Campus de São José do Rio Preto Ronisio Moises Ribeiro Persistence of periodic orbits from planar piecewise linear Hamiltonian differential systems with two or three zones São José do Rio Preto 2023 Ronisio Moises Ribeiro Persistence of periodic orbits from planar piecewise linear Hamiltonian differential systems with two or three zones Tese apresentada como parte dos requisitos para obtenção do t́ıtulo de Doutor em Matemática, junto ao Programa de Pós-Graduação em Matemática, do Instituto de Biociências, Letras e Ciências Exatas da Universidade Estadual Paulista “Júlio de Mesquita Filho”, Câmpus de São José do Rio Preto. Financiadora: CAPES Orientador: Prof. Dr. Claudio Gomes Pessoa São José do Rio Preto 2023 R484p Ribeiro, Ronisio Moises Persistence of periodic orbits from planar piecewise linear Hamiltonian differential systems with two or three zones / Ronisio Moises Ribeiro. -- São José do Rio Preto, 2023 207 f. : il., tabs. Tese (doutorado) - Universidade Estadual Paulista (Unesp), Instituto de Biociências Letras e Ciências Exatas, São José do Rio Preto Orientador: Claudio Gomes Pessoa 1. Ciclo limite. 2. Sistema diferencial linear por partes. 3. Sistema Hamiltoniano. 4. Função de Melnikov. I. Título. Sistema de geração automática de fichas catalográficas da Unesp. Biblioteca do Instituto de Biociências Letras e Ciências Exatas, São José do Rio Preto. Dados fornecidos pelo autor(a). Essa ficha não pode ser modificada. Ronisio Moises Ribeiro Persistence of periodic orbits from planar piecewise linear Hamiltonian differential systems with two or three zones Tese apresentada como parte dos requisitos para obtenção do t́ıtulo de Doutor em Matemática, junto ao Programa de Pós-Graduação em Matemática, do Instituto de Biociências, Letras e Ciências Exatas da Universidade Estadual Paulista “Júlio de Mesquita Filho”, Câmpus de São José do Rio Preto. BANCA EXAMINADORA Prof. Dr. Claudio Gomes Pessoa (Orientador) UNESP - São José do Rio Preto Prof. Dr. Weber Flavio Pereira UNESP - São José do Rio Preto Profa. Dra. Luci Any Francisco Roberto UNESP - São José do Rio Preto Prof. Dr. Luis Fernando de Osório Mello Universidade Federal de Itajubá Prof. Dr. Luiz Fernando Gonçalves Universidade Federal de Goiás São José do Rio Preto 09 de Março de 2023 A minha mãe Sônia; A minha famı́lia e amigos; A meus mestres; Dedico. Agradecimentos Agradeço, primeiramente, a minha mãe Sônia, que sempre me apoiou, incentivou e nunca me deixou desistir dos meus objetivos, além de os tornarem posśıveis. Agradeço aos meus amigos Pablo e Ezequiel por todo o apoio que me deram e pelos momentos de descontração. Agradeço também a todos os meus colegas de pós-graduação por todo o conhecimento que compartilhamos durante o curso e por toda a ajuda. Agradeço aos meus professores de graduação e mestrado da Unifei e de doutorado do IBILCE/Unesp pela dedicação e empenho que demonstraram durante os cursos. Um agradecimento especial ao Prof. Dr. Claudio Pessoa pela paciência, conselhos, apoio e por ter acreditado em mim durante a minha tese e curso, como um todo, até o fim. O presente trabalho foi realizado com apoio da Coordenação de Aperfeiçoamento de Pessoal de Nı́vel Superior - Brasil (CAPES) - Código de Financiamento 001. “Feliz aquele que transfere o que sabe e aprende o que ensina.” (Cora Coralina, 1983, p.136) Resumo Neste trabalho, nosso objetivo é estimar o número de ciclos limites do tipo costura em sistemas diferenciais Hamiltonianos lineares por partes planares com duas ou três zonas separadas por retas de modo que os sistemas lineares que definem o por partes têm pontos singulares isolados, ou seja, centros ou selas. Mais precisamente, começaremos determinando o número de ciclos limites de sistemas diferenciais Hamiltonianos lineares por partes cont́ınuos ou descont́ınuos com duas ou três zonas. Neste caso, mostraremos que se o sistema for descont́ınuo com três zonas, então ele tem no máximo um ciclo limite, e forneceremos exemplos com um ciclo limite. Em seguida, estimaremos o número de ciclos limites que podem bifurcar de um anel de órbitas periódicas de um sistema diferencial Hamiltoniano linear descont́ınuo por partes com três zonas, após perturbações polinomiais de grau n, para n = 1, 2, 3. Para estes casos, denotando por H(n) o número de ciclos limites que podem bifurcar do anel de órbitas periódicas do sistema, provaremos que se o sistema diferencial linear definido na região entre as duas retas paralelas (chamado de subsistema central) possui um centro na origem e os demais subsistemas possuem centros ou selas, então H(1) ≥ 3, H(2) ≥ 4 e H(3) ≥ 7. Agora, para o caso particular em que o subsistema central possui um centro e os demais subsistemas possuem apenas selas reais, se o centro for real (não necessariamente na origem) ou se estiver sobre a fronteira da região central, então H(1) ≥ 6, e se for virtual, então H(1) ≥ 4. Finalmente, se o subsistema central possui uma sela real e os demais subsistemas possuem centros ou selas, então H(1) ≥ 6. Para isso, estudaremos o número de zeros de suas funções de Melnikov definidas em duas e três zonas. Além disso, fornecemos métodos anaĺıticos detalhados para estudar o número de zeros das funções de Melnikov. Palavras-chave: Ciclo limite. Sistema diferencial linear por partes. Sistema Hamiltoniano. Função de Melnikov. Abstract In this work, our goal is estimate the number of crossing limit cycles in planar piecewise linear Hamiltonian differential systems with two or three zones separated by straight lines such that the linear systems that define the piecewise one have isolated singular points, that is, centers or saddles. More precisely, we will start with the study of the number of limit cycles for continuous or discontinuous piecewise linear Hamiltonian differential systems with two or three zones. In this case, we show that if the system is discontinuous with three zones then it has at most one limit cycle, and we will provide examples with one limit cycle. Next, we will estimate the number of limit cycles that can bifurcate from a periodic annulus in a discontinuous piecewise linear Hamiltonian differential system with three zones, after polynomials functions perturbations of degree n, for n = 1, 2, 3. For theses cases, denoting by H(n) the number of limit cycles that can bifurcate from this periodic annulus, we prove that if the linear differential system defined in the region between the two parallel lines (called of central subsystem) has a center at the origin and the others subsystems have centers or saddles then H(1) ≥ 3, H(2) ≥ 4 and H(3) ≥ 7. Now, for the particular case where the central subsystem has a center and the others subsystems have only real saddles, if the central subsystem has a real (not necessarily at the origin) or boundary center then H(1) ≥ 6 and if it has a virtual center then H(1) ≥ 4. Finally, if the central subsystem has a real saddle and the others subsystems have centers or saddles then H(1) ≥ 6. For this, we study the number of zeros of its Melnikov functions defined in two or three zones. Moreover, we provide detailed analytical methods to study the number of zeros from Melnikov functions defined in two or three zones. Keywords: Limit cycle. Piecewise linear differential system. Hamiltonian system. Melnikov function. List of Figures 1.1 (a) Crossing set; (b) Sliding set; (c) Escaping set. . . . . . . . . . . . . . . . . . . 23 1.2 (a) Escaping periodic orbit; (b) Crossing periodic orbit. . . . . . . . . . . . . . . 25 1.3 (a) Homoclinic loop; (b) Two heteroclinic orbits. . . . . . . . . . . . . . . . . . . 26 1.4 The crossing periodic orbits of system (1.4)|ε=0 with three and two zones, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 1.5 Poincaré maps of system (1.4) with three and two zones, respectively. . . . . . . 31 2.1 The limit cycle of vector field (2.3) with aL = 4, bL = 8, αL = 3/2, cL = −5/2, βL = 11/4, aC = 0, bC = 2, αC = βC = 2/3, cC = −2, aR = 4, bR = 2, cR = −10 and αR = βR = −4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.2 The limit cycle of vector field (2.3) with aL = bL = 1, αL = 2/3, cL = 35, βL = 214/3, aC = 0, bC = 2, αC = βC = 2/3, cC = −2, aR = 4, bR = 2, αR = βR = −4 and cR = −10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.3 The limit cycle of vector field (2.3) with aL = bL = 1, αL = 3/5, cL = 35, βL = 357/5, aC = 0, bC = 2, αC = βC = 1, cC = −2, aR = bR = 1, αR = −1, cR = 15 and βR = −31. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 2.4 The limit cycle of vector field (2.3) with aL = 4, bL = 8, αL = 2, cL = −5/2, βL = 5/2, aC = 2/5, bC = 24/5, αC = −9/5, cC = 4/5, βC = −4/15, aR = 8, bR = 10 and αR = cR = βR = −8. . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 2.5 The limit cycle of vector field (2.3) with aL = αL = −2/3, bL = 4/3, cL = 8/3, βL = 35/3, aC = 2/11, bC = 120/11, αC = −41/11, cC = 4/11, βC = −4/33, aR = −2/11, bR = 4/11, αR = 1/5, cR = 120/11 and βR = −749/55. . . . . . . . . 59 2.6 The limit cycle of vector field (2.3) with aL = αL = −2/3, bL = 4/3, cL = 8/3, βL = 35/3, aC = 2/11, bC = 120/11, αC = −41/11, cC = 4/11, βC = −4/33, aR = 8, bR = 10, αR = −7 and cR = βR = −8. . . . . . . . . . . . . . . . . . . . . 61 2.7 The limit cycle of vector field (2.3) with aL = αL = −2/3, bL = 4/3, cL = 8/3, βL = 35/3, aC = 2/11, bC = 120/11, αC = −41/11, cC = 4/11, βC = −4/33, aR = 8, bR = 10, αR = −7 and cR = βR = −8. . . . . . . . . . . . . . . . . . . . . 64 3.1 Periodic orbits of system (3.1)|ε=0 with αL = aL, αR = −aR, bC = cC = 1 and aC = αC = 0 when (a) βC = 0, (b) 0 < βC < 1 and (c) βC ≥ 1. . . . . . . . . . . . 70 3.2 Phase portraits of system (3.1)|ε=0 of the type : (a) SCS with τRS 6= τLS; (b) SCS with τRS = τLS; (c) CCS when left subsystem has a virtual center; (d) CCS when left subsystem has a real center. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.3 Phase portraits of system (3.1)|ε=0 of the type CCC when: (a) the left and right subsystems have virtual centers; (b) the left subsystem has a real center and right subsystem has a virtual center; (c) the left and right subsystems have real centers. 74 5.1 Crossing periodic orbits of system (5.1)|ε=0. . . . . . . . . . . . . . . . . . . . . . 121 5.2 Phase portraits of system (5.1)|ε=0 with (a) βC = τ2R/4 < 1, (b) βC = τ2R/4 ≥ 1. . 123 5.3 Phase portraits of system (5.1)|ε=0 with (c) τ2R/4 < βC < 1, (d) τ2R/4 < βC and βC ≥ 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 5.4 Phase portraits of system (5.1)|ε=0 with (e) 0 < βC < 1, βC < τ2R/4 and√ τ2R − 4βC < τL, (f) 1 ≤ βC < τ2R/4 and √ τ2R − 4βC < τL. . . . . . . . . . . . . . 124 5.5 Phase portraits of system (5.1)|ε=0 with (g) 0 < βC < 1, βC < τ2R/4 and√ τ2R − 4βC = τL, (h) 1 ≤ βC < τ2R/4 and √ τ2R − 4βC = τL. . . . . . . . . . . . . . 124 6.1 The crossing periodic orbits of system (6.1)|ε=0. . . . . . . . . . . . . . . . . . . . 141 6.2 Phase portrait of system (6.1)|ε=0 of the type SSS with τRS 6= τLS. . . . . . . . . 144 6.3 Phase portrait of system (6.1)|ε=0 of the type SSS with τRS = τLS. . . . . . . . . 144 6.4 Phase portrait of system (6.1)|ε=0 of the type CSS with a virtual center in the left subsystem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 6.5 Phase portrait of system (6.1)|ε=0 of the type CSS with a real center in the left subsystem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 6.6 Phase portraits of system (6.1)|ε=0 of the type CSC when (a) left and right subsystems have virtual centers; (b) left subsystem has a real center and right subsystems has a virtual center; (c) left and right subsystems have real centers. . 147 List of Tables 1 Lower bounds (Upper bounds*) of the maximum number of limit cycles of discontinuous piecewise linear differential systems with two zones separated by a straight line. Here Fr, Fv, Fb, Sr, S0 r , Nv, iNv, C and Cb denote real focus, virtual focus, boundary focus, real saddle, real saddle with zero trace, virtual node, improper node, center and boundary center, respectively. . . . . . . . . . . 15 2.1 Parameter values for that vector field (2.3) of the type CCC has one limit cycle. 48 2.2 Parameter values for that vector field (2.3) of the type SCC and SCS has one limit cycle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 2.3 Parameter values for that vector field (2.3) of the type CSC has one limit cycle. . 56 2.4 Parameter values for that vector field (2.3) of the type SSC and SSS has one limit cycle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 2.5 Parameter values for that vector field (2.3) with boundary singular points has one limit cycle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.1 Parameter values for which system (4.1)|ε=0 satisfies the hypothesis (H1). For the case SRCRSR, when τL 6= τR system (4.1)|ε=0 has a homoclinic loop and when τL = τR system (4.1)|ε=0 has two heteroclinic orbits. . . . . . . . . . . . . . . . . 101 Summary Introduction 14 1 Basic results 21 1.1 Piecewise smooth vector fields with two or three zones . . . . . . . . . . . . . . . 21 1.2 Piecewise smooth Hamiltonian vector fields with two or three zones . . . . . . . . 26 1.3 The Melnikov function method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2 Limit cycles of planar piecewise linear Hamiltonian differential system with two or three zones 37 2.1 Preliminaries and main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.2 Proof of Theorems 2-4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3 Persistence of periodic solutions from system (1)|ε=0 having a center in the central region 66 3.1 Preliminaries and main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.2 A normal form to system (3.1)|ε=0 . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.3 A classification to system (3.1)|ε=0 . . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.4 Melnikov functions associated with system (3.1) . . . . . . . . . . . . . . . . . . . 75 3.5 Proof of Theorem 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 3.6 Another method to prove Theorem 5 . . . . . . . . . . . . . . . . . . . . . . . . . 92 4 Bifurcation of limit cycles from a periodic annulus, by quadratic/cubic perturbations, from system (1)|ε=0 having a center in the central region 99 4.1 Preliminaries and main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 4.2 A classification to system (4.1)|ε=0 . . . . . . . . . . . . . . . . . . . . . . . . . . 102 4.3 Melnikov functions associated with system (4.1) . . . . . . . . . . . . . . . . . . . 103 4.4 Proof of Theorem 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 5 Persistence of periodic solutions from periodic annulus formed by a center and two saddles of system (1)|ε=0 119 5.1 Preliminaries and main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 5.2 A classification to system (5.1)|ε=0 . . . . . . . . . . . . . . . . . . . . . . . . . . 121 5.3 Melnikov functions associated with system (5.1) . . . . . . . . . . . . . . . . . . . 126 5.4 Proof of Theorem 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 6 Persistence of periodic solutions from system (1)|ε=0 having a saddle in the central region 139 6.1 Preliminaries and main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 6.2 A normal form to system (6.1)|ε=0 . . . . . . . . . . . . . . . . . . . . . . . . . . 141 6.3 A classification to system (6.1)|ε=0 . . . . . . . . . . . . . . . . . . . . . . . . . . 142 6.4 Melnikov functions associated with system (6.1) . . . . . . . . . . . . . . . . . . . 148 6.5 Proof of Theorem 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 7 Appendix 160 7.1 Proof of Theorem 11 from Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . 160 7.2 Proof of Theorem 12 from Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . 165 7.3 Proof of Theorem 16 from Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . 172 7.4 Proof of Theorem 17 from Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . 177 7.5 Proof of Theorem 18 from Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . 182 7.6 Coefficients of functions (4.44) and (4.45) from Chapter 4 . . . . . . . . . . . . . 187 7.7 Coefficients of functions (4.46)–(4.49) from Chapter 4 . . . . . . . . . . . . . . . 190 7.8 Coefficients of functions (6.38)–(6.40) from Chapter 6 . . . . . . . . . . . . . . . 193 7.9 Coefficients of functions (6.41)–(6.43) from Chapter 6 . . . . . . . . . . . . . . . 195 7.10 Parameter values to function (4.44) from Chapter 4 . . . . . . . . . . . . . . . . . 196 7.11 Parameter values to function (4.45) from Chapter 4 . . . . . . . . . . . . . . . . . 197 7.12 Parameter values to functions (4.46) – (4.49) from Chapter 4 . . . . . . . . . . . 197 8 Conclusions 201 Bibliography 202 Introduction The qualitative theory of differential systems is important tools for investigate properties inherent to the solutions of the equations that define the problem, without worrying about the possible expressions that such solutions may have. One of the most famous problems of this theory, formulated for the planar case, consist in to determine the number of limit cycles of polynomial differential systems, which was proposed by Hilbert in 1900 as part of Hilbert’s 16th problem, (see [25]). Nowadays, this problem has been considered for piecewise differential systems. The study of piecewise differential systems begins with the pioneering work of Andronov [1]. After Filippov [12] establishes the conventions that regulate the transitions of solutions of systems between different regions, piecewise differential systems have piqued the attention of researchers in qualitative theory of differential equations, mainly because many phenomena, for instance in mechanics, electrical circuits, control theory, neurobiology, etc, can be described by models that involve this kind of differential equations (see the books [10, 23, 48] and the papers [7, 13, 45, 47]). Piecewise linear differential systems are an interesting class of piecewise differential systems and, unlike the smooth case, have a rich dynamic that is far from being fully understood. In addition to numerous applications in various areas of knowledge. In 1990, Lum and Chua [8] conjectured that a continuous piecewise differential systems in the plane with two zones has at most one limit cycle. In 1998 this conjecture was proved by Freire, Ponce, Rodrigo and Torres in [15]. On the other hand, when the piecewise linear differential system is discontinuous, that is, the subsystems do not coincide on the switching curve, is known that the maximum number of limit cycles is at least three. Apparently, for it case, determining the exact number of limit cycles is a hard task. However, important partial results about this problem have been obtained. In summary, the results about the number of limit cycles of discontinuous piecewise linear differential systems with two zones separated by a straight line are given in Table . The symbol “—” indicates that those cases appear repeated in the table and the empty entries on it correspond to cases not studied in the literature, at least as far as we know. 14 Introduction 15 Fr Fv Fb Sr S0 r Nv iNv C Cb Fr 3 2∗ 3 2∗ 3 3 2∗ 2∗ Fv — 2 2∗ 2 1∗ 2 1∗ 1∗ Fb — — 1∗ 2∗ 1∗ 2∗ 2∗ 1∗ 1∗ Sr — — — 2∗ 1∗ 2 2 1∗ 1∗ S0 r — — — — 0∗ 1∗ 1∗ 0∗ 0∗ Nv — — — — — 1∗ 1∗ iNv — — — — — — 2 1∗ 1∗ C — — — — — — — 0∗ 0∗ Cb — — — — — — — — 0∗ Table 1: Lower bounds (Upper bounds*) of the maximum number of limit cycles of discontinuous piecewise linear differential systems with two zones separated by a straight line. Here Fr, Fv, Fb, Sr, S0 r , Nv, iNv, C and Cb denote real focus, virtual focus, boundary focus, real saddle, real saddle with zero trace, virtual node, improper node, center and boundary center, respectively. We denote the lower bounds in the entrances from Table by the symbols that indicate its position on the table. For example, the lower bound for the case with a real focus Fr and a virtual focus Fv is detonated by FrFv, that is, FrFv = 3. A proof for the lower bound FrFv can be found in [32]. A proof for the lower bound FrSr can be found in [29]. A proof for the lower bounds FrNv and FriNv can be found in [17]. A proof for the lower bound FvFv can be found in [16]. A proof for the lower bound FvSr can be found in [55]. A proof for the lower bound FviNv can be found in [56]. A proof for the upper bound SrSr can be found in [2]. A proof for the lower bounds SrNv and SriNv can be found in [33]. A proof for the lower bound iNviNv can be found in [27]. The other cases listed in Table can be found in [35]. In the papers [3, 5, 18, 41, 54] we can also find proofs for some lower bounds of Table . If the curve between two linear zones is not a straight line it is possible to obtain as many cycles as you want. This fact has been conjectured by Braga and Mello in [4] and firstly proved by Novaes and Ponce in [49]. Exact number of limit cycles, for discontinuous piecewise linear systems with two zones separated by a straight line, were obtained in particular cases. Llibre and Teixeira [39] proved that if the linear systems, that define the piecewise one, has no singular point, then it has at most one limit cycle. Medrado and Torregrossa [46] proved that if the straight line has only crossing sewing points and the piecewise linear system has only a monodromic singular point on it, then the system has at most one limit cycle. Recently, piecewise differential system defined in regions with more than two zones have attracted the attention of researchers, see for instance [11, 26, 36, 53, 60]. Results imposing restrictive hypotheses on the systems, such as symmetry and linearity, have been obtained. Introduction 16 For instance, conditions for nonexistence and existence of one, two or three limit cycles for symmetric continuous piecewise linear differential systems with three zones can be found in [34]. Now, for the nonsymmetric case, examples with two limit cycles surrounding a unique singular point at the origin was found in [38, 42]. When we remove these restrictions, that is, when we consider piecewise discontinuous differential system with three zones, there are few works in the literature. In fact, the works available deal with planar piecewise linear near-Hamiltonian differential systems with three zones, given by ẋ = Hy(x, y) + εf(x, y), ẏ = −Hx(x, y) + εg(x, y), (1) with H(x, y) =  HL(x, y) = bL 2 y2 − cL 2 x2 + aLxy + αLy − βLx, x ≤ −1, HC(x, y) = bC 2 y2 − cC 2 x2 + aCxy + αCy − βCx, −1 ≤ x ≤ 1, HR(x, y) = bR 2 y2 − cR 2 x2 + aRxy + αRy − βRx, x ≥ 1, f(x, y) =  fL(x, y), x ≤ −1, fC(x, y), −1 ≤ x ≤ 1, fR(x, y), x ≥ 1, g(x, y) =  gL(x, y), x ≤ −1, gC(x, y), −1 ≤ x ≤ 1, gR(x, y), x ≥ 1, where the functions fi and gi are C∞, for i = L,C,R, and 0 ≤ ε << 1. When ε = 0 (shortly (1)|ε=0) we say that system (1) is a piecewise linear Hamiltonian differential system. We call system (1) of left subsystem for x ≤ −1, right subsystem for x ≥ 1 and central subsystem for −1 ≤ x ≤ 1. In [40], Llibre and Teixeira study the existence of limit cycles for system (1)|ε=0 when the subsystems that define the piecewise one have a unique singular point, which is a center. More precisely, in the continuous case, they prove that system (1)|ε=0 has no limit cycles. Now, in the discontinuous case, system (1)|ε=0 has at most one limit cycle and there are examples with one. Fonseca, Llibre and Mello, in [14], proved that if system (1)|ε=0 is discontinuous and Introduction 17 without singular points, then the maximum number of limit cycles is also one. Many authors have contributed to estimate the number of limit cycles of system (1) that can bifurcate, after polynomial perturbations, from the periodic annulus of the unperturbed system (1)|ε=0. For instance, in [59] the authors showed that at least seven limit cycles can bifurcate from a periodic annulus, after linear perturbations, from system (1)|ε=0 with subsystems without singular points and a boundary pseudo–focus. On the other hand, in [61], 5, 7 and 12 limit cycles were obtained by linear, quadratic and cubic polynomials perturbations, respectively. But, in this paper, the periodic annulus of the unperturbed system (1)|ε=0 was obtained from a real saddle in the central subsystem and two virtual centers in the others subsystems. For the same type of periodic annulus, in [58], 10 limit cycles were obtained through cubic perturbations. The search for upper bound for the number of limit cycles that a piecewise linear system with three zones can have is what motivates most of the recently works found in the literature about this topic. However, all cases are interesting in themselves, that is, the search for better quotas for the number of limit cycles can not be used to discourage the study of particular families. So the type of singular points of the subsystems and their positions, that is, whether they are real, virtual or boundary, are important questions in this study and should be considered for all subclasses of piecewise linear systems with three zones. Motivated by the previously mentioned works, in this thesis we contribute along these study lines. Our goal is to estimate the number of crossing limit cycles that can bifurcate from periodic annulus of piecewise linear near-Hamiltonian differential systems with two or three zones. For this purpose, we have the text organized as follows. In Chapter 1, we introduce some definitions and basic results about piecewise smooth vector fields with two or three zones that will be essential in the development of this work. Moreover, in Section 1.3 we will present the Melnikov function method, that is an important tool to investigate the number of limit cycles that emerge from a periodic annulus of a piecewise Hamiltonian differential system. In Chapter 2, motivated by the paper [40], we study the existence of crossing limit cycles in continuous and discontinuous planar piecewise linear Hamiltonian differential system with two or three zones separated by straight lines, and parallel if there are more than one, such that the linear systems that define the piecewise one have isolated singular points, that is, centers or saddles. In this case, we show that if the planar piecewise linear Hamiltonian differential system is either continuous or discontinuous with two zones or continuous with three zones, then it has no crossing limit cycles. Now, if the planar piecewise linear Hamiltonian differential system is discontinuous with three zones, that is, if system (1)|ε=0 is discontinuous, then it has at most Introduction 18 one crossing limit cycle, and there are examples with one limit cycle. More precisely, without taking into account the position of the singular points in the zones, we present examples with the unique crossing limit cycle for all possible combinations of saddles and centers. This chapter is the content of paper I of the list of published and submitted papers derived from this thesis available at the end of this introduction. In Chapter 3, we study the number of crossing limit cycles that can bifurcated, after linear perturbations, from a periodic annulus formed by periodic orbits of system (1)|ε=0 that passes through the three zones. We prove that if the central subsystem has a center at the origin and the others subsystems have centers or saddles, then we have at least three crossing limit cycles visiting the three zones. Our results are obtained by studying the number of zeros of the first order Melnikov function associated to system (1). Moreover, we will present, in Section 1, a normal form for system (1)|ε=0 in order to simplify the compute. This normal form will also be used in Chapters 4 and 5. This study resulted in paper II of the list of published and submitted papers derived from this thesis. In Chapter 4, motivated by the results obtained in Chapter 3, we estimated the lower bounds for the number of crossing limit cycles that can bifurcate from periodic annulus of piecewise system (1), assuming similar hypotheses to system (1)|ε=0 from Chapter 3, but with the difference that we perturb system (1)|ε=0 by polynomial functions of degree n, for n = 2, 3. Denoting by H(n) the number of crossing limit cycles that can bifurcate from this periodic annulus, we prove that H(2) ≥ 4 and H(3) ≥ 7. A part of these results are published in the paper III of the list of published and submitted papers derived from this thesis. In Chapter 5, we study the number of crossing limit cycles that can bifurcated, after linear perturbations, from periodic annulus of system (1)|ε=0, assuming that the central subsystem has a center and the other subsystems have only saddles. What differs the results of this chapter from the one of Chapter 3 is that now the center of the central subsystem is not fixed at the origin and therefore, we will have periodic annulus formed by crossing periodic orbits passing through two and three zones. We prove that if the central subsystem from system (1)|ε=0 has a real or a boundary center, then the maximum number of limit cycles that bifurcate from the periodic annulus is at least six. Four passing through the three zones and two passing through two zones. Now, if the central subsystem has a virtual center, then we have at least four limit cycles bifurcating from the periodic annulus, three passing through the three zones and one passing through two zones. This study resulted in preprint IV of the list of published and submitted papers derived from this thesis. Finally, in Chapter 6, we study the number of crossing limit cycles that can bifurcated from Introduction 19 periodic annulus of system (1)|ε=0, assuming that the central subsystem has a real saddle and the other subsystems have centers or saddles. We prove that the maximum number of limit cycles that can bifurcate from the periodic annulus of this kind of piecewise linear Hamiltonian differential systems, after linear perturbations, is at least five, when the right and left subsystems have saddles, and at least six, when the right subsystem has a saddle or center and the left subsystem has a center (vice-versa). Our study is concentrated in the neighborhood of the double homoclinic loop which separates the periodic annulus of orbits that pass through three zones from the two periodic annulus of orbits that pass through only two zones. Thus, to estimate the zeros of the Melnikov functions we consider its expansions at the point corresponding to this orbit. Moreover, in order to simplifies the computation, we obtain a normal form to system (1)|ε=0 when the central subsystem has a real saddle. This study resulted in preprint V of the list of published and submitted papers derived from this thesis. Introduction 20 Published and submitted papers derived from this thesis I. Pessoa, C., Ribeiro, R.: Limit cycles of planar piecewise linear Hamiltonian differential systems with two or three zones, Electron. J. Qual. Theory Differ. Equ., 27, 1-19 (2022). doi:10.14232/ejqtde.2022.1.27 II. Pessoa, C., Ribeiro, R.: Limit cycles bifurcating from a periodic annulus in discontinuous planar piecewise linear Hamiltonian differential system with three zones, Int. J. Bifur. Chaos Appl. Sci. Eng., 32, 2250114, 16 pp (2022). doi:10.1142/S0218127422501140 III. Pessoa, C., Ribeiro, R.: Persistence of periodic solutions from discontinuous planar piecewise linear Hamiltonian differential systems with three zones, São Paulo J. Math. Sci., 16, 932-956 (2022). doi:10.1007/s40863-022-00313-z IV. Pessoa, C., Ribeiro, R.: Bifurcation of limit cycles from a periodic annulus formed by a center and two saddles in piecewise linear differential system with three zones, Preprint (2022). Available in doi:10.48550/arXiv.2207.05177 V. Euzébio, R., Gouveia, M., Novaes, D., Pessoa, C., Ribeiro, R.: On cyclicity in discontinuous piecewise linear near-Hamiltonian differential systems with three zones having a saddle in the central one, Preprint (2022). Available in doi:10.48550/arXiv.2212.00828 https://doi.org/10.14232/ejqtde.2022.1.27 https://doi.org/10.1142/S0218127422501140 https://doi.org/10.1007/s40863-022-00313-z https://doi.org/10.48550/arXiv.2207.05177 https://doi.org/10.48550/arXiv.2212.00828 Chapter 1 Basic results In this chapter, we present the definitions, notations and some basic results about piecewise smooth vector fields with two or three zones that will be used throughout this work. Moreover, we present a method, that is the first order Melnikov function, to investigate the number of limit cycles that can emerging from a period annulus of a piecewise Hamiltonian differential system. 1.1 Piecewise smooth vector fields with two or three zones Let us consider a codimension one manifolds Σi ⊂ R2, for i = C,L,R, given by Σi = f−1i (0), where fi : R2 → R is a smooth function such that 0 ∈ R is a regular value, i.e., ∇fi(p) 6= 0, for all p ∈ f−1i (0). We call Σi, for i = C,L,R, of switching line and we assume that ΣC separates R2 into two disjoint regions given by RR = {q ∈ R2 : fC(q) > 0} and RL = {q ∈ R2 : fC(q) < 0}, that is, R2 = RL∪ΣC ∪RR, and we also assume that ΣL and ΣR separates R2 into three disjoint regions given by RR = {q ∈ R2 : fR(q) > 0}, RC = {q ∈ R2 : fR(q) < 0 < fL(q)} and RL = {q ∈ R2 : fL(q) < 0}, that is, R2 = RL ∪ ΣL ∪RC ∪ ΣR ∪RR. Denote by X (U) the space of Cr-vector fields on U ⊂ R2 endowed with the Cr-topology, 21 1.1 Piecewise smooth vector fields with two or three zones 22 with r ≥ 1. Definition 1. A planar piecewise smooth vector field with two zones can be written as Z : U ⊂ R2 → R2 such that Z(q) =  XR(q), if fC(q) ≥ 0, XL(q), if fC(q) ≤ 0, (1.1) where XR, XL ∈ X (U). For simplify the notation, we denote vector field (1.1) by Z = (XL, XR). We call vector field (1) of left vector field when fC(q) ≤ 0 and right vector field when fC(q) ≥ 0. Definition 2. A planar piecewise smooth vector field with three zones can be written as Z : U ⊂ R2 → R2 such that Z(q) =  XR(q), if fR(q) ≥ 0, XC(q), if fR(q) ≤ 0 ≤ fL(q), XL(q), if fL(q) ≤ 0, (1.2) where XR, XC , XL ∈ X (U). For simplify the notation, we denote vector field (1.2) by Z = (XL, XC , XR). We call vector field (2) of left vector field when fL(q) ≤ 0, right vector field when fR(q) ≥ 0 and central vector field when fR(q) ≤ 0 ≤ fL(q). Next we present some basic concepts of piecewise smooth vector field theory, most of which were introduced by Filippov in [12]. We will use the vector field XL and the switching line ΣC , from piecewise vector field (1), in the next definitions. However, we can easily adapt the definitions to the vector fields XC and XR and the switching lines ΣL and ΣR from piecewise smooth vector field (2). The derivative of function fC in the direction of the vector field XL, i.e., the expression XLfC(p) = 〈XL(p),∇fC(p)〉, also known as the Lie derivative of fC with respect to XL, where 〈·, ·〉 is the usual inner product in R2, characterizes the contact between the vector field XL and the switching line ΣC . Analogously, we have Xi LfC(p) = 〈XL(p),∇Xi−1 L fC(p)〉, for i ≥ 2. We can distinguish the followings subsets of ΣC (see Figure 1.1) by Crossing set : Σc C = {p ∈ ΣC : XLfC(p) ·XRfC(p) > 0}; Sliding set : Σs C = {p ∈ ΣC : XLfC(p) > 0, XRfC(p) < 0}; 1.1 Piecewise smooth vector fields with two or three zones 23 Escaping set : Σe C = {p ∈ ΣC : XLfC(p) < 0, XRfC(p) > 0}. ΣCΣCΣC (a) (b) (c) Figure 1.1: (a) Crossing set; (b) Sliding set; (c) Escaping set. On the sets Σs C and Σe C we define the Filippov vector field FZ associated to Z = (XL, XR), as follows: if p ∈ Σs C ∪Σe C , then FZ(p) denotes the vector tangent to ΣC in the cone spanned by XL(p) and XR(p). A point p ∈ ΣC is called a ΣC-regular point of Z if p ∈ Σc C ∪ Σs C ∪ Σe C , then FZ(p) 6= 0. Otherwise, p is a ΣC-singular point of Z. When p ∈ ΣC and XLfC(p) = 0 we say that p is a tangent point of XL. A tangent point p is a fold point of XL if XLfC(p) = 0 and X2 LfC(p) 6= 0. Moreover, p is a visible (resp. invisible) fold point of XL if XLfC(p) = 0 and X2 LfC(p) < 0 (resp. X2 LfC(p) > 0). We can classify a singular point p of vector field XL according to its position in relation to the vector field. More precisely, we say that p is a real singular point of XL if XL(p) = 0 and p ∈ RL, a virtual singular point of XL if XL(p) = 0 and p ∈ (RL ∪ ΣC)c, where (RL ∪ ΣC)c denotes the complement of RL ∪ ΣC in R2, and a boundary singular point of XL if XL(p) = 0 and p ∈ ΣC . Definition 3. Piecewise smooth vector field (1.1) is called continuous if XL(p) = XR(p), ∀ p ∈ ΣC . Otherwise, it is called discontinuous. Similarly, piecewise smooth vector field (1.2) is called continuous if XL(p) = XC(p), ∀ p ∈ ΣL and XC(q) = XR(q), ∀ q ∈ ΣR. Otherwise, it is called discontinuous. The following definitions describe the concepts about local and global trajectories of a 1.1 Piecewise smooth vector fields with two or three zones 24 piecewise smooth vector field with two zones, and can be extended to three zones. Denote by φXL(t, p) the flow of vector field XL ∈ X (U), such that  d dt φXL(t, p) = XL ( φXL(t, p) ) , φXL(0, p) = p, where t ∈ I ⊂ R and I depends on the point p ∈ U and the vector field XL. Definition 4. A local trajectory (or local orbit) φZ(t, p) of vector field (1), through the point p, is obtained by concatenating the flows of the vector fields XR, XL and FZ . Definition 5. The global trajectory (or global orbit) ΦZ(t, p) passing through p is a union ΦZ(t, p) = ⋃ i∈Z {σi(t, pi) : ti ≤ t ≤ ti+1} of preserving-orientation local trajectories σi(t, pi) satisfying σi(ti, pi) = pi and σi(ti+1, pi) = pi+1. An global trajectory is positive (resp. negative) if i ∈ N (resp. −i ∈ N) and t0 = 0. Definition 6. Let ΦZ(t, p) a global trajectory of the piecewise smooth vector field (1). We say that ΦZ is periodic orbit if ΦZ is periodic in the variable t, that is, if exists T > 0 such that ΦZ(t+ T, p) = ΦZ(t, p) for all t ∈ R. In a piecewise smooth vector field, a periodic orbit can be of two types: sliding/escaping periodic orbit or crossing periodic orbit; the first one contains some segment of sliding or escaping sets (see Figure 1.2 (a)), and the second one does not contain any segments of sliding or escaping sets (see Figure 1.2 (b)). A limit cycle of vector field (1) is an isolated periodic orbit in the set of all periodic orbits of vector field (1). Therefore, a limit cycle can be of two types: sliding/escaping limit cycle or crossing limit cycle. In this work, unless mentioned otherwise, when we talk about limit cycles, we are talking about crossing limit cycles. Consider a point p ∈ ΣC such that the orbit φXL(t, p) of the vector field XL passing through p returns to ΣC after a positive time tL(p), called flight time. We define the half return map associated with XL by πXL(p) = φXL(tL(p), p) = pL ∈ ΣC . Similarly, when the orbit φXR(t, pL) of the vector field XR passing through pL ∈ ΣC returns to ΣC after a positive time tR(pL), we define the half return map associated with XR by πXR(pL) = φXR(tR(pL), pL) = pR ∈ ΣC . 1.1 Piecewise smooth vector fields with two or three zones 25 ΣCΣC (a) (b) Figure 1.2: (a) Escaping periodic orbit; (b) Crossing periodic orbit. The first return map (or Poincaré map) associated with Z = (XL, XR) is defined by πZ(p) = πXR ◦ πXL(p) = φXR ( tR(pL), φXL(tL(p), p) ) , or in reverse order, applying φXR and after φXL . Moreover, if p ∈ ΣC is a isolated fixed point to map πZ , then the global trajectory passing through p is a limit cycle of the piecewise smooth vector field (1). Instead of studying the fixed point of the first return map, some times it is more convenient to study the zeros of the correspondent displacement map, given by d(p) = πZ(p)− p. In the following, we introduce the notions of separatrix, separatrix connections and equivalence between two piecewise smooth vector fields. For more details see [19]. Definition 7. An unstable separatrix of a saddle point p of Z = (XL, XR) is the invariant unstable manifold W u(p) given by W u(p) = {q ∈ U : ΦZ(t, q) is defined for t ∈ (−∞, 0) and lim t→−∞ ΦZ(t, q) = p}. A stable separatrix of a saddle point p of Z = (XL, XR) is the invariant stable manifold W s(p) given by W s(p) = {q ∈ U : ΦZ(t, q) is defined for t ∈ (0,∞) and lim t→∞ ΦZ(t, q) = p}. If a separatrix is simultaneously stable and unstable it is a separatrix connection. Here we distinguish two types of separatrix connections: 1.1 Piecewise smooth vector fields with two or three zones 26 � Homoclinic loop : for this type we have W u(p) = W s(p), see Figure 1.3 (a). � Heteroclinic orbits : for this case we have two saddle points p 6= q such that W u(p) = W s(q) and W s(p) = W u(q), see Figure 1.3 (b). ΣCΣC (a) (b) pqp Figure 1.3: (a) Homoclinic loop; (b) Two heteroclinic orbits. In the follow, we introduce the notion of Σ-equivalence between two piecewise smooth vector fields, which allows comparing phase portraits. Definition 8. Two piecewise smooth vector fields Z1 : U1 ⊂ R2 → R2 and Z2 : U2 ⊂ R2 → R2 with switching lines Σ1 ⊂ U1 and Σ2 ⊂ U2, respectively, are Σ1-equivalent if there exists an orientation preserving homeomorphism h : U1 → U2 which sends Σ1 to Σ2 and sends orbits of Z1 to orbits of Z2. We can see that any Σ-equivalence sends regular orbits to regular orbits and distinguished singular point to distinguished singular point. Moreover, as it sends arrival and departing points to arrival and departing points, Σc , Σs and Σe are preserved, and thus it also sends sliding orbits to sliding orbits and preserves separatrices, separatrix connections, periodic orbits and cycles. 1.2 Piecewise smooth Hamiltonian vector fields with two or three zones In this section, some basic concepts about piecewise smooth Hamiltonian vector fields are presented. Definition 9. Consider H : U → R a Cr function, with U an open subset of R2 and r > 1. Fixed a coordinate system (x, y) ∈ R2, a smooth Hamiltonian vector field is a vector field given by X(x, y) = (Hy(x, y),−Hx(x, y)), (1.3) 1.2 Piecewise smooth Hamiltonian vector fields with two or three zones 27 where Hi(x, y) denotes the derivative of the function H with respect the variable i = x, y. We say that the function H is the Hamiltonian function of vector field (1.3). Unless there is a coordinate change (symplectic), every Hamiltonian vector field is like this. Now, let us remember some important properties of Hamiltonian vector fields. 1. H is constant along the solutions of vector field (1.3), that is, H is a first integral of the system  ẋ = Hy(x, y), ẏ = −Hx(x, y), where the dot denotes the derivative with respect to the independent variable t, here called the time. 2. X : R2 → R2 is a Hamiltonian vector field if and only if div (X) = 0. 3. The only non-degenerate singular points of a planar Hamiltonian vector field are centers or saddles. Definition 10. Consider H i : U ⊂ R2 → R, Cr, r > 1, and fi : R2 → R a smooth function such that 0 ∈ R is a regular value, for i = R,C,L. A piecewise smooth Hamiltonian vector field with two zones is a vector field given by Z(x, y) =  XR(x, y) = (HR y (x, y),−HR x (x, y)), if fC(x, y) ≥ 0, XL(x, y) = (HL y (x, y),−HL x (x, y)), if fC(x, y) ≤ 0. A piecewise smooth Hamiltonian vector field with three zones is a vector field given by Z(x, y) =  XR(x, y) = (HR y (x, y),−HR x (x, y)), if fR(x, y) ≥ 0, XC(x, y) = (HC y (x, y),−HC x (x, y)), if fR(x, y) ≤ 0 ≤ fL(x, y), XL(x, y) = (HL y (x, y),−HL x (x, y)), if fL(x, y) ≤ 0. When the vector fields Xi, i = R,C,L that define the piecewise smooth vector field Z = (XR, XL) (or Z = (XR, XC , XL)) are Hamiltonian, then the solutions curves of the respective differential equations are contained in the level sets of the Hamiltonian functions. Thus, the first return map can be studied by seeking by points in ΣC (or ΣR and ΣL) that are in the same level curves of these Hamiltonian functions. 1.3 The Melnikov function method 28 1.3 The Melnikov function method There are two standard methods for studying the number of limit cycles that bifurcate from a periodic annulus of a smooth vector field by small perturbations. The first one is the Averaging method, which has an extension to the piecewise smooth case, see for example [22, 31, 37], and the second one is the Melnikov function method, which also has an extension to piecewise smooth case, see [20, 43]. In [21], the authors proved that these two methods are equivalent to each other. The main goal of both methods is to transform the problem of estimate the number of limit cycles of a piecewise system into a problem of finding zeros of a given function. In the planar case, there are situations in which some limit cycles are not detected by the Melnikov function method when we perturb a periodic annulus of a piecewise Hamiltonian differential system (see [6, 44]). These limit cycles in this context are called alien limit cycles. In this work, we will use the number of zeros of the Melnikov function to estimate the number of limit cycles that can bifurcate from a periodic annulus. Furthermore, in this section, we will present expressions for the Melnikov functions associated with a discontinuous planar piecewise Hamiltonian differential system with two or three zones. To establish the Melnikov functions, consider H i : U ⊂ R2 → R, a Cr function with r > 1 and hi : R2 → R, with hi(x, y) = x− xi (note that 0 ∈ R is a regular value), for i = R,C,L. A discontinuous planar piecewise near–Hamiltonian differential system with n zones, for n = 2, 3, is a piecewise differential system given by ẋ = Hy(x, y) + εf(x, y), ẏ = −Hx(x, y) + εg(x, y), (1.4) where, for n = 2, we have that H(x, y) =  HL(x, y), hC(x, y) ≤ 0, HR(x, y), hC(x, y) ≥ 0, f(x, y) =  fL(x, y), hC(x, y) ≤ 0, fR(x, y), hC(x, y) ≥ 0, 1.3 The Melnikov function method 29 g(x, y) =  gL(x, y), hC(x, y) ≤ 0, gR(x, y), hC(x, y) ≥ 0, and, for n = 3, we have that H(x, y) =  HL(x, y), hL(x, y) ≤ 0, HC(x, y), hR(x, y) ≤ 0 ≤ hL(x, y), HR(x, y), hR(x, y) ≥ 0, f(x, y) =  fL(x, y), hL(x, y) ≤ 0, fC(x, y), hR(x, y) ≤ 0 ≤ hL(x, y), fR(x, y), hR(x, y) ≥ 0, g(x, y) =  gL(x, y), hL(x, y) ≤ 0, gC(x, y), hR(x, y) ≤ 0 ≤ hL(x, y), gR(x, y), hR(x, y) ≥ 0, where the functions fi and gi are C∞, for i = L,C,R, and 0 ≤ ε << 1. When ε = 0 system (1.4) is a piecewise Hamiltonian differential system. Regarding system (1.4)|ε=0, we make some assumptions: (H1) For the case n = 3, suppose that unperturbed system (1.4)|ε=0 has an open interval J0 = (α0, β0) and a periodic annulus consisting of a family of crossing periodic orbits L0 h, with h ∈ J0, such that each orbit of this family crosses the straight lines x = xR and x = xL in four points, A(h) = (xR, a(h)), A1(h) = (xR, a1(h)), A2(h) = (xL, a2(h)) and A3(h) = (xL, a3(h)), with a1(h) < a(h) and a2(h) < a3(h), through the three zones with clockwise orientation (see Figure 1.4), satisfying the following equations HR(A(h)) = HR(A1(h)), HC(A1(h)) = HC(A2(h)), HL(A2(h)) = HL(A3(h)), HC(A3(h)) = HC(A(h)), 1.3 The Melnikov function method 30 and, for h ∈ J0, HR y (A(h))HR y (A1(h))HL y (A2(h))HL y (A3(h)) 6= 0, HC y (A(h))HC y (A1(h))HC y (A2(h))HC y (A3(h)) 6= 0. Therefore, for each h ∈ J0, the crossing periodic orbit L0 h from system (1.4)|ε=0 is given by L0 h = ÂA1 ∪ Â1A2 ∪ Â2A3 ∪ Â3A (see Figure 1.4). (H2) For the case n = 2, suppose that unperturbed system (1.4)|ε=0 has a open interval J1 = (α1, β1) and a periodic annulus consisting of a family of crossing periodic orbits L1 h, h ∈ J1, such that each orbit of this family crosses the straight lines x = xC in two points, B(h) = (xC , b(h)) and B1(h) = (xC , b1(h)), with b1(h) < b(h), through the two zones with clockwise orientation (see Figure 1.4), satisfying the following equations HR(B(h)) = HR(B1(h)), HL(B1(h)) = HL(B(h)), and, for h ∈ J1, HR y (B(h))HR y (B1(h))HL y (B(h))HL y (B1(h)) 6= 0. Therefore, for each h ∈ J1, the crossing periodic orbit L1 h from system (1.4)|ε=0 is given by L1 h = B̂B1 ∪ B̂1B (see Figure 1.4). x = xRx = xL A A1A2 A3 B B1 L1 h L0 h x = xC Figure 1.4: The crossing periodic orbits of system (1.4)|ε=0 with three and two zones, respectively. For the perturbed system (1.4) when n = 3, consider, for h ∈ J0, the solution of right subsystem from (1.4) starting from point A(h). Let A1ε(h) = (xR, a1ε(h)) be the first intersection point of this orbit with straight line x = xR. Denote by A2ε(h) = (xL, a2ε(h)) the first intersection point of the orbit from central subsystem from (1.4) starting at A1ε(h) with straight line x = xL, 1.3 The Melnikov function method 31 A3ε(h) = (xL, a3ε(h)) the first intersection point of the orbit from left subsystem from (1.4) starting at A2ε(h) with straight line x = xL and Aε(h) = (xR, aε(h)) the first intersection point of the orbit from central subsystem from (1.4) starting at A3ε(h) with straight line x = xR (see Figure 1.5). Similarly, for the perturbed system (1.4) when n = 2, consider, for h ∈ J1, the solution of right subsystem from (1.4) starting from point B(h). Let B1ε(h) = (xC , b1ε(h)) be the first intersection point of this orbit with straight line x = xC and Bε(h) = (xC , bε(h)) the first intersection point of the orbit from central subsystem from (1.4) starting at B1ε(h) with straight line x = xC (see Figure 1.5). x = xRx = xL Aε A1ε A2ε A3ε A Bε B1ε B x = xC Figure 1.5: Poincaré maps of system (1.4) with three and two zones, respectively. Thus, we can define the displacement maps of piecewise system (1.4) as B0(h, ε) = HR(Aε(h))−HR(A(h)) = εF0(h, ε) = εM0(h) +O(ε2), ∀h ∈ J0, (1.5) B1(h, ε) = HR(Bε(h))−HR(B(h)) = εF1(h, ε) = εM1(h) +O(ε2), ∀h ∈ J1, (1.6) where M0 and M1 are called the first order Melnikov functions associated to piecewise system (1.4) with three or two zones, respectively. Note that, by the Mean Value Theorem, B0(h, ε) = DHR ( A(h) +O ( Aε(h)−A(h) )) (Aε(h)−A(h)) = ( DHR(A(h)) +O(ε) ) (Aε(h)−A(h)) = ( HR y (A(h)) +O(ε) ) (aε(h)− a(h)). As, by hypothesis (H1), HR y (A(h)) 6= 0, we have that B0(h, ε) = 0 if and only if a(h) = aε(h). Similarly, B1(h, ε) = 0 if and only if b(h) = bε(h). Therefore, for 0 < ε � 1, system (1.4) has a limit cycle in Lh if and only if Fi, i = 0, 1, has an isolated zero in h. Thus, by Implicit Function Theorem, the number of simple zeros of the function Mi, i = 0, 1, can be used to estimate the number of limit cycles emerging from the periodic annulus from system (1.4). 1.3 The Melnikov function method 32 In what follows, we provide an expression for the first order Melnikov functions M0 and M1, that we will use frequently in the text. Then, following the steps of the proof of Theorem 1.1 in [43] for the case with two zones and, doing the obvious adaptations, for the case with three zones, we can prove the following theorem. Theorem 1. Consider system (1.4) with 0 ≤ ε � 1 and suppose that the unperturbed system (1.4)|ε=0 satisfies the hypotheses (Hi), i = 1, 2. Then, the first order Melnikov functions associated to system (1.4) with three and two zones can be expressed, respectively, as M0(h) = HR y (A) HC y (A) I0C + HR y (A)HC y (A3) HC y (A)HL y (A3) I0L + HR y (A)HC y (A3)H L y (A2) HC y (A)HL y (A3)HC y (A2) Ī0C + HR y (A)HC y (A3)H L y (A2)H C y (A1) HC y (A)HL y (A3)HC y (A2)HR y (A1) I0R, h ∈ J0, (1.7) and M1(h) = HR y (B) HL y (B) I1L + HR y (B)HL y (B1) HL y (B)HR y (B1) I1R, h ∈ J1, (1.8) where I0C = ∫ Â3A gCdx− fCdy, I0L = ∫ Â2A3 gLdx− fLdy, Ī0C = ∫ Â1A2 gCdx− fCdy, I0R = ∫ ÂA1 gRdx− fRdy, I1L = ∫ B̂1B gLdx− fLdy and I1 R = ∫ B̂B1 gRdx− fRdy, Furthermore, if Mi, for i = 0, 1, has a simple zero at h∗, then for 0 < ε << 1, system (1.4) has a unique limit cycle near Lh∗. Proof. Firstly, let us obtain the expression of the Melnikov function (1.7). From (1.5), the displacement map is given by HR(Aε(h))−HR(A(h)) = δ1 + δ2 + δ3 + δ4 + δ5 + δ6 + δ7 + δ8, (1.9) where δ1 = HR(Aε(h)) − HC(Aε(h)), δ2 = HC(Aε(h)) − HC(A3ε(h)), δ3 = HC(A3ε(h)) − HL(A3ε(h)), δ4 = HL(A3ε(h))−HL(A2ε(h)), δ5 = HL(A2ε(h))−HC(A2ε(h)), δ6 = HC(A2ε(h))− HC(A1ε(h)), δ7 = HC(A1ε(h))−HR(A1ε(h)) and δ8 = HR(A1ε(h))−HR(A(h)). Note that δ2 = HC(Aε(h))−HC(A3ε(h)) = ∫ Â3εAε dHC(x, y) = ∫ Â3εAε HC x dx+HC y dy. From the right side of (1.4), we can substitute dx by (HC y + εfC)dt and dy by (−HC x + εgC)dt, 1.3 The Melnikov function method 33 respectively. Then, δ2 becomes δ2 = ∫ Â3εAε HC x (HC y + εfC)dt+ ∫ Â3εAε HC y (−HC x + εgC)dt = ε ∫ Â3εAε (HC x fC +HC y gC)dt = ε ∫ Â3εAε fC(−dy + εgCdt) + ε ∫ Â3εAε gC(dx− εfCdt) = ε ∫ Â3A gCdx− fCdy +O(ε2). Thus, we have that ∂δ2 ∂ε |ε=0 = ∫ Â3A gCdx− fCdy. (1.10) With the same argument, we obtain ∂δ4 ∂ε |ε=0 = ∫ Â2A3 gLdx− fLdy, (1.11) ∂δ6 ∂ε |ε=0 = ∫ Â1A2 gCdx− fCdy, (1.12) and ∂δ8 ∂ε |ε=0 = ∫ ÂA1 gRdx− fRdy. (1.13) Moreover, since A = (xR, a(h)), A1ε = (xR, a1ε(h)), A2ε = (xL, a2ε(h)), A3ε = (xL, a3ε(h)) and Aε = (xR, aε(h)), we obtain, from δ2 = HC(Aε(h))−HC(A3ε(h)), δ4 = HL(A3ε(h))−HL(A2ε(h)), δ6 = HC(A2ε(h))−HC(A1ε(h)) and δ8 = HR(Aε(h))−HR(A(h)), ∂δ2 ∂ε |ε=0 = HC y (A(h)) ∂aε ∂ε |ε=0 −HC y (A3(h)) ∂a3ε ∂ε |ε=0, (1.14) ∂δ4 ∂ε |ε=0 = HL y (A3(h)) ∂a3ε ∂ε |ε=0 −HL y (A2(h)) ∂a2ε ∂ε |ε=0, (1.15) ∂δ6 ∂ε |ε=0 = HC y (A2(h)) ∂a2ε ∂ε |ε=0 −HC y (A1(h)) ∂a1ε ∂ε |ε=0, (1.16) and ∂δ8 ∂ε |ε=0 = HR y (A1(h)) ∂a1ε ∂ε |ε=0. (1.17) From (1.13) and (1.17), we conclude that ∂a1ε ∂ε |ε=0 = 1 HR y (A1(h)) ∫ ÂA1 gRdx− fRdy. (1.18) 1.3 The Melnikov function method 34 From (1.18), (1.16) and (1.12), we have that ∂a2ε ∂ε |ε=0 = 1 HC y (A2(h)) ∫ Â1A2 gCdx− fCdy + HC y (A1(h)) HC y (A2(h))HR y (A1(h)) ∫ ÂA1 gRdx− fRdy. (1.19) From (1.19), (1.15) and (1.11), we have that ∂a3ε ∂ε |ε=0 = 1 HL y (A3(h)) ∫ Â2A3 gLdx− fLdy + HL y (A2(h)) HL y (A3(h)) ∂a2ε ∂ε |ε=0. (1.20) From (1.20), (1.14) and (1.10), we have that ∂aε ∂ε |ε=0 = 1 HC y (A(h)) ∫ Â3A gCdx− fCdy + HC y (A3(h)) HC y (A(h)) ∂a3ε ∂ε |ε=0. Thus, since δ1 = HR(Aε(h))−HC(Aε(h)), it follows that ∂δ1 ∂ε |ε=0 = ( HR y (A(h))−HC y (A(h)) )∂aε ∂ε |ε=0. (1.21) Similarly, since δ3 = HC(A3ε(h)) − HL(A3ε(h)), δ5 = HL(A2ε(h)) − HC(A2ε(h)) and δ7 = HC(A1ε(h))−HR(A1ε(h)), we obtain ∂δ3 ∂ε |ε=0 = ( HC y (A3(h))−HL y (A3(h)) )∂a3ε ∂ε |ε=0, (1.22) ∂δ5 ∂ε |ε=0 = ( HL y (A2(h))−HC y (A2(h)) )∂a2ε ∂ε |ε=0, (1.23) and ∂δ7 ∂ε |ε=0 = ( HC y (A1(h))−HR y (A1(h)) )∂a1ε ∂ε |ε=0. (1.24) Therefore, from (1.5) and (1.9), we have that M0(h) = ∂ ∂ε ( HR(Aε(h))−HR(A(h)) ) = ∂ ∂ε ( δ1 + δ2 + δ3 + δ4 + δ5 + δ6 + δ7 + δ8 ) |ε=0. (1.25) Replacing (1.10), (1.11), (1.12), (1.13), (1.21), (1.22), (1.23), (1.24) into (1.25), we obtain the expression (1.7). 1.3 The Melnikov function method 35 Now, let us obtain the expression of the Melnikov function (1.8). From (1.6), the displacement map is given by HR(Bε(h))−HR(B(h)) = δ1 + δ2 + δ3 + δ4, (1.26) where δ1 = HR(Bε(h)) − HL(Bε(h)), δ2 = HL(Bε(h)) − HL(B1ε(h)), δ3 = HL(B1ε(h)) − HR(B1ε(h)) and δ4 = HR(B1ε(h))−HR(B(h)). Note that δ2 = HL(Bε(h))−HL(B1ε(h)) = ∫ B̂1εBε dHL(x, y) = ∫ B̂1εBε HL xdx+HL y dy. From the right side of (1.4), we can substitute dx by (HL y + εfL)dt and dy by (−HL x + εgL)dt, respectively. Then, δ2 becomes δ2 = ∫ B̂1εBε HL x (HL y + εfL)dt+ ∫ B̂1εBε HL y (−HL x + εgL)dt = ε ∫ B̂1εBε (HL x fL +HL y gL)dt = ε ∫ B̂1εBε fL(−dy + εgLdt) + ε ∫ B̂1εBε gL(dx− εfLdt) = ε ∫ B̂1B gLdx− fLdy +O(ε2). Thus, we have that ∂δ2 ∂ε |ε=0 = ∫ B̂1B gLdx− fLdy. (1.27) With the same argument, we obtain ∂δ4 ∂ε |ε=0 = ∫ B̂B1 gRdx− fRdy, (1.28) On the other hand, since Bε = (xC , bε(h)) and B1ε = (xC , b1ε(h)), we obtain, from δ2 = HL(Bε(h))−HL(B1ε(h)), ∂δ2 ∂ε |ε=0 = HL y (B(h)) ∂bε ∂ε |ε=0 −HL y (B1(h)) ∂b1ε ∂ε |ε=0. (1.29) Moreover, since B = (xC , b(h)) and δ4 = HR(B1ε(h))−HR(B(h)), we obtain ∂δ4 ∂ε |ε=0 = HR y (B1(h)) ∂b1ε ∂ε |ε=0. (1.30) From (1.28) and (1.30), we have that ∂b1ε ∂ε |ε=0 = 1 HR y (B1(h)) ∫ B̂B1 gRdx− fRdy. (1.31) 1.3 The Melnikov function method 36 From (1.27), (1.29) and (1.31), we have that ∂bε ∂ε |ε=0 = 1 HL y (B(h)) ∫ B̂1B gLdx− fLdy + HL y (B1(h)) HL y (B(h))HR y (B1(h)) ∫ B̂B1 gRdx− fRdy. Moreover, we have that ∂δ1 ∂ε |ε=0 = ( HR y (B(h))−HL y (B(h)) )∂bε ∂ε |ε=0, (1.32) ∂δ3 ∂ε |ε=0 = ( HL y (B1(h))−HR y (B1(h)) )∂b1ε ∂ε |ε=0. (1.33) Therefore, from (1.6) and (1.26), we have that M1(h) = ∂ ∂ε ( HR(Bε(h))−HR(B(h)) ) = ∂ ∂ε ( δ1 + δ2 + δ3 + δ4 ) |ε=0. (1.34) Replacing (1.27), (1.28), (1.32), (1.33) into (1.34), we obtain the expression (1.8). The rest of the statement from theorem is a straightforward consequence of Implicit Function Theorem. Chapter 2 Limit cycles of planar piecewise linear Hamiltonian differential system with two or three zones In [40], Llibre and Teixeira studied the existence and the number of crossing limit cycles for continuous and discontinuous planar piecewise linear differential systems with two or three zones such that the linear systems involved have a unique singular point which are centers. More precisely, they prove that if the planar piecewise linear differential system is continuous or discontinuous with two zones or continuous with three zones, then it has no limit cycles. Now, if the planar piecewise linear differential system is discontinuous with three zones, then it has at most one limit cycle, and they provided an example with one limit cycle. Motivated by these results, the main purpose of this chapter is to show that if the singular points of the subsystems that define the discontinuous planar piecewise linear differential system are saddles or centers, then the number of crossing limit cycles is also at most one. Moreover, without taking into account the position of the singular points in the zones, we present examples in Section 2.3 of discontinuous planar piecewise linear Hamiltonian differential system with tree zones with this unique limit cycle for all possible combinations of saddles and centers. 2.1 Preliminaries and main results As we are interested in studying the existence and the number of limit cycles of piecewise linear Hamiltonian differential systems with two or three zones in the plane, we assume the following 37 2.1 Preliminaries and main results 38 hypotheses: (H1) The switching curves are straight lines, and parallel if there are more than one. (H2) The vector fields which define the piecewise one are linear. (H3) The vector fields which define the piecewise one are Hamiltonian. (H4) The vector fields which define the piecewise one have isolated singular points, i.e., centers or saddles. Moreover, we can classify the systems that satisfy the above hypotheses according to the configuration of their singular points. Thus, denoting the centers by the capital letter C and by S the saddles, in the case of two zones we have systems of the type CC, SC and SS. Thus, CC indicates that the singular points of the linear systems that define the piecewise linear differential system are centers and so on. Following this idea, for three zones, we have the following six classes of piecewise linear Hamiltonian differential systems: CCC, SCC, SCS, CSC, SSS and SSC. We already know that the case with two zones has been studied in the literature, i.e., the next theorem is already proved. Theorem 2. A continuous or discontinuous planar piecewise linear Hamiltonian differential system with two zones separated by a straight line and such that the linear systems that define it have isolated singular points, i.e., centers or saddles, has no limit cycles. A proof of Theorem 2 is contained in the proofs of Theorem 2 and 4 from [35]. Alternative proofs can also be found in other papers. See the proof of Theorem 1 from [41] for the cases CC and CS, and see the proof of Theorem 3.4 from [28] for the case SS. We include in Section 2.2 a proof of Theorem 2 just for the sake of completeness. Assuming hypotheses (H1)–(H4), the main results of this chapter are the following. Theorem 3. A continuous planar piecewise linear Hamiltonian differential system with three zones separated by two parallel straight lines and such that the linear systems that define it have isolated singular points, i.e., centers or saddles, has no limit cycles. Theorem 4. A discontinuous planar piecewise linear Hamiltonian differential system with three zones separated by two parallel straight lines and such that the linear systems that define it have isolated singular points, i.e., centers or saddles, has at most one limit cycle. 2.1 Preliminaries and main results 39 Theorems 3 and 4 have been proved for the particular case CCC, see [40]. Theorem 3 has also been proved for the particular case SCS, see the proof of Lemma 11 from [51]. For the other possibilities, as far as we know, the results of Theorems 3 and 4 are new. In order to prove Theorems 3 and 4, we will make some considerations about the hypotheses (H1), (H2) and (H3). Let hi : R2 → R, i = C,L,R, be the functions hC(x, y) = x, hL(x, y) = x+1 and hR(x, y) = x−1. By the hypothesis (H1), we can assume, by means of rotations, translations and homotheties, that the switching line ΣC of a piecewise linear system with two zones in the plane is defined as ΣC = h−1C (0) = {(x, y) ∈ R2 : x = 0}. These straight line decomposes the plane into two regions RL = {(x, y) ∈ R2 : x < 0} and RR = {(x, y) ∈ R2 : x > 0}. Assuming the hypotheses (H2) and (H3), the piecewise linear Hamiltonian vector field with two zones is given by XL(x, y) = (aLx+ bLy + αL, cLx− aLy + βL), x ≤ 0, XR(x, y) = (aRx+ bRy + αR, cRx− aRy + βR), x ≥ 0. (2.1) Note that the Hamiltonian functions that determine the vector field (2.1) are HL(x, y) = bL 2 y2 − cL 2 x2 + aLxy + αLy − βLx, x ≤ 0, HR(x, y) = bR 2 y2 − cR 2 x2 + aRxy + αRy − βRx, x ≥ 0. (2.2) For the case with three zones, the switching lines ΣL and ΣR are given by ΣL = h−1L (0) = {(x, y) ∈ R2 : x = −1}, and ΣR = h−1R (0) = {(x, y) ∈ R2 : x = 1}. This straight lines decomposes the plane into three regions RL = {(x, y) ∈ R2 : x < −1}, RC = {(x, y) ∈ R2 : −1 < x < 1}, 2.1 Preliminaries and main results 40 and RR = {(x, y) ∈ R2 : x > 1}. Assuming the hypotheses (H2) and (H3), the piecewise linear Hamiltonian vector field with three zones is given by XL(x, y) = (aLx+ bLy + αL, cLx− aLy + βL), x ≤ −1, XC(x, y) = (aCx+ bCy + αC , cCx− aCy + βC), −1 ≤ x ≤ 1, XR(x, y) = (aRx+ bRy + αR, cRx− aRy + βR), x ≥ 1. (2.3) The Hamiltonian functions that determine the vector field (2.3) are HL(x, y) = bL 2 y2 − cL 2 x2 + aLxy + αLy − βLx, x ≤ −1, HC(x, y) = bC 2 y2 − cC 2 x2 + aCxy + αCy − βCx, −1 ≤ x ≤ 1, HR(x, y) = bR 2 y2 − cR 2 x2 + aRxy + αRy − βRx, x ≥ 1. (2.4) 2.2 Proof of Theorems 2-4 This section is devoted to present the proofs of the main results. Proof of Theorem 2. Consider a discontinuous piecewise linear Hamiltonian vector field with two zones separated by a straight line, such that the linear vector fields, that define it, have isolated singular points. That is, we have piecewise linear vector field (2.1), with a2i + bici 6= 0, for i = L,R. If the piecewise linear vector field has a periodic orbit, then it intersect the straight line x = 0 at two points, (0, y0) and (0, y1), with y1 < y0, satisfying HR(0, y1) = HR(0, y0), HL(0, y0) = HL(0, y1), where HL and HR are given by (2.2). More precisely, we have the equations −1 2 (y0 − y1)(bR(y0 + y1) + 2αR) = 0, 1 2 (y0 − y1)(bL(y0 + y1) + 2αL) = 0. As y1 < y0, if bR = 0 and αR 6= 0 or bL = 0 and αL 6= 0 the above system has no solutions. If bR = αR = 0 and bL 6= 0 the solution (y0, y1) of the above system with y1 < y0 satisfies y0 = −(bLy1 + 2αL)/bL, with arbitrary y1. If bL = αL = 0 and bR 6= 0 the solution (y0, y1) of the 2.2 Proof of Theorems 2-4 41 above system with y1 < y0 satisfies y0 = −(bRy1 + 2αR)/bR, with arbitrary y1. If bLbR 6= 0, then the above system has a solution (y0, y1) with y1 < y0 only when bL = bR = b and αL = αR = α. Moreover, y0 = (−by1 + 2α)/b with arbitrary y1. If bR = bL = αR = αL = 0, then the system has infinitely many solutions. Therefore, the piecewise linear vector field (2.1) has no periodic orbits or has a continuum of periodic orbits, and consequently, it has no limit cycle. Note that the continuous case is a constraint of the discontinuous one. In fact, the continuous condition is given by XR(0, y) = XL(0, y), ∀ y ∈ R, which implies aR = aL = a, bR = bL = b, αR = αL = α and βR = βL = β. Proof of Theorem 3. Consider a continuous piecewise linear Hamiltonian vector field with three zones separated by two parallel straight lines, such that the linear vector fields that define it, have isolated singular points. That is, we have piecewise linear vector field (2.3), with a2i + bici 6= 0, for i = L,C,R, and due to continuity XR(1, y) = XC(1, y) and XC(−1, y) = XL(−1, y), ∀ y ∈ R. These equalities imply that aR = aC = aL = a, bR = bC = bL = b, αR = αC = αL = α and βR − βC − cC + cR = βL − βC − cL + cC = 0. By Theorem 2, the piecewise linear vector field has no limit cycles contained in two zones. Thus, if the piecewise linear vector field has a periodic orbit, then it intersect the straight lines x = ±1 at four points, (1, y0), (1, y1), with y1 < y0, and (−1, y2), (−1, y3), with y2 < y3, respectively, satisfying HR(1, y1) = HR(1, y0), HC(1, y0) = HC(−1, y3), HL(−1, y3) = HL(−1, y2), HC(−1, y2) = HC(1, y1), (2.5) 2.2 Proof of Theorems 2-4 42 where HL, HC and HR are given by (2.4). More precisely, we have the equations −1 2 (y0 − y1)(b(y0 + y1) + 2(a+ α)) = 0, (2.6) a(y0 + y3) + 1 2 (y0 − y3)(b(y0 + y3) + 2α)− 2βC = 0, (2.7) −1 2 (y2 − y3)(b(y2 + y3)− 2(a− α)) = 0, (2.8) −a(y1 + y2)− 1 2 (y1 − y2)(b(y1 + y2) + 2α) + 2βC = 0. (2.9) As y1 < y0, y2 < y3 and a2 + bci 6= 0, for i = L,C,R, if either b = 0 and a+ α 6= 0 or b = 0 and a + α = 0 the above system has no solutions. If b 6= 0, as y1 < y0 and y2 < y3, from equation (2.6), we can obtain y0 as a function of y1, i.e., y0 = −by1 − 2(a+ α) b . (2.10) Now, from equation (2.8), we can obtain y2 as a function of y3, i.e., y2 = −by3 − 2(α− a) b . (2.11) Substituting (2.10) and (2.11) into equations (2.7) and (2.9), respectively, we obtain a solution (y0, y1, y2, y3) of system (2.5) satisfying y1 < y0 and y2 < y3, given by (ϕ1(y1), y1, ϕ2(y1), ϕ3(y1)), where ϕ1(y1) = −by1 − 2(a+ α) b , ϕ2(y1) = a− α+ √ b2y21 + 2b(a+ α)y1 + (a− α)2 − 4bβC b , ϕ3(y1) = a− α− √ b2y21 + 2b(a+ α)y1 + (a− α)2 − 4bβC b , with arbitrary y1. Note that the inequality b2y21 + 2b(a + α)y1 + (a − α)2 − 4bβC ≤ 0 for all y1 ∈ R is not possible. Therefore, piecewise linear vector field (2.3) has no periodic orbits or has a continuum of periodic orbits, and consequently, it has no limit cycle. Proof of Theorem 4. Consider a discontinuous piecewise linear Hamiltonian vector field with three zones separated by two parallel straight lines, such that the linear vector fields that define it, have isolated singular points. That is, we have piecewise linear vector field (2.3), with −a2i − bici 6= 0, for i = L,C,R. By Theorem 2, the piecewise linear vector field has no limit cycles contained in two zones. Thus, if the piecewise linear vector field has a periodic orbit, then it intersect the straight lines x = ±1 at four points, (1, y0), (1, y1), with y1 < y0, and (−1, y2), (−1, y3), with y2 < y3, respectively, satisfying 2.2 Proof of Theorems 2-4 43 HR(1, y1) = HR(1, y0), HC(1, y0) = HC(−1, y3), HL(−1, y3) = HL(−1, y2), HC(−1, y2) = HC(1, y1), (2.12) where HL, HC and HR are given by (2.4). More precisely, we have the equations 1 2 (y1 − y0)(bR(y0 + y1) + 2(aR + αR)) = 0, (2.13) 1 2 (y0 − y3)(bC(y0 + y3) + 2αC)− 2βC + aC(y0 + y3) = 0, (2.14) 1 2 (y3 − y2)(bL(y2 + y3)− 2(aL − αL)) = 0, (2.15) 1 2 (y2 − y1)(bC(y1 + y2) + 2αC) + 2βC − aC(y1 + y2) = 0. (2.16) To determine all the solutions of the above system, restricted to the conditions y1 < y0, y2 < y3 and a2i + bici 6= 0, for i = L,C,R, we distinguish two cases. In the first case we assume that bRbLbC = 0. For this cases, system (2.13)–(2.16) has no solutions when � bR = 0 and aR + αR 6= 0; � bL = 0 and aL − αL 6= 0; � bR = aR + αR = bL = aL − αL = bC = αC − aC = 0; � bR = aR + αR = bC = αC − aC = 0 and bL 6= 0; � bL = aL − αL = bC = αC − aC = 0 and bR 6= 0; � bC = 0, bRbL 6= 0 and bRαC(aL−αL)+aCbR(αL−aL)+bL(aR+αR)(aC +αC)+2bLbRβC 6= 0; and it has infinitely many solutions when � bR = aR + αR = bL = aL − αL = bC = 0 and αC − aC 6= 0; � bR = aR + αR = bL = aL − αL = 0 and bC 6= 0; � bR = aR + αR = bC = 0, αC − aC 6= 0 and bL 6= 0; � bR = aR + αR = 0 and bLbC 6= 0; � bL = aL − αL = bC = 0, bR 6= 0 and αC − aC 6= 0; � bL = aL − αL = 0 and bRbC 6= 0; 2.2 Proof of Theorems 2-4 44 � bC = 0, bRbL 6= 0 and bRαC(aL−αL)+aCbR(αL−aL)+bL(aR+αR)(aC +αC)+2bLbRβC = 0. In the second case, we assume that bLbCbR 6= 0. From equation (2.13), we can obtain y0 as a function of y1, i.e., y0 = −bRy1 − 2(aR + αR) bR . (2.17) Now, from equation (2.15), we can obtain y2 as a function of y3, i.e., y2 = −bLy3 − 2(αL − aL) bL . (2.18) Substituting (2.17) and (2.18) into equations (2.14) and (2.16), respectively, we obtain the equations of two hyperbolas in the plane y1y3, given by (y1 −A)2 K − (y3 −B)2 K − C = 0, (y1 −D)2 K − (y3 − E)2 K − C = 0, (2.19) with K = 2 bC , A = bR(aC + αC)− 2bC(aR + αR) bCbR , B = aC − αC bC , C = 2(aCαC + bCβC) bC , D = −(aC + αC) bC and E = bL(αC − aC)− 2bC(αL − aL) bCbL . Note that system (2.19) is equivalent to the system y21 − 2Ay1 +A2 − y23 + 2By3 −B2 −KC = 0, 2(A−D)y1 + 2(E −B)y3 +D2 − E2 +B2 −A2 = 0. (2.20) The above system could have infinitely many solutions (y1, y3), for instance when A = D and B = E. In this case, the piecewise linear vector field (2.3) has a continuum of periodic orbits, and consequently, it has no limit cycle. Suppose that system (2.20) has finitely many solutions. According to Bezout’s Theorem (see [52, p. 144]), if a system of polynomial equations has finitely many solutions, then the number of its solutions is at most the product of the degrees of the polynomials, that for system (2.20) is two. Therefore, the two above hyperbolas intersect at most two points. Note that, by (2.13)–(2.16), if (y0, y1, y2, y3) is solution of system (2.12) then (y1, y0, y3, y2) is also a solution. However, for y1 < y0 and y2 < y3 we have at most a single solution. Therefore, the piecewise linear vector field (2.3) can have at most one limit cycle. 2.3 Examples 45 2.3 Examples In the proof of Theorem 4 it was not necessary to consider the position of the singular points from subsystems that define the piecewise one, i.e., if the singular points are real, virtual or boundary. Therefore, a natural question is if there are examples of discontinuous planar piecewise linear Hamiltonian differential systems with three zones separated by two parallel straight lines with at most one limit cycle, for each position of the singular points. In this section, we answer this question. More precisely, we give examples of piecewise linear Hamiltonian systems of the type CCC, SCC, SCS, CSC, SSS and SSC with exactly one limit cycle, for different positions of the singular points. Denoting a real singular point by the letter R, a boundary singular point by the letter B and a virtual singular point by the letter V, we say that a system is of the type CRSBCV if the singular point of the right subsystem is a virtual center, of the left subsystem is a real center and of the central subsystem is a boundary saddle. Example 1. (Case CVCRCV) Consider the discontinuous planar piecewise linear Hamiltonian vector field (2.3) with aL = 4, bL = 8, αL = 3/2, cL = −5/2, βL = 11/4, aC = 0, bC = 2, αC = βC = 2/3, cC = −2, aR = 4, bR = 2, cR = −10 and αR = βR = −4. We can see that the central vector field from (2.3) has a center at the point (1/3,−1/3), the right vector field has a center at the point (−6, 14) and the left vector field has a center at the point (7,−59/16). Therefore, a candidate to limit cycle of vector field (2.3), in this case, corresponds to the solution of system (2.12), i.e., (y1 − y0)(y1 + y0) = 0, 1 3 (y0(2 + 3y0)− y3(2 + 3y3)− 4) = 0, 1 2 (y3 − y2)(8(y2 + y3)− 5) = 0, 1 3 (4− y1(2 + 3y1) + y2(2 + 3y2)) = 0. After some computation, the unique solution (y0, y1, y2, y3) of the above system, satisfying the conditions y1 < y0 and y2 < y3, is given by( 31 48 √ 1259 235 ,−31 48 √ 1259 235 , 5 16 − 1 3 √ 1259 235 , 5 16 + 1 3 √ 1259 235 ) . The points (−1, y2), (−1, y3) ∈ ΣL and (1, y0), (1, y1) ∈ ΣR are crossing points because 2.3 Examples 46 〈XL(−1, y2), (1, 0)〉 · 〈XC(−1, y2), (1, 0)〉 ≈ 1.5518 > 0, 〈XL(−1, y3), (1, 0)〉 · 〈XC(−1, y3), (1, 0)〉 ≈ 17.4969 > 0, 〈XC(1, y0), (1, 0)〉 · 〈XR(1, y0), (1, 0)〉 ≈ 10.9315 > 0, 〈XC(1, y1), (1, 0)〉 · 〈XR(1, y1), (1, 0)〉 ≈ 6.9452 > 0. The orbit (xR(t), yR(t)) of XR, such that (xR(0), yR(0)) = (1, y0), is given by xR(t) = 7 cos(2t) + 31 48 √ 1259 235 sin(2t)− 6, yR(t) = ( 31 48 √ 1259 235 − 14 ) cos(2t)− ( 7 + 31 24 √ 1259 235 ) sin(2t) + 14. The orbit (xC1 (t), yC1 (t)) of XC, such that (xC1 (0), yC1 (0)) = (1, y1), is given by xC1 (t) = 2 3 cos(2t) + ( 1 3 − 31 48 √ 1259 235 ) sin(2t) + 1 3 , yC1 (t) = ( 1 3 − 31 48 √ 1259 235 ) cos(2t)− 1 3 (1 + 4 cos(t) sin(t)). The orbit (xL(t), yL(t)) of XL, such that (xL(0), yL(0)) = (−1, y2), is given by xL(t) = −8 cos(2t)− 4 3 √ 1259 235 sin(2t) + 7, yL(t) = ( 4− 1 3 √ 1259 235 ) cos(2t) + ( 4 + 4 3 √ 1259 235 ) cos(t) sin(t)− 59 16 . The orbit (xC2 (t), yC2 (t)) of XC, such that (xC2 (0), yC2 (0)) = (−1, y3), is given by xC2 (t) = −4 3 cos(2t) + ( 31 48 + 1 3 √ 1259 235 ) sin(2t) + 1 3 , yC2 (t) = ( 31 48 + 1 3 √ 1259 235 ) cos(2t) + 1 3 (8 cos(t) sin(t)− 1). The flight time of the orbit (xR(t), yR(t)), from (1, y0) ∈ ΣR to (1, y1) ∈ ΣR, is tR = 1 2 arctan ( 20832 √ 295865 25320661 ) . The flight time of the orbit (xC1 (t), yC1 (t)), from (1, y1) ∈ ΣR to (−1, y2) ∈ ΣL, is tC1 = π 2 − 1 2 arctan ( 96(10810 + √ 295865) 496001 ) . 2.3 Examples 47 The flight time of the orbit (xL(t), yL(t)), from (−1, y2) ∈ ΣL to (−1, y3) ∈ ΣL, is tL = 1 2 arctan ( 12 √ 295865 7201 ) . Finally, the flight time of the orbit (xC2 (t), yC2 (t)), from (−1, y3) ∈ ΣL to (1, y0) ∈ ΣR, is tC2 = −1 2 arctan ( 96( √ 295865− 10810) 496001 ) . The point (−1, 5/16) ∈ ΣL is an invisible fold point of XL and the point (−1,−1/3) ∈ ΣL is a visible fold point of XC. They are endpoints of the escaping set in ΣL. The point (1, 0) ∈ ΣR is an invisible fold point of XR and the point (1,−1/3) ∈ ΣR is a visible fold point of XC. They are endpoints of the sliding set in ΣR. Using the Mathematica software (see [57]), we can draw the orbits (xi(t), yi(t)) for the time t ∈ [0, ti], i = R,L,C1, C2, i.e., we obtain the limit cycle illustrated in Fig. 2.1 (a). Fig. 2.1 (b) has been made with the help of P5 software (see [24]), and provides the phase portrait of vector field (2.3) in this case (the symbol ◦ indicates a virtual singular point). - 1.5 - 0.5 0.5 - 1.5 - 0.5 0.5 1.5 1 1- ΣRΣL (1, y0) (1, y1) (−1, y2) (−1, y3) ΣRΣL (a) (b) Figure 2.1: The limit cycle of vector field (2.3) with aL = 4, bL = 8, αL = 3/2, cL = −5/2, βL = 11/4, aC = 0, bC = 2, αC = βC = 2/3, cC = −2, aR = 4, bR = 2, cR = −10 and αR = βR = −4. In order to simplify the text, in Table 2.1 we give the parameter values for that the discontinuous planar piecewise linear vector field (2.3) has at most one limit cycle, passing through the points (1, y0), (1, y1), (−1, y2) and (−1, y3), for the other possible configurations of the type CCC. Example 2. (Case SRCRCV) Consider the discontinuous planar piecewise linear Hamiltonian vector field (2.3) with aL = bL = 1, αL = 2/3, cL = 35, βL = 214/3, aC = 0, bC = 2, 2.3 Examples 48 CVCRCR CRCRCR CVCVCV CVCVCR CRCVCR aC 0 0 0 0 0 bC 2 2 2 2 2 cC -2 -2 -2 -2 -2 αC 1 1 1 1 1 βC 1 1 3 3 3 aR 8 8 8 8 8 bR 10 10 10 10 10 cR -8 -8 -8 -8 -8 αR -8 -8 -8 -8 -8 βR 10 10 -8 10 10 aL 16 16 16 16 16 bL 65 2 65 2 65 2 65 2 65 2 cL -8 -8 -8 -8 -8 αL 8 8 8 8 8 βL 0 10 -8 0 -92 y0 97 2340 √ 4873 2 97 2340 √ 4873 2 97 780 √ 4441 6 97 780 √ 4441 6 97 780 √ 4441 6 y1 − 97 2340 √ 4873 2 − 97 2340 √ 4873 2 − 97 780 √ 4441 6 − 97 780 √ 4441 6 − 97 780 √ 4441 6 y2 16 45 − 1 36 √ 4873 2 16 45 − 1 36 √ 4873 2 16 65 − 1 12 √ 4441 6 16 65 − 1 12 √ 4441 6 16 65 − 1 12 √ 4441 6 y3 16 45 + 1 36 √ 4873 2 16 45 + 1 36 √ 4873 2 16 65 + 1 12 √ 4441 6 16 65 + 1 12 √ 4441 6 16 65 + 1 12 √ 4441 6 Table 2.1: Parameter values for that vector field (2.3) of the type CCC has one limit cycle. αC = βC = 2/3, cC = −2, aR = 4, bR = 2, αR = βR = −4 and cR = −10. We can see that the central vector field from (2.3) has a center at the point (1/3,−1/3), the right vector field has a center at the point (−6, 14) and the left vector field has a saddle at the point (−2, 4/3). Therefore, a candidate to limit cycle of system (2.3), in this case, corresponds to the solution of system (2.12), i.e., (y1 − y0)(y1 + y0) = 0, 1 3 (y0(2 + 3y0)− y3(2 + 3y3)− 4) = 0, 1 6 (y3 − y2)(3(y2 + y3)− 2) = 0, 1 3 (4− y1(2 + 3y1) + y2(2 + 3y2)) = 0. The only solution (y0, y1, y2, y3) of the above system, satisfying the conditions y1 < y0 and y2 < y3, is given by ( 2 √ 5 3 ,−2 √ 5 3 , 1− √ 5 3 , 1 + √ 5 3 ) . 2.3 Examples 49 The points (−1, y2), (−1, y3) ∈ ΣL and (1, y0), (1, y1) ∈ ΣR are crossing points because 〈XL(−1, y2), (1, 0)〉 · 〈XC(−1, y2), (1, 0)〉 ≈ 0.1173 > 0, 〈XL(−1, y3), (1, 0)〉 · 〈XC(−1, y3), (1, 0)〉 ≈ 2.1049 > 0, 〈XC(1, y0), (1, 0)〉 · 〈XR(1, y0), (1, 0)〉 ≈ 10.8765 > 0, 〈XC(1, y1), (1, 0)〉 · 〈XR(1, y1), (1, 0)〉 ≈ 6.9013 > 0. The orbit (xR(t), yR(t)) of XR, such that (xR(0), yR(0)) = (1, y0), is given by xR(t) = 7 cos(2t) + 2 √ 5 3 sin(2t)− 6, yR(t) = 2 3 ( √ 5− 21) cos(2t)− ( 7 + 4 √ 5 3 ) sin(2t) + 14. The orbit (xC1 (t), yC1 (t)) of XC, such that (xC1 (0), yC1 (0)) = (1, y1), is given by xC1 (t) = 1 3 ( 2 cos(2t) + ( 1− 2 √ 5 ) sin(2t) + 1 ) , yC1 (t) = 1 3 (( 1− 2 √ 5 ) cos(2t)− 2 sin(2t)− 1 ) . The orbit (xL(t), yL(t)) of XL, such that (xL(0), yL(0)) = (−1, y2), is given by xL(t) = 1 36 ( 18 + √ 5 ) e−6t + 1 36 ( 18− √ 5 ) e6t − 2, yL(t) = − 1 36 ( 126 + 7 √ 5 ) e−6t + 1 36 ( 90− 5 √ 5 ) e6t + 4 3 . The orbit (xC2 (t), yC2 (t)) of XC, such that (xC2 (0), yC2 (0)) = (−1, y3), is given by xC2 (t) = 1 3 ( 1− 4 cos(2t) + ( 2 + √ 5 ) sin(2t) ) , yC2 (t) = 1 3 ( (2 + √ 5) cos(2t) + 4 sin(2t)− 1 ) . The flight time of the orbit (xR(t), yR(t)), from (1, y0) ∈ ΣR to (1, y1) ∈ ΣR, is tR = 1 2 arctan ( 84 √ 5 421 ) . The flight time of the orbit (xC1 (t), yC1 (t)), from (1, y1) ∈ ΣR to (−1, y2) ∈ ΣL, is tC1 = π 2 − 1 2 arctan(2). 2.3 Examples 50 The flight time of the orbit (xL(t), yL(t)), from (−1, y2) ∈ ΣL to (−1, y3) ∈ ΣL, is tL = 1 6 log ( 329 + 36 √ 5 319 ) . Finally, the flight time of the orbit (xC2 (t), yC2 (t)), from (−1, y3) ∈ ΣL to (1, y0) ∈ ΣR, is tC2 = 1 2 arctan(2). The point (−1, 1/3) ∈ ΣL is an invisible fold point of XL and the point (−1,−1/3) ∈ ΣL is a visible fold point of XC. They are endpoints of the escaping set in ΣL. The point (1, 0) ∈ ΣR is an invisible fold point of XR and the point (1,−1/3) ∈ ΣR is a visible fold point of XC. They are endpoints of the sliding set in ΣR. Hence, using the Mathematica software, we can draw the orbits (xi(t), yi(t)) for the time t ∈ [0, ti], i = R,L,C1, C2, i.e., we obtain the limit cycle given in Fig. 2.2 (a). Fig. 2.2 (b) has been made with the help of P5 software, and provides the phase portrait of vector field (2.3) in this case. - -1.5 - -0.5 0.5 1 1.5 1 0.5 0.5 ΣRΣL (1, y0) (1, y1) (−1, y2) (−1, y3) ΣRΣL (a) (b) Figure 2.2: The limit cycle of vector field (2.3) with aL = bL = 1, αL = 2/3, cL = 35, βL = 214/3, aC = 0, bC = 2, αC = βC = 2/3, cC = −2, aR = 4, bR = 2, αR = βR = −4 and cR = −10. Example 3. (Case SRCRSR) Consider the discontinuous planar piecewise linear Hamiltonian vector field (2.3) with aL = bL = 1, αL = 3/5, cL = 35, βL = 357/5, aC = 0, bC = 2, αC = βC = 1, cC = −2, aR = bR = 1, αR = −1, cR = 15 and βR = −31. We can see that the central vector field from (2.3) has a center at the point (1/2,−1/2), the right vector field has a saddle at the point (2,−1) and the left vector field has a saddle at the point (−2, 7/5). Therefore, a candidate to limit cycle of system (2.3), in this case, corresponds to the solution of 2.3 Examples 51 system (2.12), i.e., 1 2 (y1 + y0)(y1 − y0) = 0, y20 + y0 − y3(1 + y3)− 2 = 0, 1 10 (y3 − y2)(5(y2 + y3)− 4) = 0, y22 + y2 − y1(1 + y1) + 2 = 0. The only solution (y0, y1, y2, y3) of the above system, satisfying the condition y1 < y0 and y2 < y3, is given by ( 18 5 √ 2 7 ,−18 5 √ 2 7 , 2 5 − 2 √ 2 7 , 2 5 + 2 √ 2 7 ) . The points (−1, y2), (−1, y3) ∈ ΣL and (1, y0), (1, y1) ∈ ΣR are crossing points because 〈XL(−1, y2), (1, 0)〉 · 〈XC(−1, y2), (1, 0)〉 ≈ 0.3614 > 0, 〈XL(−1, y3), (1, 0)〉 · 〈XC(−1, y3), (1, 0)〉 ≈ 4.21 > 0, 〈XC(1, y0), (1, 0)〉 · 〈XR(1, y0), (1, 0)〉 ≈ 9.33 > 0, 〈XC(1, y1), (1, 0)〉 · 〈XR(1, y1), (1, 0)〉 ≈ 5.4814 > 0. The orbit (xR(t), yR(t)) of XR, such that (xR(0), yR(0)) = (1, y0), is given by xR(t) = − ( 70 + 9 √ 14 ) 140 e−4t + ( 9 √ 14− 70 ) 140 e4t + 2, yR(t) = ( 350 + 45 √ 14 ) 140 e−4t + ( 27 √ 14− 210 ) 140 e4t − 1. The orbit (xC1 (t), yC1 (t)) of XC, such that (xC1 (0), yC1 (0)) = (1, y1), is given by xC1 (t) = cos(t) ( cos(t) + sin(t)− 36 5 √ 2 7 sin(t) ) , yC1 (t) = ( 35− 36 √ 14 ) 70 cos(2t)− cos(t) sin(t)− 1 2 . The orbit (xL(t), yL(t)) of XL, such that (xL(0), yL(0)) = (−1, y2), is given by xL(t) = ( 21 + √ 14 ) 42 e−6t − (√ 14− 21 ) 42 e6t − 2, yL(t) = − ( 735 + 35 √ 14 ) 210 e−6t − ( 25 √ 14− 525 ) 210 e6t + 7 5 . 2.3 Examples 52 The orbit (xC2 (t), yC2 (t)) of XC, such that (xC2 (0), yC2 (0)) = (−1, y3), is given by xC2 (t) = −3 2 cos(2t) + 9 5 cos(t) sin(t) + 2 √ 2 7 sin(2t) + 1 2 , yC2 (t) = ( 9 10 + 2 √ 2 7 ) cos(2t) + 3 cos(t) sin(t)− 1 2 . The flight time of the orbit (xR(t), yR(t)), from (1, y0) ∈ ΣR to (1, y1) ∈ ΣR, is tR = 1 4 log ( 431 + 90 √ 14 269 ) . The flight time of the orbit (xC1 (t), yC1 (t)), from (1, y1) ∈ ΣR to (−1, y2) ∈ ΣL, is tC1 = arctan ( 2(7 + 4 √ 14) 35 ) . The flight time of the orbit (xL(t), yL(t)), from (−1, y2) ∈ ΣL to (−1, y3) ∈ ΣL, is tL = 1 6 log ( 65 + 6 √ 14 61 ) . Finally, the flight time of the orbit (xC2 (t), yC2 (t)), from (−1, y3) ∈ ΣL to (1, y0) ∈ ΣR, is tC2 = arctan ( 2(−7 + 4 √ 14) 35 ) . The point (−1, 2/5) ∈ ΣL is an invisible fold point of XL and the point (−1,−1/2) ∈ ΣL is a visible fold point of XC. They are endpoints of the escaping set in ΣL. The point (1, 0) ∈ ΣR is an invisible fold point of XR and the point (1,−1/2) ∈ ΣR is a visible fold point of XC. They are endpoints of the sliding set in ΣR. Hence, using the Mathematica software, we can draw the orbits (xi(t), yi(t)) for the time t ∈ [0, ti], i = R,L,C1, C2, i.e., we obtain the limit cycle illustrated in Fig. 2.3 (a). Fig. 2.3 (b) has been made with the help of P5 software, and provides the phase portrait of vector field (2.3) in this case. In Table 2.2 we give the parameter values for that the discontinuous planar piecewise linear vector field (2.3) has at most one limit cycle, passing through the points (1, y0), (1, y1), (−1, y2) and (−1, y3), for the others possibles configurations of the type SCC and SCS. Example 4. (Case CVSRCV) Consider the discontinuous planar piecewise linear Hamiltonian vector field (2.3) with aL = 4, bL = 8, αL = 2, cL = −5/2, βL = 5/2, aC = 2/5, bC = 24/5, 2.3 Examples 53 - 0.5 0.5 - 2 - 1 1 2 ΣRΣL (1, y0) (1, y1) (−1, y2) (−1, y3) ΣRΣL (a) (b) Figure 2.3: The limit cycle of vector field (2.3) with aL = bL = 1, αL = 3/5, cL = 35, βL = 357/5, aC = 0, bC = 2, αC = βC = 1, cC = −2, aR = bR = 1, αR = −1, cR = 15 and βR = −31. SRCRCR SRCVCV SRCVCR SRCVSR aC 0 0 0 0 bC 2 2 2 2 cC -2 -2 -2 -2 αC 1 1 1 1 βC 1 3 3 3 aR 8 8 8 1 bR 10 10 10 1 cR -8 -8 -8 15 αR -8 -8 -8 -1 βR 10 -8 10 -31 aL 1 1 1 1 bL 1 1 1 1 cL 35 35 35 35 αL 3 5 3 5 3 5 3 5 βL 357 5 357 5 357 5 357 5 y0 18 5 √ 2 7 9 5 √ 41 14 9 5 √ 41 14 9 5 √ 41 14 y1 −18 5 √ 2 7 −9 5 √ 41 14 −9 5 √ 41 14 −9 5 √ 41 14 y2 2 5 − 2 √ 2 7 2 5 − √ 41 14 2 5 − √ 41 14 2 5 − √ 41 14 y3 2 5 + 2 √ 2 7 2 5 + √ 41 14 2 5 + √ 41 14 2 5 + √ 41 14 Table 2.2: Parameter values for that vector field (2.3) of the type SCC and SCS has one limit cycle. αC = −9/5, cC = 4/5, βC = −4/15, aR = 8, bR = 10 and αR = cR = βR = −8. We can see that the central vector field from (2.3) has a saddle at the point (1/2, 1/3), the right vector field 2.3 Examples 54 has a center at the point (−9, 8) and the left vector field has a center at the point (−7, 15/4). Therefore, a candidate to limit cycle of system (2.3), in this case, corresponds to the solution of system (2.12), i.e., 5(y1 − y0)(y1 + y0) = 0, 1 15 (3y0(12y0 − 7) + 33y3 − 36y23 + 8) = 0, 2(y2 − 2y22 + y3(2y3 − 1)) = 0, 1 15 (3y1(7− 12y1)− 33y2 + 36y22 − 8) = 0. The only solution (y0, y1, y2, y3) of the above system, satisfying the conditions y1 < y0 and y2 < y3, is given by ( 5 12 √ 7 3 ,− 5 12 √ 7 3 , 1 4 − 7 √ 21 36 , 1 4 + 7 √ 21 36 ) . The points (−1, y2), (−1, y3) ∈ ΣL and (1, y0), (1, y1) ∈ ΣR are crossing points because 〈XL(−1, y2), (1, 0)〉 · 〈XC(−1, y2), (1, 0)〉 ≈ 37.617 > 0, 〈XL(−1, y3), (1, 0)〉 · 〈XC(−1, y3), (1, 0)〉 ≈ 23.360 > 0, 〈XC(1, y0), (1, 0)〉 · 〈XR(1, y0), (1, 0)〉 ≈ 10.534 > 0, 〈XC(1, y1), (1, 0)〉 · 〈XR(1, y1), (1, 0)〉 ≈ 28.355 > 0. The orbit (xR(t), yR(t)) of XR, such that (xR(0), yR(0)) = (1, y0), is given by xR(t) = 10 cos(4t) + 25 24 √ 7 3 sin(4t)− 9, yR(t) = ( 5 12 √ 7 3 − 8 ) cos(4t)− ( 5 6 √ 7 3 + 4 ) sin(4t) + 8. The orbit (xC1 (t), yC1 (t)) of XC, such that (xC1 (0), yC1 (0)) = (1, y1), is given by xC1 (t) = ( 18 + 5 √ 21 ) 30 e−2t − ( 3 + 5 √ 21 ) 30 e2t + 1 2 , yC1 (t) = − ( 54 + 15 √ 21 ) 180 e−2t − ( 6 + 10 √ 21 ) 180 e2t + 1 3 . The orbit (xL(t), yL(t)) of XL, such that (xL(0), yL(0)) = (−1, y2), is given by xL(t) = −8 cos(2t)− 7 3 √ 7 3 sin(2t) + 7, yL(t) = ( 4− 7 12 √ 7 3 ) cos(2t) + ( 2 + 7 6 √ 7 3 ) sin(2t)− 15 4 . 2.3 Examples 55 The orbit (xC2 (t), yC2 (t)) of XC, such that (xC2 (0), yC2 (0)) = (−1, y3), is given by xC2 (t) = − ( 15 + 7 √ 21 ) 30 e−2t − ( 30− 7 √ 21 ) 30 e2t + 1 2 , yC2 (t) = ( 45 + 21 √ 21 ) 180 e−2t − ( 60− 14 √ 21 ) 180 e2t + 1 3 . The flight time of the orbit (xR(t), yR(t)), from (1, y0) ∈ ΣR to (1, y1) ∈ ΣR, is tR = 1 4 arctan ( 480 √ 21 6737 ) . The flight time of the orbit (xC1 (t), yC1 (t)), from (1, y1) ∈ ΣR to (−1, y2) ∈ ΣL, is tC1 = 1 2 log ( 5 + √ 21 4 ) . The flight time of the orbit (xL(t), yL(t)), from (−1, y2) ∈ ΣL to (−1, y3) ∈ ΣL, is tL = 1 2 arctan ( 336 √ 21 1385 ) . Finally, the flight time of the orbit (xC2 (t), yC2 (t)), from (−1, y3) ∈ ΣL to (1, y0) ∈ ΣR, is tC2 = 1 2 log(5 + √ 21). The point (−1, 1/4) ∈ ΣL is an invisible fold point of XL and the point (−1, 11/24) ∈ ΣL is a visible fold point of XC. They are endpoints of the sliding set in ΣL. The point (1, 0) ∈ ΣR is an invisible fold point of XR and the point (1, 7/24) ∈ ΣR is a visible fold point of XC. They are endpoints of the escaping set in ΣR. Hence, using the Mathematica software, we can draw the orbits (xi(t), yi(t)) for the time t ∈ [0, ti], i = R,L,C1, C2, i.e., we obtain the limit cycle illustrated in Fig. 2.4 (a). Fig. 2.4 (b) has been made with the help of P5 software, and provides the phase portrait of vector field (2.3) in this case. In Table 2.3 we give the parameter values for that the discontinuous planar piecewise linear vector field (2.3) has at most one limit cycle, passing through the points (1, y0), (1, y1), (−1, y2) and (−1, y3), for the others possibles configurations of the type CSC. Example 5. (Case SRSRSR) Consider the discontinuous planar piecewise linear Hamiltonian vector field (2.3) with aL = αL = −2/3, bL = 4/3, cL = 8/3, βL = 35/3, aC = 2/11, bC = 120/11, αC = −41/11, cC = 4/11, βC = −4/33, aR = −2/11, bR = 4/11, αR = 1/5, cR = 120/11 and 2.3 Examples 56 - 0.5 0.5 1 0.5 0.5- -1.5 ΣRΣL (1, y0) (1, y1)(−1, y2) (−1, y3) ΣRΣL (a) (b) Figure 2.4: The limit cycle of vector field (2.3) with aL = 4, bL = 8, αL = 2, cL = −5/2, βL = 5/2, aC = 2/5, bC = 24/5, αC = −9/5, cC = 4/5, βC = −4/15, aR = 8, bR = 10 and αR = cR = βR = −8. CVSRCR CRSRCR CVSVCV CVSVCR CRSVCR aC 2 5 2 5 2 5 2 5 2 5 bC 24 5 24 5 24 5 24 5 24 5 cC 4 5 4 5 4 5 4 5 4 5 αC −9 5 −9 5 −9 5 −9 5 −9 5 βC − 4 15 − 4 15 -2 -2 -2 aR 8 8 8 8 8 bR 10 10 10 10 10 cR -8 -8 -8 -8 -8 αR -8 -8 -8 -8 -8 βR 9 9 -8 9 9 aL 4 8 4 4 4 bL 8 8 8 8 8 cL −5 2 −5 2 −5 2 −5 2 −5 2 αL 2 2 2 2 2 βL 5 2 −5 2 5 2 5 2 −5 2 y0 5 12 √ 7 3 5 12 √ 7 3 5 12 √ 11 5 12 √ 11 5 12 √ 11 y1 − 5 12 √ 7 3 − 5 12 √ 7 3 − 5 12 √ 11 − 5 12 √ 11 5 12 √ 11 y2 1 4 − 7 36 √ 21 1 4 − 7 36 √ 21 1 4 − 7 12 √ 11 1 4 − 7 12 √ 11 1 4 − 7 12 √ 11 y3 1 4 + 7 36 √ 21 1 4 + 7 36 √ 21 1 4 + 7 12 √ 11 1 4 + 7 12 √ 11 1 4 + 7 12 √ 11 Table 2.3: Parameter values for that vector field (2.3) of the type CSC has one limit cycle. βR = −749/55. We can see that the central vector field from (2.3) has a saddle at the point (1/2, 1/3), the right vector field has a saddle at the point (1509/1210, 89/1210) and the left vector field has a saddle at the point (−4,−3/2). Therefore, a candidate to limit cycle of system (2.3), 2.3 Examples 57 in this case, corresponds to the solution of system (2.12), i.e., 1 55 (10y21 + y1 − y0(1 + 10y0)) = 0, 1 33 (9y0(20y0 − 13) + 3y3(43− 60y3) + 8) = 0, 2 3 (y3 − y2)(y3 + y2) = 0, 1 33 (9y1(13− 20y1) + 3y2(60y2 − 43)− 8) = 0. The only solution (y0, y1, y2, y3) of the above system, satisfying the conditions y1 < y0 and y2 < y3, is given by ( 43 √ 26− 12 240 ,−43 √ 26 + 12 240 ,−3 8 √ 13 2 , 3 8 √ 13 2 ) . The points (−1, y2), (−1, y3) ∈ ΣL and (1, y0), (1, y1) ∈ ΣR are crossing points because 〈XL(−1, y2), (1, 0)〉 · 〈XC(−1, y2), (1, 0)〉 ≈ 18.2786 > 0, 〈XL(−1, y3), (1, 0)〉 · 〈XC(−1, y3), (1, 0)〉 ≈ 8.3123 > 0, 〈XC(1, y0), (1, 0)〉 · 〈XR(1, y0), (1, 0)〉 ≈ 1.9518 > 0, 〈XC(1, y1), (1, 0)〉 · 〈XR(1, y1), (1, 0)〉 ≈ 4.6699 > 0. The orbit (xR(t), yR(t)) of XR, such that (xR(0), yR(0)) = (1, y0), is given by xR(t) = − ( 3588 + 473 √ 26 ) 29040 e−2t + ( 473 √ 26− 3588 ) 29040 e2t + 1509 1210 , yR(t) = ( 17940 + 2365 √ 26 ) 29040 e−2t + ( 2838 √ 26− 21528 ) 29040 e2t + 89 1210 . The orbit (xC1 (t), yC1 (t)) of XC, such that (xC1 (0), yC1 (0)) = (1, y1), is given by xC1 (t) = ( 112 + 43 √ 26 ) 88 e−2t − ( 68 + 43 √ 26 ) 88 e2t + 1 2 , yC1 (t) = − ( 672 + 258 √ 26 ) 2640 e−2t − ( 340 + 215 √ 26 ) 2640 e2t + 1 3 . The orbit (xL(t), yL(t)) of XL, such that (xL(0), yL(0)) = (−1, y2), is given by xL(t) = ( 24 + √ 26 ) 16 e−2t − (√ 26− 24 ) 16 e2t − 4, yL(t) = − ( 24 + √ 26 ) 16 e−2t − (√ 26− 24 ) 8 e2t − 3 2 . 2.3 Examples 58 The orbit (xC2 (t), yC2 (t)) of XC, such that (xC2 (0), yC2 (0)) = (−1, y3), is given by xC2 (t) = (20− 45 √ 26) 88 e−2t + (45 √ 26− 152) 88 e2t + 1 2 , yC2 (t) = (54 √ 26− 24) 528 e−2t + (45 √ 26− 152) 528 e2t + 1 3 . The flight time of the orbit (xR(t), yR(t)), from (1, y0) ∈ ΣR to (1, y1) ∈ ΣR, is tR = 1 2 log ( 718873 + 130548 √ 26 271415 ) . The flight time of the orbit (xC1 (t), yC1 (t)), from (1, y1) ∈ ΣR to (−1, y2) ∈ ΣL, is tC1 = log (√ 23 + 2 √ 26 5 ) . The flight time of the orbit (xL(t), yL(t)), from (−1, y2) ∈ ΣL to (−1, y3) ∈ ΣL, is tL = 1 2 log ( 301 + 24 √ 26 275 ) . Finally, the flight time of the orbit (xC2 (t), yC2 (t)), from (−1, y3) ∈ ΣL to (1, y0) ∈ ΣR, is tC2 = 1 2 log ( 23 + 2 √ 26 17 ) . The point (−1, 0) ∈ ΣL is an invisible fold point of XL and the point (−1, 43/120) ∈ ΣL is a invisible fold point of XC. They are endpoints of the sliding set in ΣL. The point (1,−1/20) ∈ ΣR is an invisible fold point of XR and the point (1, 13/40) ∈ ΣR is a visible fold point of XC. They are endpoints of the escaping set in ΣR. Hence, using the Mathematica software, we can draw the orbits (xi(t), yi(t)) for the time t ∈ [0, ti], i = R,L,C1, C2, i.e., we obtain the limit cycle illustrated in Fig. 2.5 (a). Fig. 2.5 (b) has been made with the help of P5 software, and provides the phase portrait of vector field (2.3) in this case. Example 6. (Case SRSRCV) Consider the discontinuous planar piecewise linear Hamiltonian vector field (2.3) with aL = αL = −2/3, bL = 4/3, cL = 8/3, βL = 35/3, aC = 2/11, bC = 120/11, αC = −41/11, cC = 4/11, βC = −4/33, aR = 8, bR = 10, αR = −7 and cR = βR = −8. We can see that the central vector field from (2.3) has a saddle at the point (1/2, 1/3), the right vector field has a center at the point (−17/2, 15/2) and the left vector field has a saddle at the point (−4,−3/2). Therefore, a candidate to limit cycle of system (2.3), in this case, corresponds to 2.3 Examples 59 - 0.5 - - 1 0.5 0.5 0.5 1 ΣRΣL (1, y0) (1, y1)(−1, y2) (−1, y3) ΣRΣL (a) (b) Figure 2.5: The limit cycle of vector field (2.3) with aL = αL = −2/3, bL = 4/3, cL = 8/3, βL = 35/3, aC = 2/11, bC = 120/11, αC = −41/11, cC = 4/11, βC = −4/33, aR = −2/11, bR = 4/11, αR = 1/5, cR = 120/11 and βR = −749/55. the solution of system (2.12), i.e., 5y21 + y1 − y0(1 + 5y0) = 0, 1 33 (9y0(20y0 − 13) + 3y3(43− 60y3) + 8) = 0, 2 3 (y3 − y2)(y3 + y2) = 0, 1 33 (9y1(13− 20y1) + 3y2(60y2 − 43)− 8) = 0. Making a calculation, the only solution (y0, y1, y2, y3) of the above system, satisfying the conditions y1 < y0 and y2 < y3, is given by( 43 24 √ 43 470 − 1 10 ,−43 24 √ 43 470 − 1 10 ,−17 8 √ 43 470 , 17 8 √ 43 470 ) . The points (−1, y2), (−1, y3) ∈ ΣL and (1, y0), (1, y1) ∈ ΣR are crossing points because 〈XL(−1, y2), (1, 0)〉 · 〈XC(−1, y2), (1, 0)〉 ≈ 9.3593 > 0, 〈XL(−1, y3), (1, 0)〉 · 〈XC(−1, y3), (1, 0)〉 ≈ 2.6591 > 0, 〈XC(1, y0), (1, 0)〉 · 〈XR(1, y0), (1, 0)〉 ≈ 6.9128 > 0, 〈XC(1, y1), (1, 0)〉 · 〈XR(1, y1), (1, 0)〉 ≈ 57.1644 > 0. The orbit (xR(t), yR(t)) of XR, such that (xR(0), yR(0)) = (1, y0), is given by xR(t) = 19 2 cos(4t) + 43 48 √ 215 94 sin(4t)− 17 2 , yR(t) = ( 43 24 √ 43 470 − 38 5 ) cos(4t)− ( 19 5 + 43 12 √ 43 470 ) sin(4t) + 15 2 . 2.3 Examples 60 The orbit (xC1 (t), yC1 (t)) of XC, such that (xC1 (0), yC1 (0)) = (1, y1), is given by xC1 (t) = ( 5828 + 43 √ 20210 ) 4136 e−2t − ( 3760 + 43 √ 20210 ) 4136 e2t + 1 2 , yC1 (t) = − ( 34968 + 258 √ 20210 ) 124080 e−2t − ( 18800 + 215 √ 20210 ) 124080 e2t + 1 3 . The orbit (xL(t), yL(t)) of XL, such that (xL(0), yL(0)) = (−1, y2), is given by xL(t) = ( 16920 + 17 √ 20210 ) 11280 e−2t − ( 17 √ 20210− 16920 ) 11280 e2t − 4, yL(t) = − ( 16920 + 17 √ 20210 ) 11280 e−2t − ( 34 √ 20210− 33840 ) 11280 e2t − 3 2 . The orbit (xC2 (t), yC2 (t)) of XC, such that (xC2 (0), yC2 (0)) = (−1, y3), is given by xC2 (t) = ( 940− 51 √ 20210 ) 4136 e−2t + ( 51 √ 20210− 7144 ) 4136 e2t + 1 2 , yC2 (t) = ( 306 √ 20210− 5640 ) 124080 e−2t + ( 255 √ 20210− 35720 ) 124080 e2t + 1 3 . The flight time of the orbit (xR(t), yR(t)), from (1, y0) ∈ ΣR to (1, y1) ∈ ΣR, is tR = 1 4 arctan ( 39216 √ 20210 19148449 ) . The flight time of the orbit (xC1 (t), yC1 (t)), from (1, y1) ∈ ΣR to (−1, y2) ∈ ΣL, is tC1 = 1 2 log ( 605 + 4 √ 20210 805 ) . The flight time of the orbit (xL(t), yL(t)), from (−1, y2) ∈ ΣL to (−1, y3) ∈ ΣL, is tL = 1 2 log ( 621547 + 1224 √ 20210 596693 ) . Finally, the flight time of the orbit (xC2 (t), yC2 (t)), from (−1, y3) ∈ ΣL to (1, y0) ∈ ΣR, is tC2 = 1 2 log ( 605 + 4 √ 20210 53 ) . The point (−1, 0) ∈ ΣL is an invisible fold point of XL and the point (−1, 43/120) ∈ ΣL is a invisible fold point of XC. They are endpoints of the sliding set in ΣL. The point (1,−1/10) ∈ ΣR is an invisible fold point of XR and the point (1, 13/40) ∈ ΣR is a visible fold point of XC. They are endpoints of the escaping set in ΣR. 2.3 Examples 61 Hence, using the Mathematica software, we can draw the orbits (xi(t), yi(t)) for the time t ∈ [0, ti], i = R,L,C1, C2, i.e., we obtain the limit cycle illustrated in Fig. 2.6 (a). Fig. 2.6 (b) has been made with the help of P5 software, and provides the phase portrait of vector field (2.3) in this case. -0.5 - 0.6 - 0.4 - 0.2 0.2 0.4 0.6 0.5 ΣRΣL (1, y0) (1, y1)(−1, y2) (−1, y3) ΣRΣL (a) (b) Figure 2.6: The limit cycle of vector field (2.3) with aL = αL = −2/3, bL = 4/3, cL = 8/3, βL = 35/3, aC = 2/11, bC = 120/11, αC = −41/11, cC = 4/11, βC = −4/33, aR = 8, bR = 10, αR = −7 and cR = βR = −8. In Table 2.4 we give the parameter values for that the discontinuous planar piecewise linear vector field (2.3) has at most one limit cycle, passing through the points (1, y0), (1, y1), (−1, y2) and (−1, y3), for the others possibles configurations of the type SSC and SSS. Example 7. (Case CVSBCV) Consider the discontinuous planar piecewise linear Hamiltonian vector field (2.3) with aL = 4, αL = 2, bL = 5/2, cL = −5/2, βL = 35/3, aC = √ 2/11, bC = 60 √ 2/11, αC = −21 √ 2/11, cC = 2 √ 2/11, βC = −5 √ 2/33, aR = 8, bR = 10, αR = −19/2 and cR = βR = −8. We can see that the central vector field from (2.3) has a saddle at the point (1, 1/3), the right vector field has a center at the point (−39/4, 35/4) and the left vector field has a center at the point (7,−15/4). Therefore, a candidate to limit cycle of system (2.3), in this case, corresponds to the solution of system (2.12), i.e., − 1 2 (y0 − y1)(10y0 + 10y1 − 3) = 0, 2 √ 2 33 (5(1− 3y0) 2 + 33y3 − 45y23) = 0, 2(y2 − 2y22 + y3(−1 + 2y3)) = 0, − 2 √ 2 33 (5(1− 3y1) 2 + 33y2