 34C25, 34C29. Key words and phrases. Limit cycles, linear Hamiltonian saddles, continuous piecewise linear dif- ferential systems, discontinuous piecewise differential systems, first integrals. 1 2 J. LLIBRE AND C. VALLS Theorem 2. A continuous piecewise linear differential system separated by two parallel straight lines and formed by three Hamiltonian linear saddles has no limit cycles. Theorem 2 is proved in section 4. The study of discontinuous piecewise linear differential systems separated by straight lines goes back to Andronov et al. [1] and nowadays they had attracted the attention of many authors mainly because these systems appear in mechanics, electrical circuits, economy, etc, (see for instance the books [6, 23], the surveys [20, 24] and the references therein). In planar discontinuous piecewise linear differential systems we can have two kinds of limit cycles: the slidding limit cycle and the crossing limit cycle. A slidding limit cycle contains some segment of the lines of discontinuity, and a crossing limit cycle does not contain any of such segments. In the present paper, we only focus our study on the crossing limit cycles and in all the paper when we talk about limit cycles, we are referring to crossing limit cycles. As for the continuous case, the easiest discontinuous piecewise linear differential systems are the ones formed by two linear differential systems and separated by a straight line. It is known that such systems can have three limit cycles but it is not known if three is the maximum number of limit cycles that such systems can exhibit (see [4, 5, 8, 10, 13, 15]). We first show, as in the continuous time, that if both linear differential systems are linear Hamiltonian saddles, then the discontinuous piecewise linear differential system has no limit cycles. Theorem 3. A discontinuous piecewise linear differential system separated by one straight line and formed by two linear Hamiltonian saddles has no limit cycles. Theorem 3 is proved in section 5. It can be extended to the case in which the discontinuous piecewise linear differential system are separated by two parallel straight lines and in this case the upper bound on the number of limit cycles is one. Theorem 4. A discontinuous piecewise linear differential system separated by two par- allel straight lines and formed by three linear Hamiltonian saddles can have at most one limit cycle. Moreover there are systems in this class having one limit cycle, see Figure 1. Theorem 4 is proved in section 6. We remark that it is clear from the proof of Theorem 4 that there are discontinuous piecewise linear differential system separated by two parallel straight lines and formed by three linear Hamiltonian saddles that do not have limit cycles. The unique linear differential systems which are Hamiltonian are the linear centers and the linear saddles. In [16] the authors studied the limit cycles of the continuous and discontinuous piecewise linear differential systems formed only by centers and separated by one or two parallel straight lines. In the present paper we do a similar study for PIECEWISE DIFFERENTIAL SYSTEMS WITH LINEAR HAMILTONIAN SADDLES 3 Figure 1. The limit cycle of the discontinuous piecewise differential system formed by the three linear Hamiltonian saddles (16), (17) and (18). This limit cycles is travelled in counterclockwise sense. the continuous and discontinuous piecewise linear differential systems formed only by Hamiltonian linear saddles and separated by either one or two parallel straight lines. The paper is organized such that in section 2 before the proof of the main theorems of the paper we present a normal form of a linear differential system having a linear weak saddle and we characterize the continuous and discontinuous piecewise linear differential system separated by any number κ of parallel straight lines for κ ≥ 1 and formed by κ + 1 linear Hamiltonian saddles. We prove it for k = 1 and then we state it in the general case. 2. Preliminaries The following lemma, provides a normal form for an arbitrary linear differential system having a linear Hamiltonian saddle. Proposition 5. A differential system having a linear Hamiltonian saddle can be written as 4b2 − ω 2 ẋ = −bx − y + d, ẏ = ax + by + c, when a 6= 0, 4a or ẋ = −bx + By + d, ẏ = by + c, with b 6= 0 when a = 0. Proof. Consider a general linear differential system ẋ = Ax + By + d, ẏ = ax + by + c, in R2 and assume that it has a Hamiltonian saddle. The eigenvalues of this system are p A + b ± 4aB + (A − b)2 . 2 4 J. LLIBRE AND C. VALLS Since this system has a Hamiltonian saddle A + b = 0 and 4aB + (A − b)2 = ω 2 for some ω 6= 0. Hence, A = −b and 4aB + 4b2 = ω 2 . So if a 6= 0 we have B = −(4b2 − ω 2 )/(4a) and if a = 0 then b 6= 0. This completes the proof of the proposition.  Corollary 6. A differential system having a linear Hamiltonian saddle can be written as (1) ẋ = −bx − δy + d, ẏ = αx + by + c, with α ∈ {0, 1}. Proof. Note that in view of Proposition 5 any linear differential system having a Hamil- tonian saddle can be written as (2) ẋ = −bx − δy + d, ẏ = ax + by + c, where δ ∈ R if a = 0, and if a 6= 0 then δ = (4b2 − ω 2 )/(4a) with ω 6= 0. It is possible to do a rescaling of the independent variable since it does not change the orbits and so it will not change the number of crossing limit cycles. After a rescaling of the independent variable of the form τ = at if a 6= 0 we can assume that equation (2) can be written as in (1) where δ ∈ R and α = 0 if a = 0, and if a 6= 0 then δ = (4b2 − ω 2 )/(4a) with ω 6= 0 and α = 1. So, α ∈ {0, 1}.  The first integral of system (1) is α δ (3) H(x, y) = − x2 − bxy − y 2 − cx + dy. 2 2 3. Proof of Theorem 1 Assume that we have a continuous piecewise differential system separated by one straight line and formed by two linear Hamiltonian saddles. Without loss of generality we can assume that the straight line of continuity is x = 0. It follows from Corollary 6 that we can assume that the systems in x < 0 and x > 0 are written in the form (1). We have system (4) ẋ = −b1 x − δ1 y + d1 , ẏ = α1 x + b1 y + c1 , in x < 0 with the first integral α1 2 δ1 (5) H1 = − x − b1 xy − y 2 − c1 x + d1 y, 2 2 and system (6) ẋ = −b2 x − δ2 y + d2 , ẏ = α2 x + b2 y + c2 , in x > 0 with the first integral α2 2 δ2 (7) H2 = − x − b2 xy − y 2 − c2 x + d2 y. 2 2 Since we must have a continuous piecewise differential system, both systems must co- incide on x = 0 and so δ1 = δ2 , d1 = d2 , b1 = b2 and c1 = c2 . PIECEWISE DIFFERENTIAL SYSTEMS WITH LINEAR HAMILTONIAN SADDLES 5 Note that if the continuous piecewise differential system has a periodic orbit candidate to be a limit cycle, because the two differential systems are linear Hamiltonian saddles such a periodic orbit must intersect the line x = 0 in exactly two points, namely (0, y1 ) and (0, y2 ) with y1 < y2 . Since H1 and H2 are two first integrals, we have that (8) H1 (0, y1 ) = H1 (0, y2 ) and H2 (0, y1 ) = H2 (0, y2 ), that is (2d2 − δ2 (y1 + y2 ))(y1 − y2 ) = 0. So the periodic orbits of these continuous piecewise differential systems are in a con- tinuum of periodic orbits and consequently this differential system has no limit cycles. This completes the proof of the theorem. 4. Proof of Theorem 2 Assume that we have a continuous piecewise differential system separated by two parallel straight lines and formed by three linear Hamiltonian saddles. Without loss of generality we can assume that the straight lines of discontinuity are x = −1 and x = 1. It follows from Corollary 6 that we can assume that the systems in x < −1, −1 < x < 1 and x > 1 are written as in (1). We have system (4) with first integral (5) in x < −1, system (6) with first integral (7) in −1 < x < 1, and system (9) ẋ = −b3 x − δ3 y + d3 , ẏ = α3 x + b3 y + c3 , with first integral α3 2 δ3 (10) H3 = − x − b3 xy − y 2 − c3 x + d3 y, 2 2 in x > 1. Since we must have a continuous piecewise differential system, systems (4) and (6) must coincide in x = −1, and systems (6) and (9) must coincide in x = 1. Doing so we obtain b1 = b2 = b3 , d1 = d2 = d3 , δ1 = δ2 = δ3 , c1 = c3 +α1 −2α2 +α3 , c2 = c3 −α2 +α3 . Note that if the continuous piecewise differential systems has a periodic orbit candi- date to be a limit cycle, because its three differential systems are linear Hamiltonian saddles such a periodic orbit must intersect each line x = ±1 in exactly two points, namely (−1, y1 ), (−1, y2 ), (1, y3 ) and (1, y4 ), with y1 > y2 and y3 < y4 . Since H1 , H2 and H3 are three first integrals, we have that H1 (−1, y1 ) − H1 (−1, y2 ) = 0, H2 (−1, y2 ) − H2 (1, y3 ) = 0, (11) H3 (1, y3 ) − H3 (1, y4 ) = 0, H2 (1, y4 ) − H2 (−1, y1 ) = 0, 6 J. LLIBRE AND C. VALLS Doing so we get (2b3 + 2d3 − δ3 (y1 + y2 ))(y1 − y2 ) = 0, 4c3 − 4α2 + 4α3 + 2(b3 + d3 )y2 + 2(b3 − d3 )y3 − δ3 (y22 − y32 ) = 0, (2b3 − 2d3 + δ3 (y3 + y4 ))(y3 − y4 ) = 0, 4c3 − 4α2 + 4α3 + 2(b3 + d3 )y1 + 2(b3 − d3 )y4 − δ3 (y12 − y42 ) = 0. The solutions (y1 , y2 , y3 , y4 ) of these last systems satisfying the necessary condition y1 < y2 and y3 < y4 are √ √ 2(b3 + d3 ) d3 − b3 ∆ d3 − b3 ∆ y2 = − y1 , y3 = ± , y4 = ∓ , δ3 δ3 δ3 δ3 δ3 where ∆ = (b3 − d3 )2 − 4(c3 − α2 + α3 )δ3 − 2(b3 + d3 )δ3 y1 + δ32 y12 . Note that we only have two solutions taking the upper signs of y3 , y4 or the lower signs of y3 , y4 . Hence all the periodic orbits of the continuous piecewise differential system are in a continuum of periodic orbits and consequently this differential system has no limit cycles. This completes the proof of the theorem. 5. Proof of Theorem 3 Assume that we have a discontinuous piecewise differential system separated by one straight line and formed by two linear Hamiltonian saddles. Without loss of generality we can assume that the straight line of continuity is x = 0. It follows from Corollary 6 that we can assume that the systems in x < 0 and x > 0 are written in the form (1). We have system (4) with first integral (5) in x < 0, and system (6) with first integral (7) in x > 0. Note that if the discontinuous piecewise differential system has a periodic orbit can- didate to be a limit cycle, because its two differential systems are linear Hamiltonian saddles such a periodic orbit must intersect the line x = 0 in exactly two points, namely (0, y1 ) and (0, y2 ) with y1 < y2 . Since H1 and H2 are two first integrals we have that (8) must be satisfied, that is (2d1 − δ1 (y1 + y2 ))(y1 − y2 ) = 0, (2d2 − δ2 (y1 + y2 ))(y1 − y2 ) = 0 The solutions (y1 , y2 ) of this last system satisfying the necessary condition y1 < y2 either do not exist if d1 /δ1 6= d2 /δ2 , or there is a continuum of solutions. So the periodic orbits of the discontinuous piecewise linear differential systems are in a continuum of periodic orbits, and consequently this differential system has no limit cycles. This completes the proof of the theorem. 6. Proof of Theorem 4 Assume that we have a discontinuous piecewise differential system separated by two parallel straight lines and formed by three linear Hamiltonian saddles. Without loss of generality we can assume that the straight lines of discontinuity are x = −1 and x = 1. PIECEWISE DIFFERENTIAL SYSTEMS WITH LINEAR HAMILTONIAN SADDLES 7 It follows from Corollary 6 that we can assume that the systems in x < −1, −1 < x < 1 and x > 1 are written as in (1). We have system (4) with first integral (5) in x < −1, system (6) with first integral (7) in −1 < x < 1, and system (9) with Hamiltonian (10) in x > 1. Note that if the discontinuous piecewise differential systems has a periodic orbit candidate to be a limit cycle, because its three differential systems are linear Hamiltonian saddles such a periodic orbit must intersect each line x = ±1 in exactly two points, namely (−1, y1 ), (−1, y2 ), (1, y3 ) and (1, y4 ), with y1 > y2 and y3 < y4 . Since H1 , H2 and H3 are three first integrals, we have that system (11) must be satisfied. Doing so we get (2(b1 + d1 ) − δ1 (y1 + y2 ))(y1 − y2 ) = 0, 4c2 + 2(b2 + d2 )y2 + 2(b2 − d2 )y3 − δ2 (y22 − y32 ) = 0, (12) (2(b3 − d3 ) + δ3 (y3 + y4 ))(y3 − y4 ) = 0, 4c2 + 2(b2 + d2 )y1 + 2(b2 − d2 )y4 − δ2 (y12 − y42 ) = 0. Assume first that δ1 = δ3 = 0 the solutions of equations (12) are d1 = c1 − b1 , d3 = bc + c3 and y3 = f1 (y2 ), y4 = f2 (y1 ) being f1 , f2 functions in the variables y2 and y1 , respectively. In this case the periodic orbits of the discontinuous piecewise linear differential systems are in a continuum of periodic orbits and consequently this differential system has no limit cycles. Assume now that δ1 = 0 and δ3 6= 0 the solutions of equations (12) are d1 = −b1 , 2(d3 − b3 ) (13) y3 = − y4 δ3 and y2 = f1 (y4 ), y1 = f2 (y4 ) being f1 , f2 functions in the variable y4 , respectively. In this case the periodic orbits of the discontinuous piecewise linear differential systems are in a continuum of periodic orbits and consequently this differential system has no limit cycles. Assume now that δ1 6= 0 and δ3 = 0 the solutions of equations (12) are d1 = −b1 , 2(b1 + d1 ) (14) y1 = − y2 , δ1 d3 = b3 , and y2 = f1 (y2 ), y1 = f2 (y2 ) being f1 , f2 functions in the variable y2 , respec- tively. In this case the periodic orbits of the discontinuous piecewise linear differential systems are in a continuum of periodic orbits and consequently this differential system has no limit cycles. Finally, assume that δ1 δ3 6= 0. The solution of the first and third equations are (14) and (13). Introducing these solutions into the second and fourth equations in (12) we get e1 = 4((b3 − d3 )2 δ2 − (b2 − d2 )(b3 − d3 )δ3 + c2 δ32 ) + 2(b2 + d2 )δ32 y2 (15) + 2δ3 (2(b3 − d3 )δ2 − (b2 − d2 )δ3 )y4 − δ2 δ32 (y22 − y42 ) = 0 8 J. LLIBRE AND C. VALLS and e2 = 4(δ1 ((b1 + d1 )(b2 + d2 ) + c2 δ1 ) + (b1 + d1 )2 δ2 ) − 2δ1 ((b2 + d2 )δ1 − 2(b1 + d1 )δ2 )y2 + 2(b2 − d2 )δ12 y4 − δ12 δ2 (y22 − y42 ) = 0. Taking e3 = δ12 e1 − δ32 e2 = 0 and solving in y4 we get A0 A2 y4 = + y2 A1 A1 where A0 = (b3 − d3 )2 δ12 δ2 − (b2 − d2 )(b3 − d3 )δ12 δ3 + (b1 + d1 )((b1 + d1 )δ2 − (b2 + d2 )δ1 )δ32 , A1 = δ12 δ3 ((d3 − b3 )δ2 + (b2 − d2 )δ3 ), A2 = δ1 ((b2 + d2 )δ1 − (b1 + d1 )δ2 )δ32 , whenever A1 6= 0. The case with A1 = 0 yields d2 = b2 + (d3 − b3 )δ2 /δ3 . Introducing it into e3 = 0 and solving in y2 we obtain y2 = y1 = (b1 − c1 + d1 )/δ1 which is not possible. So, we an assume that A1 6= 0. Now introducing y4 into the first equation in (15) and solving in y2 we get √ (b1 + d1 ) ∆ y2 = y2± = ± δ1 A4 where A4 = δ12 δ2 δ32 (b3 δ1 δ2 − d3 δ1 δ2 + 2d2 δ1 δ3 − b1 δ2 δ3 − d1 δ2 δ3 )(b3 δ1 δ2 − d3 δ1 δ2 − 2b2 δ1 δ3 + b1 δ2 δ3 + d1 δ2 δ3 )), ∆ = 4δ14 δ2 δ32 ((b3 − d3 )δ2 + (−b2 + d2 )δ3 )2 ((b3 − d3 )δ1 δ2 + 2d2 δ1 δ3 − (b1 + d1 )δ2 δ3 )((b3 − d3 )δ1 δ2 − 2b2 δ1 δ3 + (b1 + d1 )δ2 δ3 )((b3 − d3 )2 δ12 δ2 − 2(b2 − d2 )(b3 − d3 )δ12 δ3 + (2δ1 ((b1 + d1 )(b2 + d2 ) + 2c2 δ1 ) − (b1 + d1 )2 δ2 )δ32 ), whenever A4 6= 0 and if A4 = 0 then there is at most one solution y2 . When A4 6= 0, since √ 2(d1 + b1 ) d1 + b1 ∆ y1 = y1± = − y2± = ∓ = y2∓ , δ1 δ1 A4 there is at most one solution with y1 > y2 and y3 < y4 . In summary, we have proved that at most we can have one limit cycle. Now we shall prove that the discontinuous piecewise linear differential system having a Hamiltonian saddle in each of these three pieces has also one limit cycle. This will complete the proof of Theorem 4. The Hamiltonians of the three linear Hamiltonian systems with a saddle are √ √ H1 (x, y) = 1606 + 5 20954161 + 32256x + 5(−6130 + 20954161)y + 7680x2 − 7680y 2 , √ √ √ H2 (x, y) = 4615x − 20954161x + ( 20954161 − 7645)y + (−4573 + 20954161)x2 − √ ( 20954161 − 4615)xy − 3072y 2 , √ √ H3 (x, y) = 519 + 1317937 − (1024 + 1317937)x − 505y + 512x2 − 512y 2 , PIECEWISE DIFFERENTIAL SYSTEMS WITH LINEAR HAMILTONIAN SADDLES 9 where the Hamiltonian system in the half-plane x < −1 is √ ẋ = −30650 + 5 20954161 − 15360y, (16) ẏ = −32256 − 15360x; the Hamiltonian system in the strip −1 < x < 1 is √ √ ẋ = −7645 + 20954161 + (4615 − 20954161)x − 6144y, (17) √ √ √ ẏ = −4615 + 20954161 + (9146 − 2 20954161)x − (4615 + 20954161)y; and the Hamiltonian system in the half-plane x > 1 is ẋ = −505 − 1024y, (18) √ ẏ = 1024 + 1317937 − 1024x. These three linear differential systems are saddles √ because the determinant of their lin- ear part are −235929600, 13940638 − 3058 20954161 < 0 and −1048576, respectively. The discontinuous piecewise differential system formed by the three linear differen- tial systems (16), (17) and (18) in order to have one limit cycle intersecting the two discontinuous straight lines x = ±1 at the points (−1, y1 ), (−1, y2 ), (1, y3 ) and (1, y4 ), these points must satisfy system (11), and this system has a unique solution satisfying that y1 > y2 and y3 < y4 , namely √ ! ! 1 20954161 2297 7 (y1 , y2 , y3 , y4 ) = − , −1, −1, . 3 512 256 512 Drawing the corresponding limit cycle associated to this solution we obtain the limit cycle of Figure 1. 7. Conclusions We have studied the continuous and discontinuous planar piecewise differential sys- tems formed only by linear Hamiltonian saddles separated by one or two parallel straight lines. Such continuous piecewise differential systems appear in control theory, while the discontinuous ones appear in mechanics, electrical circuits, economy, etc. When these piecewise differential systems are continuous separated by either one or two parallel straight lines, we prove that they have no limit cycles. But when the piecewise differen- tial systems are discontinuous separated two parallel straight lines, we show that they can have at most one limit cycle, and that there exist systems with either zero or one limit cycle. In the case in which the piecewise differential systems are discontinuous and separated only by one straight line, they cannot have limit cycles. Acknowledgements The first author is supported by the Ministerio de Economı́a, Industria y Compet- itividad, Agencia Estatal de Investigación grant PID2019-104658GB-I00, the Agència 10 J. LLIBRE AND C. VALLS de Gestió d’Ajuts Universitaris i de Recerca grant 2017SGR1617, and the H2020 Euro- pean Research Council grant MSCA-RISE-2017-777911. The second author is partially supported by FCT/Portugal through UID/MAT/04459/2019. References [1] Andronov, A., Vitt, A., Khaikin, S.: Theory of oscillations. Pergamon Press, Oxford (1966) [2] Atherton, D.P.: Nonlinear Control Engineering. Van Nos- trand Reinhold Co., Ltd., New York (1982) [3] Belousov, B.P.: Periodically acting reaction and its mechanism. In: Collection of Abstracts on Radiation Medicine, pp. 145–147. Moscow (1958) [4] Braga, D.C., Mello, L.F.: Limit cycles in a family of discontinuous piecewise linear differential systems with two zones in the plane. Nonlinear Dyn. 73, 1283–1288 (2013) [5] Buzzi, C., Pessoa, C., Torregrosa, J.: Piecewise linear perturbations of a linear center. Discrete Contin. Dyn. Syst. 9, 3915–3936 (2013) [6] di Bernardo, M., Budd, C. J., Champneys, A. 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Math- ematics of Complexity and Dynamical Systems, vol. 1–3, pp. 1325–1336. Springer, New York (2012) [25] van der Pol, B.: A theory of the amplitude of free and forced triode vibrations. Radio Rev. 1, 701–710 (1920) [26] van der Pol, B.: On relaxation-oscillations. Lond. Edinb. Dublin Philos. Mag. J. Sci. 2(7), 978–992 (1926) [27] Zhabotinsky, A.M.: Periodical oxidation of malonic acid in solution (a study of the Belousov reaction kinetics). Biofizika 9, 306–311 (1964) 1 Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bel- laterra, Barcelona, Catalonia, Spain Email address: jllibre@mat.uab.cat 2 Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais 1049–001, Lisboa, Portugal Email address: cvalls@math.ist.utl.pt