
RESSALVA 

Atendendo solicitação do autor, 
o texto completo desta tese será
disponibilizado somente a partir

de 09/03/2025 


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.


Chapter

8

Conclusions

In this work, we estimate the number of crossing limit cycles in planar piecewise linear

Hamiltonian differential systems 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, i.e. centers or saddles. More precisely, we started 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 showed 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 limit cycles, but if it is discontinuous with three zones then it has at most

one limit cycle, and we provide examples with one limit cycle.

Next, we study the number of crossing limit cycles that can bifurcate from periodic annulis

in a discontinuous piecewise linear Hamiltonian differential system with three zones, after

polynomials functions perturbations. For theses cases, we prove that if the linear differential

system defined in the region between the two parallel lines has a center at the origin and the

others subsystems have centers or saddles then we have at least three, four or seven limit cycles

bifurcating from a periodic annulus after linear, quadratic or cubic polynomials perturbations,

respectively. Now, when the central subsystem has a real or boundary center and the others

subsystems have only real saddles then at least six limit cycles can bifurcate simultaneously from

two periodic annuli, after linear perturbations. Finally, if the central subsystem has a real saddle

and the others subsystems have centers or saddles then at least six limit cycles can bifurcate

simultaneously from three periodic annuli, after linear perturbations.

201


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	Introduction
	Basic results
	Piecewise smooth vector fields with two or three zones
	Piecewise smooth Hamiltonian vector fields with two or three zones
	The Melnikov function method

	Limit cycles of planar piecewise linear Hamiltonian differential system with two or three zones
	Preliminaries and main results
	Proof of Theorems 2-4
	Examples

	Persistence of periodic solutions from system (1)|=0 having a center in the central region
	Preliminaries and main result
	A normal form to system (3.1)|=0
	A classification to system (3.1)|=0
	Melnikov functions associated with system (3.1)
	Proof of Theorem 5
	Another method to prove Theorem 5

	Bifurcation of limit cycles from a periodic annulus, by quadratic/cubic perturbations, from system (1)|=0 having a center in the central region
	Preliminaries and main result
	A classification to system (4.1)|=0
	Melnikov functions associated with system (4.1)
	Proof of Theorem 9

	Persistence of periodic solutions from periodic annulus formed by a center and two saddles of system (1)|=0
	Preliminaries and main result
	A classification to system (5.1)|=0
	Melnikov functions associated with system (5.1)
	Proof of Theorem 13

	Persistence of periodic solutions from system (1)|=0 having a saddle in the central region
	Preliminaries and main result
	A normal form to system (6.1)|=0
	A classification to system (6.1)|=0
	Melnikov functions associated with system (6.1)
	Proof of Theorem 15

	Appendix
	Proof of Theorem 11 from Chapter 4
	Proof of Theorem 12 from Chapter 4
	Proof of Theorem 16 from Chapter 6
	Proof of Theorem 17 from Chapter 6
	Proof of Theorem 18 from Chapter 6
	Coefficients of functions (4.44) and (4.45) from Chapter 4
	Coefficients of functions (4.46)–(4.49) from Chapter 4 
	Coefficients of functions (6.38)–(6.40) from Chapter 6
	Coefficients of functions (6.41)–(6.43) from Chapter 6
	Parameter values to function (4.44) from Chapter 4
	Parameter values to function (4.45) from Chapter 4
	Parameter values to functions (4.46) – (4.49) from Chapter 4

	Conclusions
	Bibliography