JULIANA FERNANDES LARROSA
GENERIC BIFURCATION IN PLANAR FILIPPOV SYSTEMS
BIFURCAC¸ ˜OES GEN´ERICAS EM SISTEMAS PLANARES DE FILIPPOV
CAMPINAS 2015
JULIANA FERNANDES LARROSA
GENERIC BIFURCATION IN PLANAR FILIPPOV SYSTEMS BIFURCAC¸ ˜OES GEN´ERICAS EM SISTEMAS PLANARES DE FILIPPOV
Thesis presented to the Institute of Mathe-matics, Statistics and Scientific Computing of the University of Campinas in partial ful-fillment of the requirements for the degree of Doctor in Mathematics.
Tese apresentada ao Instituto de Matem´ati-ca, Estat´ıstica e Computa¸c˜ao Cient´ıfica da Universidade Estadual de Campinas como parte dos requisitos exigidos para obten¸c˜ao do t´ıtulo de Doutora em Matem´atica.
Supervisor/Orientador: MARCO ANTONIO TEIXEIRA
Co-supervisor/Coorientadora: MARIA TERESA MARTINEZ-SEARA ALON-SO
O arquivo digital corresponde `a vers˜ao fi-nal da tese defendida pela aluna Juliana Fernandes Larrosa, e orientada pelo Prof. Dr. Marco Antonio Teixeira.
Marco Antonio Teixeira
CAMPINAS 2015
Ficha catalográfica
Universidade Estadual de Campinas
Biblioteca do Instituto de Matemática, Estatística e Computação Científica Maria Fabiana Bezerra Muller - CRB 8/6162
Larrosa, Juliana Fernandes,
L329g LarGeneric bifurcation in planar Filippov systems / Juliana Fernandes Larrosa. – Campinas, SP : [s.n.], 2015.
LarOrientador: Marco Antonio Teixeira.
LarCoorientador: Maria Tereza Martinez-Seara Alonso.
LarTese (doutorado) – Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica.
Lar1. Filippov, Sistemas de. 2. Teoria da bifurcação. 3. Estabilidade estrutural. 4. Soluções canard. I. Teixeira, Marco Antonio,1944-. II. Martinez-Seara Alonso, Maria Tereza. III. Universidade Estadual de Campinas. Instituto de Matemática, Estatística e Computação Científica. IV. Título.
Informações para Biblioteca Digital
Título em outro idioma: Bifurcações genéricas em sistemas planares de Filippov Palavras-chave em inglês:
Filippov systems Bifurcation theory Structural stability Canard solutions
Área de concentração: Matemática Titulação: Doutora em Matemática Banca examinadora:
Marco Antonio Teixeira [Orientador] Claudio Aguinaldo Buzzi
Regilene Delazari dos Santos Oliveira Ronaldo Alves Garcia
Ricardo Miranda Martins Data de defesa: 27-11-2015
Programa de Pós-Graduação: Matemática
Tese de Doutorado defendida em 27 de novembro de 2015 e aprovada
Pela Banca Examinadora composta pelos Profs. Drs.
Prof(a). Dr(a). MARCO ANTONIO TEIXEIRA
Prof(a). Dr(a). CLAUDIO AGUINALDO BUZZI
Prof(a). Dr(a). REGILENE DELAZARI DOS SANTOS OLIVEIRA
Prof(a). Dr(a). RONALDO ALVES GARCIA
Prof(a). Dr(a). RICARDO MIRANDA MARTINS
A Ata da defesa com as respectivas assinaturas dos membros
Agradecimentos
Este trabalho representa a finaliza¸c˜ao de mais uma etapa e eu agrade¸co imen-samente a todos que estiveram presentes nesse longo trajeto.
Em particular, gostaria de agradecer:
Ao meu orientador, Prof. Marco Antonio Teixeira, pelo incentivo, paciˆencia e oportunidades proporcionadas.
A minha co-orientadora, Prof. Tere M-Seara, pela receptividade no per´ıodo em que estive em Barcelona, pelo comprometimento, apoio e principalmente pelas produtivas discuss˜oes.
Ao Prof. Carles Bonet pela preciosa colabora¸c˜ao e por me apresentar ao mundo dos Canards.
A banca examinadora pelas valiosas contribui¸c˜oes.
A todos os Professores que tive at´e hoje, em especial aos Professores do IMECC e da UFSM, tenho certeza que cada um colaborou de maneira positiva para a minha forma¸c˜ao.
Aos funcion´arios do IMECC, especialmente aos da Secretaria de P´os-Gradua¸c˜ao e da Biblioteca do IMECC.
A minha fam´ılia, meu marido e meus amigos que sempre estiveram presentes fazendo com que essa caminhada se tornasse mais leve e feliz!
A FAPESP, que atrav´es do Projeto 2011/22529-8, viabilizou a realiza¸a˜ao deste trabalho.
Considere um sistema planar de Filippov Z = (X, Y ), onde X e Y são campos vetoriais suaves definidos em uma vizinhança da origem e cuja curva de descontinuidade é dada pelo conjunto de zeros da função f(x, y) = y. Neste trabalho apresentamos um estudo rigoroso das singularidades do tipo dobra-dobra de um sistema planar de Filippov. Mostramos que, sob algumas condições genéricas, o conjunto dos sistemas de Filippov que possuem uma singularidade dobra-dobra na origem é uma subvariedade mergulhada de codimensão um dentro do conjunto formado por todos os sistemas planares de Filippov definidos em torno da origem. Além disso, mostramos que para Z = (X, Y ) pertencente à esta subvariedade todos os seus desdobramentos são equivalentes. Consideramos tam-bém sistemas suaves por partes Z = (X, Y ) satisfazendo Z(x, y) = X(x, y) se x · y > 0 e
Z(x, y) = Y (x, y) se x·y < 0. Neste caso, a descontinuidade do sistema é dada pelos zeros
da função f(x, y) = x·y. Para sistemas deste tipo, apresentamos uma classificação das sin-gularidades de codimensão zero (estruturalmente estáveis) e das sinsin-gularidades genéricas de codimensão um. Além disso, apresentamos os diagramas de bifurcação de cada singu-laridade de codimensão um e mostramos que esses desdobramentos são desdobramentos versais para estes sistemas. Na sequência, estudamos a regularização Teixeira-Sotomayor de sistemas de Filippov que possuem uma singularidade dobra-dobra e cuja regulariza-ção tenha um ponto crítico próximo da origem. Neste contexto, estudamos a natureza do ponto crítico existente para o sistema regularizado e quando o mesmo apresenta uma bifurcação, estudamos a relação entre a bifurcação que ocorre para o sistema suave por partes e para o sistema regularizado.
Palavras-chave: Sistemas de Filippov, Teoria da bifurcação, Estabilidade estrutural, Soluções canard.
Abstract
Consider a planar Filippov system Z = (X, Y ), where X and Y are smooth vector fields defined in a neighborhood of the origin and whose discontinuity curve is given by the set of zeros of f(x, y) = y. In this work we present a rigorous study of the fold-fold singularities of a planar Filippov system. We show that, under some generic assumptions, the set of the planar Filippov systems having a fold-fold singularity at the origin is an embedded codimension one submanifold contained in the set of all planar Filippov systems. In addition, we show that all the unfoldings of Z = (X, Y ) belonging to this submanifold are equivalent. We also consider piecewise smooth systems of the kind
Z = (X, Y ) satisfying Z(x, y) = X(x, y) if x · y > 0 and Z(x, y) = Y (x, y) if x · y < 0 . In
this case, the discontinuity set is the set of zeros of f(x, y) = x · y. For these systems, we present a classification of structurally stable and generic codimension one singularities. In addition, we present the bifurcation diagram of each codimension one singularity and we show that they are, in fact, universal unfoldings. In the sequel we study the Teixeira-Sotomayor regularization of planar Filippov systems having a fold-fold singularity and whose regularization has a critical point around the origin. In this context, we study the nature of this critical point and when the critical point presents a bifurcation, we study the relations between the bifurcation for the planar Filippov system and for the regularized system.
Keywords: Filippov systems, Bifurcation theory, Structural stability, Canard solutions.
1 Introduction 11 2 Generic behavior of a Filippov system around a fold-fold singularity 14
2.1 The versal unfolding of a visible fold-fold singularity . . . 22
2.2 The versal unfolding of a invisible fold-fold singularity . . . 24
2.3 The versal unfolding of a visible-invisible fold-fold singularity . . . 28
3 Generic behavior of a piecewise smooth vector field with non-smooth switching surface 32 3.1 Σ−structural stability . . . 38
3.2 Codimension one generic bifurcations . . . 49
3.2.1 The double pseudo-equilibrium bifurcation . . . 49
3.2.2 The pseudo-Hopf bifurcation . . . 52
3.2.3 Regular-fold bifurcation . . . 55
4 The regularization near some generic codimension one fold-fold singu-larity 59 4.1 The regularization of Z having a generic fold-fold singularity . . . 59
4.1.1 The critical point of the regularized system Zε . . . 60
4.1.2 The critical manifold of Zε . . . 64
4.2 The regularization of a versal unfolding Zα of Z . . . 66
4.2.1 The critical points of the regularized system Zα ε . . . 67
4.2.2 The critical manifold of the regularized system Zα ε . . . 71
4.2.2.1 Folds with the same visibility . . . 73
4.2.2.2 Folds with opposite visibility . . . 74
5 The bifurcation diagram of the regularized vector field Zα ε 76 5.1 Visible-visible case . . . 76
5.2 The invisible-invisible case . . . 78
5.2.1 The supercritical Hopf bifurcation for the attractive case . . . 83
5.2.2 The subcritical Hopf bifurcation for the attractive case . . . 88
5.3 The visible-invisible case . . . 96
5.3.1 The existence of a Canard trajectory . . . 96
5.3.2 The saddle case: (det Z)x(0) > 0 . . . 101
5.3.3 The focus case: (det Z)x(0) < 0 . . . 103
5.3.3.1 The subcritical Hopf bifurcation with αH(ε) < αC(ε) . . . 104
5.3.3.2 The subcritical Hopf bifurcation with αH(ε) > αC(ε) . . . 108
Future Work 126
Introduction
Let X, Y : U ⊂ R2 → R2 be sufficiently smooth vector fields defined in a
bounded neighbourhood U of the origin. Consider f : (x, y) ∈ U ⊂ R2 → y ∈ R and
let Σ = f−1(0). Thus Σ is a regular codimension one submanifold of U ⊂ R2. The
submanifold Σ splits the open set U into two open sets Σ+ = {p ∈ U : y > 0} and
Σ− = {p ∈ U : y < 0}.
Let Z be the space of all piecewise smooth vector fields Z = (X, Y ) defined as
Z(p) = X(p), if p ∈ Σ+ Y(p), if p ∈ Σ−. (1.1)
The dynamics on each open set Σ± is given by the smooth vector fields X and
Y, respectively. The submanifold Σ is called discontinuity manifold or switching manifold
and we assume that the dynamics over Σ is given by the Filippov’s convention. More details about the Filippov’s convention can be found in [Fil03]. The piecewise smooth vector field defined in this way is called a Filippov system.
Due to its importance on applications, Filippov systems have been largely studied in the recent years. There are a huge number of works focusing on local and global aspects of these systems. For some works dedicated to planar Filippov systems, one can see [KRG03], [GST11] and [Der+11] and for examples on higher dimensions, look at [CJ11] and [DRD12].
Concerning to the study of the generic behavior for planar Filippov systems, it was firstly made by Kozlova ([Koz84]). In [KRG03], Kuznetsov at al., classified and stud-ied all the codimension one bifurcations and also some global bifurcations. Guardia, Seara and Teixeira, in [GST11], complemented these works presenting a rigorous proof of the lo-cal Σ−structural stable Filippov systems and revisiting the codimension one bifurcations. In addition, they gave a preliminary classification of codimension two bifurcations.
Using the concepts of local Σ−equivalence and weak equivalence of unfoldings, in Chapter 2 we revisit the codimension one generic bifurcation of Filippov systems, pre-senting a rigorous classification of the generic fold-fold singularities set ΛF as well a formal
study of the versal unfoldings of each singularity. In contrast to the study of codimen-sion one bifurcations presented in [KRG03], we do not used normal forms to describe the bifurcation diagrams of the fold-fold singularities, therefore, our results seem to be more general and formal. Moreover, we have proved that ΛF is a codimension one embedded
12
Using the same definition of piecewise smooth vector fields, instead of consider Σ as a regular codimension one submanifold one can consider it as the union of two codimension one submanifolds which intersect transversely at the origin. One can write
f : U ⊂ R2 → R as f = f1· f2 such that zero is a regular value for each function fi for
i = 1, 2. For instance, we consider f(x, y) = xy and then Σ = f−1(0) is a degenerate
hyperbole, thus Σ can be seen as the union of two lines that intersect at the origin. In this case, the origin is a non-degenerated critical point for f. The piecewise smooth vector field Z = (X, Y ) is given exactly as in (1.1) and in this case, we assume that the Filippov’s convention is valid in each regular part of Σ.
The motivation to the study of these systems comes from applications, one example can be found in [Bar70], where the author presents a stabilization problem that can be solved provided a discontinuity of this type is introduced in the system.
In chapter 3, we present a rigorous classification of all generic behaviors and generic codimension one bifurcations, as well the rigorous study of its bifurcation dia-grams.
A very important tool in the study of non smooth dynamical systems, called
regularization, was introduced by Teixeira and Sotomayor in [ST98]. The idea of the
re-gularization process is quite simple, it consists on approximate the discontinuous vector field Z, using a transition function ϕ, by an one–parameter family of continuous vector fields Zε such that limε→0Zε= Z, in a singular sense.
In [ST98], Teixeira and Sotomayor have shown that, in the planar case, when the discontinuity set is composed by sliding, the regularized system has an invariant manifold such that the flow over this manifold has as limit the Filippov flow. In [BST06], Buzzi, da Silva and Teixeira deal with discontinuous vector fields on R2 and prove that
the analysis of their local behavior around a typical singularity can be treated via singular perturbation.
Recently, in [RS14], Bonet and Seara have studied the Sotomayor-Teixeira reg-ularization of a general visible fold singularity of a Filippov system. Using the extending geometric Fenichel theory beyond the fold with asymptotic methods, they determined the deviation of the orbits of the regularized system from the generalized solutions of the Filippov one. This result is also applied to the regularization of some global sliding bifurcations of periodic orbits and homoclinic connections.
In Chapters 4 and 4.2, we focus our attention on the regularization of a Filippov system Z having a generic codimension one fold-fold singularity, that is, Z ∈ ΛF. For this
purpose, we have used qualitative methods of Dynamical systems, the Fenichel theory as well some asymptotics methods. More precisely, our aim is to understand the relation between the regularized and non regularized systems and also between their unfoldings, provided it exists.
In this context, we proved that when X1· Y1(0) > 0, the regularized system Z
ε
has no critical points near the origin. Therefore, by the Tubular Flow Theorem, all theses systems are equivalent around the origin. On the other hand, when X1· Y1(0) < 0, the
regularized system has a critical point P (ε) and we have restricted our attention to this case. The topological type of P (ε) depends on the fold-fold type as well of some generic conditions.
At the beginning of this work, we expected to find a direct relation between the bifurcation diagram of the non smooth system and the regularized one. However, contradicting our intuition, when we have an invisible-invisible fold-fold, the stability of
the critical point P (ε) of Zε has no direct relation with the stability of the fold-fold point
given by the first return map associated to the non-smooth vector field Z (see [GST11]). Since the regularization of Z was not sufficient to totally understand the prob-lem, the next step was to consider Zα a versal unfolding of Z0 = Z (see [CLW94]) and
study the two parameter family Zα
ε given by the Sotomayor-Teixeira regularization of Zα.
It is still true that when X1·Y1(0) < 0 the regularized system Zα
ε has a critical
point P (α, ε) for α and ε small enough. We prove that P (α, ε) is a continuation of P (ε). We study each case separately.
When both folds are visible, then for every α and ε > 0 sufficiently small, the critical point P (α, ε) is a saddle point. Therefore, the regularized system Zα
ε is always
structurally stable in this case. This fact is not surprising, since the nonsmooth system has a pseudo-saddle for α 6= 0 and it is well known (see [ST98]) that the regularization of a pseudo-saddle is saddle.
When the folds are invisible, then the topological type of P (α, ε) changes according to the values of α and ε < 0. Moreover, in this case, a Hopf bifurcation occurs. The most surprising fact in this case is that the Hopf bifurcation type (sub or supercritical) does not depend on the pseudo-Hopf bifurcation type (see [KRG03]). But in both cases, we prove that the crossing orbit which appear in the unfolding of the nonsmooth system persists in the regularization process. Generically, this orbit is born in two situations: on the Hopf bifurcation, in the case where the Hopf bifurcation agrees with the pseudo-Hopf bifurcation, or on a saddle-node bifurcation of the periodic orbits, in the case where the Hopf bifurcation does not agree with the pseudo-Hopf bifurcation. We also give concrete models where these two situations are achieved.
The last case is when the folds have opposite visibility, in this case the topolog-ical type of P (α, ε) also changes depending on the parameters (α, ε). The surprising fact in this case is the Hopf Bifurcation and the presence of periodic orbits on the regularized system Zα
ε because there is no periodic orbits in the unfolding of Z. The difficulty in this
case is to determine when the periodic orbits disappear, since we know that the system
Zεα converges to Zα, in a singular sense, when ε tends to zero and Zα has no crossing
periodic orbits.
The most challenging problem at this point is to prove rigorously the existence of a curve on the bifurcation diagram where the limit cycle which raises from the Hopf bifurcation disappear. When considering a piecewise linear transition map ϕ, we estab-lish an equivalence between the regularized system Zα
ε when α = δε with the well known
Krupa-Szmolyan system. Through this equivalence, we prove the existence of a curve where a Canard takes place and therefore, the disappearance of the limit cycle. More-over, using the Krupa-Szmolyan Theorem, see [KS01], we establish a relation between the Hopf curve, the First Lyapunov Coeficient and the Canard curve on the bifurcation diagram of the regularized system. When ϕ is an arbitrary transition map, we give the expression of the curve where a Canard takes place.
Chapter 2
Generic behavior of a Filippov
system around a fold-fold singularity
Let Z = Zr, r ≥ 1 be the set of all planar Filippov systems defined in a
bounded neightborhood U ⊂ R2 of the origin, that is
Z(x, y) = X(x, y), y > 0 Y(x, y), y < 0, (2.1)
where X = (X1, X2), Y = (Y1, Y2) ∈ Xr(U), r ≥ 1. As we want to study local
sin-gularities we can assume f(x, y) = y and that the dynamics on the discontinuity curve Σ = U ∩ f−1(0) is given by the Filippov convention. We consider Z = Xr× Xr with the
product Cr topology, which is a Banach space.
Recall that, by the Filippov convention, as can be seen in [Fil03], the discon-tinuity curve is decomposed as the closure of the following regions:
Σc = {x ∈ Σ : X2· Y2(x, 0) > 0},
Σs = {x ∈ Σ : X2(x, 0) < 0 and Y2(x, 0) > 0},
Σe = {x ∈ Σ : X2(x, 0) > 0 and Y2(x, 0) < 0}.
The flow through a point in the crossing region is the concatenation of the flow of X and
Y through p in a consistent way. Over the regions Σs,e, the flow is given by the sliding
vector field, denoted by Zs and given by
Zs(x) = Y 2· X1(x, 0) − X2· Y1(x, 0) Y2(x, 0) − X2(x, 0) Σ .
Definition 2.1. The point p ∈ Σs,e is a pseudo-equilibrium of Z if Zs(p) = 0 and it
is a hyperbolic pseudo-equilibrium of Zs, if (Zs)0(p) 6= 0. Moreover,
Σ Zs(p) X(p) Y (p) X Y p
• p is a pseudo-node if (Zs)0(p) < 0 and p ∈ Σs or (Zs)0(p) > 0 and p ∈ Σe;
• p is a pseudo-saddle if (Zs)0(p) < 0 and p ∈ Σe or (Zs)0(p) > 0 and p ∈ Σs.
When p ∈ Σc,s,e, the vector fields X and Y are transverse to Σ at the point p,
otherwise we have a tangency point, in particular, we are going to deal with fold points. Definition 2.2. p ∈ Σ is a fold point of X if Xf(p) = X2(p) = 0 and X(Xf)(p) =
X2
x(p) · X1(p) 6= 0. The fold is visible if X(Xf)(p) > 0 and it is invisible if X(Xf)(p) <
0. Analogously, a fold point p ∈ Σ of Y is visible if Y (Y f)(p) < 0 and invisible if
Y(Y f)(p) > 0.
Our purpose is to study vector fields Z ∈ Z having a fold-fold singularity, which we assume, without loss of generality, that is at the origin 0 = (0, 0) ∈ Σ. That is
Xf(0) = X2(0) = 0 X(Xf)(0) = X2 x(0) · X1(0) 6= 0 (2.2) Y f(0) = Y2(0) = 0 Y(Y f)(0) = Y2 x(0) · Y1(0) 6= 0 (2.3)
The fold-fold singularity has been studied in [KRG03] and [GST11] by consid-ering some normal forms for the Filippov vector fields and its unfoldings. In this section we present a rigorous approach to the bifurcation diagrams of these singularities, proving that the set of the fold-fold singularities, under some generic conditions, is a codimension one embedded submanifold of Z.
Let Z1 = Z \ Ξ0, where Ξ0 is the set of all locally Σ−structurally stable
Filippov systems defined on U (see [GST11]). Let ΛF be the subset of Z
1 composed by
the Filippov systems which have a locally Σ−structurally stable fold-fold at the origin. More precisely, given Z ∈ ΛF there exists a neighbouhood V
Z such that given ˜Z ∈ VZ∩Z1
then Z is locally Σ−equivalent to ˜Z and their versal unfoldings are weak equivalent, that
is
Definition 2.3. Let Z, ˜Z ∈ Z. We say that two unfoldings of Zδ and ˜Zδ˜ are weak
equivalent if there exists a homeomorphic change of parameters µ(δ), such that, for each δ the vector fields Zδ and ˜Zµ(δ) are locally Σ−equivalent. Moreover, given an unfolding Zδ
of Z it is said to be a versal unfolding if every other unfolding Zα of Z is weak equivalent
to Zδ.
In this context, this section is devoted to prove the following theorem: Theorem 2.4. Consider ΛF ⊂ Z
1 the set of all Filippov systems Z which have a
Σ−structurally stable fold-fold singularity in the induced topology on Z1. Then Z ∈ Z1
belongs to ΛF if and only if satisfies one of the following conditions:
(A) it is a visible-visible fold;
(B) it is an invisible-invisible fold which the origin is a non degenerated fixed point for the generalized Poincaré return map. See (2.18) for a precise definition;
(C) it is a visible-invisible fold and the in the case where the sliding vector field Zs(x) is
16
Zs(0) = γ 6= 0. (2.4)
In addition, ΛF is a codimension one embedded submanifold of Z. Moreover,
given Z ∈ ΛF then any two unfoldings ξ and α of Z which are transverse to ΛF at Z are
weak equivalent.
Remark 2.5. Observe that when the folds have opposite visibility the cases visible-invisible
and invisible-visible are essentially the same, then we are going to focus our attention only in the visible-invisible case.
Next lemma, whose proof is straightforward, characterizes the cases where there is a region of sliding around the fold-fold point.
Lemma 2.6. Suppose that the origin is a fold-fold point for Z ∈ Z. Then Σ is decomposed
as follows:
• If the folds have the same visibility then Σ = Σc if X1· Y1(0) < 0 and Σ = Σe∪Σs
if X1 · Y1(0) > 0.
• If the folds have the opposite visibility then Σ = Σcif X1·Y1(0) > 0 and Σ = Σe∪Σs
if X1 · Y1(0) < 0.
In the cases where the sliding vector field is defined, one can write, using x as a variable in Σ
Zs(x) = det Z(x, 0)
(Y2− X2)(x, 0). (2.5)
where
det Z(p) = (X1· Y2 − X2 · Y1)(p), p ∈ R2 (2.6)
It follows from (2.5) that p ∈ Σe,s is a pseudo-equilibrium if, and only if,
det Z(p) = 0. Moreover, the stability of a pseudo-equilibrium p ∈ Σe,s is determined by
(Zs)0(p) = (det Z)x(p)
(Y2− X2)(p). (2.7)
In the case that the sliding vector field (2.5) is defined around the fold-fold (0, 0) , even if is not defined at x = 0 (which corresponds to the fold point), one can easily extend it by using the L’Hospital rule
γ = Zs(0) = lim x→0Z s(x) = lim x→0 det Z(x, 0) (Y2− X2)(x, 0) = (det Z)x(0) (Y2 x − Xx2)(0) . (2.8)
Using Definition 2.2 and Lemma 2.6, we see that Y2
x − Xx2(0) 6= 0 provided
the sliding vector field is defined. Thus the sliding vector field Zs is well defined at the
origin.
Lemma 2.7. Suppose that the origin is a fold-fold point of Z = (X, Y ) and that
(b) have opposite visibility and X1· Y1(0) < 0 and Zs satisfies hypothesis (2.4),
then (det Z)x(0) 6= 0.
Moreover:
(a) if both folds are visible, we have (det Z)x(0) > 0 when X1·Y1(0) < 0 and (det Z)x(0) <
0 when X1· Y1(0) > 0;
(b) if both folds are invisible, we have (det Z)x(0) < 0 when X1·Y1(0) < 0 and (det Z)x(0)
>0 when X1· Y1(0) > 0;
(c) if the folds have opposite visibility, a priori, one can not establish the sign of (det Z)x(0).
Proof. Since we have a fold-fold at the origin, we get
(det Z)x(0) = (X1· Yx2− Y
1· X2
x)(0).
From (2.2) and (2.3) it follows that if both folds have the same visibility, then (X1· Y2
x) ·
(Y1 · X2
x)(0) < 0 thus (det Z)x(0) 6= 0. A straightforward computation shows the second
part of the lemma.
When the folds have opposite visibility we have (X1· Y2
x) · (Y1 · Xx2)(0) > 0,
so one can not conclude that (det Z)x(0) 6= 0. However, X1 · Y1(0) < 0 the sliding is
defined, by hypothesis (2.4) and formula (2.8) we have (det Z)x(0) 6= 0.
Remark 2.8. The results of this lemma are necessary in the proof of Theorem 2.4.
Ob-serve that in the case which we have a visible-invisible fold with X1 · Y1(0) > 0 then
(det Z)x(0) can vanish. Nevertheless this is the only case in which this information is not
necessary for the proof of the theorem.
Corolary 2.9. In the cases where sliding vector field Zs is defined around the fold-fold
point, we get the following statements.
• If the folds have the same visibility, then sgn (γ) = sgn (Zs(0)) = sgn (X1(0));
• If the folds have opposite visibility, then sgn (γ) = sgn (Zs(0)) = −sgn
X1(0) · (det Z)x(0)
.
Proof. Using Lemma 2.6 we get conditions to the sliding vector field Zs to be defined.
Then this result can be obtained by combining (2.2), (2.3) and (2.8).
Corolary 2.10. Let Z0 = (X0, Y0) ∈ Z having a fold-fold at the origin satisfying the same
hypotheses of Lemma 2.7. Then there exist neighbourhoods Z0 ∈ U0 ⊂ Z and 0 ∈ I0 ⊂Σ
such that for each Z ∈ U0 there exist a unique P (Z) ∈ I0 such that det Z(P (Z), 0) = 0
and sgn ((det Z)x(P (Z), 0)) = sgn ((det Z0)x(0)) .
Proof. Let Z0 ∈ Z satisfying the hypothesis of Lemma 2.7. Let ξ be the Frechet
differen-tiable map
ξ : Z × R → R
(Z, x) 7→ det Z(x, 0)
As Z0 has a fold-fold at the origin, then ξ(Z0,0) = det Z0(0) = 0 and by the
Lemma 2.7 we have ξx(Z0,0) = (det Z0)x(0) 6= 0. Then by the Implicit Function Theorem
we obtain neighborhoods Z0 ∈ U0 ⊂ Z and 0 ∈ I0 ⊂ Σ and a Frechet differentiable map
P : U0 → I0 satisfying ξ(Z, x) = 0 if, and only if, x = P (Z). That is ξ(Z, P (Z)) = 0 for
all Z ∈ U0. Moreover, we can assume without loss of generality that in this neighbourhood
18
In the case of the invisible fold-fold it is natural to consider the first return map, as can be seen in [Tei81]. In [GST11] it was observed that in the invisible-invisible case with Σ = Σc, for the fold-fold to be generic, one needs to assume some non degeneracy
condition for the generalized return map φZ. However, no concrete expression for this
map was given. In the following propositions we compute explicitly the Poincaré map associated to an invisible fold singularity (x0, y0) of a general vector field X.
Proposition 2.11 (Poincaré map for X at a point (x0, y0) ∈ Σy0). Let Z = (X, Y ) ∈ Z
having a invisible quadratic tangency (invisible fold point) at the point p0 = (x0, y0) ∈
Σy0 = {(x, y0) : (x, y0) ∈ U}, where U is a neighborhood of p0. Then the Poincaré map
φp0 X is given by φp0 X : Σy0 → Σy0 x 7→ φp0 X(x) = 2x0− x+ βXp0(x − x0)2+ O(x − x0)3 (2.9) where βp0 X = 1 3 " −X 2 xx X2 x + 2Xx1 X1 + 2 X2 y X1 # (p0)
Proof. Let p = (x, y0) ∈ Σy0 and consider ϕ (t; p) = (ϕ
1(t; p) , ϕ2(t; p)) its flow. Since we
want to find the point in which the flow ϕ (t; p) intersects the section Σy0 for t 6= 0, the first step is to solve the equation
ϕ2(t; p) − y0 = 0. (2.10)
By Taylor expanding ϕ2(t; p) we see that to solve Equation (2.10) is equivalent
to solve the equation S(x, t) = 0 where
S(x, t) = a1(x) + 1 2a2(x)t + 1 6a3(x)t2+ O(t3), and ai(x) = di dtiϕ 2(t; (x, y 0)) t=0 .
An easy computation gives:
a1(x) = X2(x, y0) a2(x) = Xx2X1+ Xy2X2(x, y0) a3(x) = Xxx2 (X 1)2+ 2X2 xyX 1 X2+ Xx2Xx1X1+ Xx2Xy1X2 + X2 yy(X 2)2 + X2 yX 2 xX 1+ (X2 y) 2 X2
Observe that S(x0,0) = a1(x0) = X2(p0) = 0 and dtdS(x0, t0) = a2(x20) = 12X1·
X2
x(p0) 6= 0 since p0 ∈Σy0 is a quadratic tangency. Then by the Implicit Function Theorem there exist a differentiable function t(x) defined in a neighbourhood x0 ∈ Iy0 ⊂Σy0 such that
S(x, t(x)) = 0 ∀x ∈ Iy0.
Using the chain rule one can obtain the Taylor expansion of t(x) near the origin and since t(x0) = 0 we have
t(x) = b1(x − x0) +
1
and some easy but tedious computations give b1 = −2 X1(p 0) and b 2 = 2 3 " − (X 2 xx) X2 x · X1 + 2 Xx1 (X1)2 + 2 X2 y (X1)2 # (p0)
Once we have the expression to t(x) for each p = (x, y0) ∈ Σy0, we need to compute the point where the intersection happens, that is, the point
(φp0
X(x), y0) = (ϕ1(t(x); p) , y0) ∈ Σy0. By Taylor expanding φp0
X(x) around x = x0 and computing its coeficients, we
obtain φp0 X(x) = 2x0− x+ βXp0(x − x0)2+ O(x − x0)3 (2.12) where βp0 X = 1 3 " −X 2 xx X2 x + 2Xx1 X1 + 2 X2 y X1 # (p0) (2.13)
With an analogous computation we obtain the same formulas for the Poincaré map for Y around a quadratic tangency p0 = (x0, y0) ∈ Σy0.
Proposition 2.12 (Poincaré map for Y at a point (x0, y0) ∈ Σy0). Let Z = (X, Y ) ∈ Z
having a invisible quadratic tangency (invisible fold point) at the point p0 = (x0, y0) ∈
Σy0 = {(x, y0) : (x, y0) ∈ U}. Then the Poincaré map φ
p0 Y is given by φp0 Y : Σy0 → Σy0 x 7→ φp0 Y (x) = 2x0− x+ β p0 Y (x − x0)2+ O(x − x0)3 (2.14) where βp0 Y = 1 3 " −Y 2 xx Y2 x + 2Yx1 Y1 + 2 Y2 y Y1 # (p0)
Proof. The proof can be done exactly as in Proposition 2.11.
Suppose that the vector field Z has an invisible fold-fold point at 0 ∈ Σ. In the case that we are dealing with the invisible fold-fold point with Σ = Σc it has sense to
consider the first return map that, for convenience, we define on Σ− = {x ∈ Σ : x < 0}
φZ : Σ− →Σ−, (2.15)
by setting p0 = 0 and composing appropriately the Poincaré maps obtained in
Proposi-tions 2.11 and 2.12. Then we have
φZ(x) = φY ◦ φX(x) = x + (βY − βX)x2+ O(x3), if X1(0) > 0 (2.16)
φZ(x) = φX ◦ φY(x) = x + (βX − βY)x2+ O(x3), if X1(0) < 0 (2.17)
The generic condition for the invisible fold-fold is that
20
Remark 2.13. If X1(0) > 0, by the above equations the origin is an attractor fixed point
for φZ if µZ = βY − βX >0 and it is repellor in case µZ = βY − βX <0. Analogously for
the case X1(0) < 0. Nevertheless, it is important to stress that, as φ0
Z(0) = 1, the origin
is never a hyperbolic fixed point of the first return map φZ even in the generic case. This
will have consequences latter in Chapter 4 when we study the regularization of the vector field Z.
Remark 2.14. An important detail that had not been observed in [GST11] and [KRG03]
is that, even in the case Σ = Σs∪Σe, one needs to consider the first return map and
the same generic condition needs to be imposed. Even though the first return map has no dynamical meaning in this case, the pseudo-cycles, which correspond to fixed points of φZ
must be preserved by Σ−equivalences. This map will be used in Section 2.2 when we study the unfolding of a invisible fold-fold satisfying X1· Y1(0) > 0 and in this case we consider
φZ = φY ◦ φX independently of the sign of X1(0).
Now, we are able to state and prove the theorem which proves that conditions (A) to (C) in Theorem 2.15 characterize a codimension one embedded submanifold in Z. Theorem 2.15. The set ΛF ⊂ Z of all Filippov systems which have a fold-fold at the
ori-gin satisfying the hypothesis (A), (B) or (C) in Theorem 2.4 is an embedded co-dimension one submanifold of Z. That is, for each Z0 ∈ ΛF there exist a map λ : V
0 → R where
V0 ⊂ Z is a neighborhood of Z0 and Z0 ∈ λ−1(0) = V0∩ΛF and DλZ0 6= 0.
Proof. Consider Z0 ∈ ΛF. Let U
0 ⊂ Z be a neighborhood of Z0 sufficiently small such
that in this neighborhood the sign of X1(x, 0), Y1(x, 0), X2
x(x, 0) and Yx2(x, 0) is constant
for x ∈ I0 ⊂ R. Moreover, if Z0 satisfies the hypothesis of Corolary 2.10, suppose that
sgn ((det Z)x(x, 0)) = sgn ((det Z0)x(0))) for all Z ∈ U0 and x ∈ I0.
Consider the following Frechet differentiable map
ξ: U0× R2 → R2
(Z, (p, q)) 7→ (X2(p, 0), Y2(q, 0)) .
Since {0} ∈ Σ is a fold-fold point ξ(Z0, 0) = 0 and by (2.2) and (2.3),
det D(p,q)ξ(Z0, 0) = (X0)2x·(Y
0)2
x(0) 6= 0.
Using the Implicit Function Theorem for ξ there exist V∗
0 ⊂ U0 and a Frechet
differentiable map
T : Z = (X, Y ) ∈ V0∗ ⊂ U0 7→(TX, TY) ∈ I0× I0 ⊂ R2 (2.19)
defined in a path connected open set such that ξ(Z, (p, q)) = (0, 0) if, and only if, (p, q) = (TX, TY). In other words, ξ(Z, T (Z)) = 0 for every Z ∈ V0∗. That is, X and Y have a fold
point (TX,0) and (TY,0) near the origin with the same visibility as the origin has for X0
and Y0.
To show that ΛF is a submanifold let consider the Frechet differentiable map
λ∗ : V0∗ → R
Z 7→ TX − TY
.
It is clear that Z ∈ V∗
0 has a fold-fold point near the origin if, and only if,
every Z ∈ λ−1
∗ (0) also belongs to ΛF and the fold-fold type is preserved. In this case, set
V0 = V0∗.
When Z0 satisfies (B) with µ
Z0 6= 0 (2.18), then there exists a neighborhood ˜V1
0 = V01 ∩ Z1, with V01 ⊂ Z, such that sgn (µZ) = sgn (µZ0), for all Z ∈ V01. Therefore the fold-fold point has the same attractivity to φZ as the origin has to φZ0. In this case set V0 = V0∗∩ V01.
Consider then the map λ = λ∗
V 0. It follows that λ −1(0) = V 0∩ΛF. To finish
our proof, observe that
DλZ0(Z) = D(TX − TY)Z0(Z) = (DTX)Z0(Z) − (DTY)Z0(Z) = (XX22(0) 0)x(0) − Y 2(0) (Y2 0)x(0) .
The last equality can be obtained by using the chain rule to ξ(Z, T (Z)) = 0 for all Z ∈ V0. The resulting map is a non-vanishing linear map, what proves the desired
result.
Let Λ0 = λ−1(0). Since it is a codimension one embedded submanifold and V0is
path connected the set Λ0 splits V0in two connected components, namely, V0± = λ
−1(R±).
Then if Z ∈ V+
0 we have TX > TY and Z ∈ V0− we have TX < TY. Thus in
these connected components, the relative position of the folds change.
Our next step is to study the generic unfoldings of this singularity and to show that all of them have the same topological behavior. More concretely, we will show that given a smooth curve γ : (−α0, α0) → V0 transverse to ΛF at γ(0) ∈ Λ0 the behavior of
γ(α) does not depend on the chosen curve γ. To avoid a cumbersome notation we will
call γ(0) = Z0.
Let Z0 ∈ ΛF and γ : (−α
0, α0) → V0, α0 1, where V0 is the neighborhood
given in Theorem 2.15. Since γ(α) is a vector field near Z0 one can write γ(α) = Z0+
˜
Zαα = Z0+ ˜Zα+ O(α2) with ˜Z = ( ˜X, ˜Y). By Theorem 2.15, applied to the particular
curve γ(α), for each α ∈ (−α0, α0) there exist TXα, TYα ∈Σ near the origin given by
TXα = −X˜ 2(0) X2 x(0) α+ O(α2), (2.20) TYα = −˜Y 2(0) Y2 x(0) α+ O(α2). (2.21)
Since γ is transverse to ΛF at Z0 the values ˜X2(0) or ˜Y2(0) can not be zero
simultaneously. Without loss of generality, one can suppose that γ(−α0,0) ⊂ V0− and
γ(0, α0) ⊂ V0+. Using that TXα− TYα = −X˜ 2(0) X2 x(0) + ˜Y2(0) Y2 x(0) ! α+ O(α2),
this is equivalent to assume
˜Y2(0) Y2 x(0) − X˜ 2(0) X2 x(0) >0. (2.22)
The next subsections are devoted to study the versal unfoldings of the vector fields in ΛF.
22
2.1
The versal unfolding of a visible fold-fold
singu-larity
In this section we will describe the phase portrait of the unfoldings of a Filippov vector field Z which has a visible fold-fold at the origin. As we will see, the phase portrait depends on the sign of X1·Y1(0), but the study is completely analogous for X1(0) positive
or negative. For this reason and to avoid the enumeration of long lists of cases, we will assume during the section that X1 >0. We split the study of the two cases depending of
the sign of X1· Y1(0) in two propositions. Once we show that any unfolding has the same
phase portrait a systematic construction of the homeomorphism giving the topological equivalences between them can be easily done using the ideas of [GST11].
Proposition 2.16. Let Z ∈ ΛF satisfying condition (A) (visible-visible fold) and
X1· Y1(0) > 0. Let V
0 be the neighbourhood given by Theorem 2.15. Then any smooth
curve
γ : α ∈ (−α0, α0) 7→ Zα ∈ V0
which is transverse to ΛF at γ(0) = Z leads to the same topological behaviors in V+ 0 and
in V−
0 . Any vector field Zα has two visible fold points with a crossing region between them.
In the sliding and escaping regions there are no pseudo-equilibria. Therefore, there exists a weak equivalence between any two unfoldings of Z.
Proof. We are going to focus on the case X1(0) > 0. Assume that (2.22) holds and
therefore Zα ∈ V−
0 if α < 0 and Zα ∈ V0+ if α > 0, otherwise we can make the change
α 7→ −α.
Since X1 · Y1(0) > 0, the discontinuity curve is decomposed as Σ = Σs∪Σe
for α = 0.
For α 6= 0, we know the existence of the folds Tα
X and TYα given in (2.20) and
(2.21), respectively. Moreover, by (2.22) we know that Tα
X < TYα if α < 0 and TXα > TYα
if α > 0. Observe that for α 6= 0 we have Xα2(x, 0) < 0 if x < Tα
X and Yα2(x, 0) < 0
if x > Tα
Y. Analogously, Xα2(x, 0) > 0 if x > TXα and Yα2(x, 0) > 0 if x < TYα, see
Figure 2.2. Therefore, a crossing region appears between the folds and the discontinuity curve is decomposed as Σ = Σs∪Σc∪Σe, as follows:
Σs= {x ∈ Σ : x < min{Tα X, T α Y}}, Σc= {x ∈ Σ : x ∈ (min{Tα X, T α Y},max{T α X, T α Y})}, Σe= {x ∈ Σ : x > max{Tα X, TYα}}. (2.23) Observe that as det Zα(Tα X,0) = Xα1· Yα2(TXα,0) det Zα(Tα Y,0) = −Xα2· Yα1(TYα,0), (2.24) We get det Zα(Tα
X,0) · det Zα(TYα,0) < 0 for α 6= 0. Then there is a point
Q(α) ∈ Σcsuch det Zα(Q(α), 0) = 0. Moreover, by Lemma 2.7 we know that (det Z0)
x(0) <
0 and therefore by Corolary 2.10, for α small enough Q(α) is unique. Then no pseudo-equilibrium appears for the sliding vector field for α 6= 0. Moreover, by definition (2.5) of the sliding vector field, the fact of Tα
X,Y are fold points and sgn (Xα1(x, 0)) =
sgn (Yα1(x, 0)) > 0, we have sgn ((Zα)s(Tα X)) = sgn ((Z α)s(Tα Y)) = sgn Xα1(0)>0.
Then the sliding vector field of Zαnear Z0satisfies sgn ((Zα)s(x)) = sgn (X1(0)),
in particular there are no pseudo-equilibria.
By (2.23), for α < 0, the sliding vector field is defined for x < Tα
X and for
x > TYα. In addition, between the folds we have that Xα2, Yα2>0.
For α > 0, the sliding vector field is defined for x < Tα
Y and x > TXα and
between the folds Xα2, Yα2 <0.
This proves that any unfolding of Z ∈ ΛF satisfying (A) leads to vector fields
with exactly the same behavior, that is, the same Σ−regions and singularities. A sketch of a versal unfolding of the visible fold-fold satisfying X1 · Y1(0) > 0 can be seem in
Figure 2.2.
Moreover, by using the same arguments presented in [GST11] one can construct the homeomorphism which gives the Σ−equivalence between any two Fillipov vector fields
Z, ˜Z ∈ V±. Then given two unfoldings γ and γ∗ one can establish an weak equivalence
between them. Σ Σ Σ Z ∈ VZ− 0 Z ∈ VZ0∩ Σ F Z ∈ V+ Z0
Figure 2.2: Versal unfolding for a visible fold-fold: X1· Y1(0) > 0
Proposition 2.17. Let Z ∈ ΛF satisfying condition (A) and X1· Y1(0) < 0. Let V 0 be
the neighborhood given by Theorem 2.15. Then any smooth curve γ : α ∈ (−α0, α0) 7→ Zα ∈ V0
which is transverse to ΛF at γ(0) = Z leads to the same behaviors in V+
0 and in V − 0 .
If Zα ∈ V−
0 (Zα ∈ V0+), it has two visible fold points with a sliding (escaping) region
between them, whose sliding vector field has a pseudo-saddle. Therefore, there exists a weak equivalence between any two unfoldings of Z.
Proof. We are going to focus on the case X1(0) > 0. Once again assume that Zα ∈ V− 0 if
α <0 and Zα∈ V+
0 if α > 0.
Since X1 · Y1(0) < 0, the discontinuity curve is decomposed as Σ = Σc for
α= 0.
For α 6= 0, we know the existence of the folds Tα
X and TYα given in (2.20) and
(2.21). Observe that Xα2(x, 0) < 0 if x < Tα
X and Yα2(x, 0) > 0 if x > TYα. Analogously,
Xα2(x, 0) > 0 if x > TXα and Yα2(x, 0) < 0 if x < TYα, see Figure 2.3. Therefore, a piece
of sliding or escaping region appear between the folds. Σc = {x ∈ Σ : x < min{Tα X, T α Y}} ∪ {x ∈Σ : x > max{T α X, T α Y}} (2.25) Σ \ Σc, = Σe = {x ∈ Σ : x ∈ (Tα X, TYα)}}, α < 0, Σs = {x ∈ Σ : x ∈ (Tα Y, TXα)}}, α > 0. (2.26)
24
Using the same argument as in Proposition 2.16 for α 6= 0, there exists a unique point Q(α) ∈ Σe,s such det Zα(Q(α)) = 0. The difference is now the point Q(α) ∈ Σe,s,
and therefore it is a pseudo-equilibrium of (Zα)s. Moreover, by (2.7)
((Zα)s)0(Q(α)) = (det Zα)x(Q(α), 0)
(Yα2− Xα2)(Q(α), 0) (2.27)
By Lemma 2.7 we have that (det Z)x(0) > 0 and so (det Zα)x(Q(α), 0) > 0.
Therefore, ((Zα)s)0(Q(α)) is positive if α > 0 and it is negative if α < 0. This fact implies
that the point Q(α) is a pseudo-saddle of the sliding vector field.
This proves that any unfolding of Z ∈ ΛF satisfying (A) with X1 · Y1(0) < 0
leads to vector fields with exactly the same topological invariants. A sketch of a versal unfolding in this case can be seem in Figure 2.3.
By the same argument as in Proposition 2.17, we show that the unfoldings of
Z0 are C0−equivalent. Σ Σ Σ Z ∈ VZ− 0 Z ∈ VZ0∩ Σ F Z ∈ V+ Z0
Figure 2.3: Versal unfolding for a visible fold-fold: X1· Y1(0) < 0
2.2
The versal unfolding of a invisible fold-fold
sin-gularity
In this section we study the unfoldings of a Filippov vector field Z having an invisible fold-fold. To study these unfoldings we need to consider the generalized first return map (2.15). In this case, we will have four different types of bifurcations depending on the sign of X1· Y1(0) and the attracting or repelling character of the return
map around the fold-fold point. The case where Σ = Σc is the so called pseudo-Hopf
bifurcation (see [KRG03]). This bifurcation has also been studied by [GST11] by the use of a normal form of the vector field and its unfolding. In the case of a pseudo-Hopf bifurcation, they observed that the condition µZ 6= 0 (see (2.18)) was necessary to obtain
a generic codimension one bifurcation.
We reinforce now that the condition µZ 6= 0 also needs to be imposed in the
case Σ = Σs∪Σe if we want to make a rigorous study of the unfoldings of this singularity.
In Proposition 2.20 we prove that, under this hypothesis, a unique pseudo-cycle appears in the unfolding of such bifurcation.
Proposition 2.18. Let Z ∈ ΛF satisfying condition (B) of Theorem 2.4, X1· Y1(0) < 0
and µZ 6= 0 (see (2.18)). Let V0 be the neighborhood given in Theorem 2.15. Then any
smooth curve
which is transverse to ΛF at γ(0) = Z leads to the same topological behavior in V+ 0 and
in V−
0 . This behavior depends of the sign of µZ:
1. If µZ >0:
• Every Z ∈ V−
0 has two invisible fold points and there exists a region of sliding
between them. The sliding vector field has a stable pseudo-node in Σ which is a global attractor.
• Every Z ∈ V+
0 has two invisible fold points and there exists a region of escaping
between them. The sliding vector field has a unstable pseudo-node in Σ and a stable periodic orbit Γα which is a global attractor.
2. If µZ <0:
• Every Z ∈ V−
0 has two invisible fold points and there exists a region of sliding
between them. The sliding vector field has a stable pseudo-node in Σ and a unstable periodic Γα orbit which is a global repellor.
• Every Z ∈ V+
0 has two invisible fold points and there exists a region of escaping
between them. The sliding vector field has a unstable pseudo-node in Σ which is a global repellor.
Therefore, there exists a weak equivalence between any two unfoldings of Z. Proof. We are going to focus on the case X1(0) > 0. Assume that Zα ∈ V−
0 if α < 0 and
Zα ∈ V+
0 if α > 0.
For α = 0, since Z0 = Z by hypothesis X1 · Y1(0) < 0. Therefore, the
discontinuity curve is Σ = Σc.
For α 6= 0, for the all points (x, 0) between the folds Tα
X and TYα the vector field
Zα satisfies Xα2· Yα2(x, 0) < 0. Therefore, a piece of sliding (for α > 0) or escaping (for
α <0) region appears between the folds. The discontinuity curve becomes Σ = Σe,s∪Σc,
where Σc = {x ∈ Σ : x < min{Tα X, T α Y}} ∪ {x ∈Σ : x > max{T α X, T α Y}} (2.28) Σ \ Σc, = Σs = {x ∈ Σ : x ∈ (Tα X, TYα)}}, α < 0, Σe = {x ∈ Σ : x ∈ (Tα Y, TXα)}}, α > 0. (2.29)
Since we have sliding motion defined on one side of the tangencies and crossing on the other, the fold points are singular tangency points. Using the same argument as in Proposition 2.16, there exists a unique Q(α) ∈ Σe,s such that det Zα(Q(α), 0) = 0. By
Lemma 2.7, we have (det Z)x(0) < 0. Using the formulas for (Zs)0(Q(α)) given in (2.7)
the pseudo-equilibrium Q(α) is an stable pseudo-node when α < 0 and it is a unstable pseudo-node when α > 0.
To give a complete description of the dynamics one needs to analyze the first return map to the fold-fold singularity. Let φα
X and φαY be the Poincaré maps associated
to the fold points Tα
X and TYα, as in Propositions 2.11 and 2.12. Therefore the first return
map φα Z becomes: φα Z : Dα → Iα x 7→ φα Z(x) = φαY ◦ φαX(x) .
26 where Dα = {x ∈ Σ : x < (φα X)−1(TYα)}, Iα = {x ∈ Σ : x < TYα}, if α < 0 and Dα = {x ∈ Σ : x < Tα X}, Iα = {x ∈ Σ : x < (φαY)(TXα)}, if α > 0.
More precisely, using the expresions given for φα
X and φαY in propositions 2.11 and 2.12
respectively, the Poincaré map is given by
φα(x) = 2(TYα− Tα X) + x − βX(x − TXα) 2+ β Y(2TXα − T α Y − x) 2+ O 3(x, TXα, T α Y) (2.30) where Tα
X and TYα are given by formulas (2.20) and (2.21) respectively.
TXα TYα F (α)
TXα TYα
α < 0 α = 0 α > 0
F (0) = 0
Figure 2.4: The graphic of Φ(α, x) = φα(x) − x when µZ > 0, for different values of the parameter α fixed.
Now we will look for the appearance of periodic orbits near the fold-fold point. Equivalently, we look for fixed points of φα
Z, that is, x ∈ Dα satisfying φαZ(x) = x. Consider
the auxiliary map
Φ : (α, x) ∈ W ⊂ (−α0, α0) × R 7→ Φ(α, x) = φαZ(x) − x ∈ R. (2.31)
The map Φ satisfies Φ(0, 0) = 0, ∂
∂xΦ(0, 0) = 0 and ∂2
∂x2Φ(0, 0) = 2µZ 6= 0.
Then by the Implicit Function Theorem for each α sufficiently small there exists a unique
C(α) near 0 ∈ Σ such that ∂x∂Φ(α, C(α)) = 0. Moreover, as Tα
X, TYα = O(α) also C(α) =
O(α).
Thus the map Φ(α, x) has a critical point at C(α) which is a maximum or minimum depending on the sign of µZ.
Lets begin with the attracting case (see Figure 2.4), that is, µZ >0. Therefore,
C(α) is a local minimum of Φ(α, .). If α < 0, then Φ(α, C(α)) = 2(Tα
Y − TXα) + O(α2) > 0,
being C(α) a minimum this means that Φ(α, x) > 0, for x ∈ Σ. Therefore there are no fixed points for φα if α < 0.
On the other hand, if α > 0 we obtain Φ(α, C(α)) < 0 and therefore Φ(α, x) has two zeros. Moreover, Φ(α, Tα
Y) < 0 and we call F (α) ∈ Dα the zero of Φ satisfying
F(α) < TYα. Therefore, the map φαZ has a fixed point which corresponds to an attracting
crossing cycle Γα, since
∂ ∂xΦ(α, F (α)) = ∂ ∂xφ α Z(F (α)) − 1 < 0.
Summarizing the case µZ > 0: for α < 0 the vector field Zα has an stable
pseudo-node and no crossing cycles Γα exist. When α > 0, the point Q(α) is a unstable
pseudo-node. Moreover, an attracting crossing cycle through the point (F (α), 0) appears. Using that Tα
X and TYα is O(α) one can compute that
F(α) = − s 2(Tα X − TYα) µZ + O(α), 0 .
An analogous reasoning leads to the unfolding when the origin is a repelling fixed point of φα
Z. As µZ < 0 the point C(α) is a local maximum of Φ(α, x). Thus an
repelling crossing cycle exists for α < 0 and no crossing cycles appear for α > 0. The nature of the pseudo-equilibrium Q(α) remains the same as in the case µZ <0, since their
stability does not depend on µZ.
Then any unfolding of Z ∈ ΛF satisfying (B), X1 · Y1(0) < 0 with µ
Z 6= 0
leads to vector fields with exactly the same topological invariants, that is, the same Σ−regions and singularities. A sketch of a versal unfolding of the invisible fold-fold satisfying X1· Y1(0) < 0 can be seen in Figure 2.5 and Figure 2.6.
Σ Σ Σ
Z ∈ V−
Z0 Z ∈ VZ0∩ ΣF Z ∈ V+Z0
Γα
Figure 2.5: The unfolding of Z ∈ ΛF satisfying (B), X1· Y1(0) < 0 and µ
Z > 0. Σ Σ Σ Z ∈ V− Z0 Z ∈ VZ0∩ Σ F Z ∈ V+ Z0 Γα
Figure 2.6: The unfolding of Z ∈ ΛF satisfying satisfying (B), X1· Y1(0) < 0 and µ
Z < 0.
Proposition 2.19. Let Z ∈ ΛF satisfying condition (B), X1 · Y1(0) > 0 and µ
Z 6= 0.
Let V0 be the neighborhood given in Theorem 2.15. Then any smooth curve
γ : α ∈ (−α0, α0) 7→ Zα ∈ V0
which is transverse to ΛF at γ(0) = Z leads to the same behaviors in V+
0 and in V − 0 .
For any Z ∈ V±
0 has two invisible folds with a crossing region between them. In both
cases, the sliding vector field has no pseudo-equilibria. Moreover, in the case µZ >0 then
Z ∈ V0+ has an “stable” pseudo-cycle and when µZ < 0 then Z ∈ V0− has a “unstable”
pseudo-cycle. Therefore, there exists a weak equivalence between any two unfoldings of Z. Proof. We are going to focus on the case X1(0) > 0. Assume that Zα ∈ V−
0 if α < 0 and
Zα ∈ V0+ if α > 0.
For α = 0, since Z0 = Z by hypothesis X1 · Y1(0) > 0. Therefore, the
discontinuity curve is decomposed as Σ = Σs∪Σe.
For α 6= 0, since for all points (x, 0) between the folds Tα
X and TYα the vector
field Zα satisfies Xα2· Yα2(x, 0) > 0. Therefore, a crossing region appears between the
folds. The discontinuity curve becomes Σ = Σe∪Σc∪Σs, where
Σe = {x ∈ Σ : x < min{Tα X, T
α
28 Σc = {x ∈ Σ : x ∈ (min{Tα X, TYα},max{TXα, TYα})}, (2.33) Σs = {x ∈ Σ : x > max{Tα X, T α Y}}. (2.34)
Using the same argument as in Proposition 2.16, there exists a unique Q(α) ∈ Σc such that det Zα(Q(α), 0) = 0. Moreover, in the escaping and sliding regions we have
(Zα)s(x) > 0. Thus no pseudo-equilibrium appear for α 6= 0.
As mentioned in Remark 2.13, one need to consider the generalized first return map for this case. For convenience, we set
φαZ = φαY ◦ φα
X = 2(TXα − TYα) + x − βX(x − TXα)2+ βY(2TXα− TYα− x)2+ O3(x, TXα, TYα)
By the same arguments of Proposition 2.18, if µZ >0, there exists an attracting
pseudo-cycle for α > 0 and for α < 0 no pseudo-cycles appear. On the other hand, if
µZ <0 then a reppelor pseudo-cycle appears for α < 0.
Then any unfolding of Z ∈ ΛF satisfying (B), X1 · Y1(0) > 0 with µ
Z 6= 0
leads to vector fields with exactly the same topological invariants, that is, the same Σ−regions and singularities. A sketch of a versal unfolding of the invisible fold-fold satisfying X1· Y1(0) > 0 can be seen in Figure 2.7 and Figure 2.8.
Σ Σ Σ Z ∈ V− Z0 Z ∈ VZ0∩ Σ F Z ∈ V+ Z0 Γα
Figure 2.7: The unfolding of Z ∈ ΛF satisfying (B), X1· Y1(0) > 0 and µ
Z < 0.
Σ Σ Σ
Z ∈ V−
Z0 Z ∈ VZ0∩ ΣF Z ∈ V+Z0
Γα
Figure 2.8: The unfolding of Z ∈ ΛF satisfying satisfying (B), X1· Y1(0) > 0 and µ
Z > 0.
2.3
The versal unfolding of a visible-invisible
fold-fold singularity
This section is devoted to the study of the unfoldings of a Filippov vector field having a visible-invisible fold point. We have essentially three different bifurcations, two of them occur when the vector fields X and Y point at opposite directions at the fold-fold point and differ in the sign of (det Z)x(0). The third occurs when both vector fields point
The main difference between the two first cases and the third one is that in the first and second ones, one needs to impose the condition (2.4) and in the third one, no generic conditions are needed, as we will see in Propositions 2.20 and 2.21.
Proposition 2.20. Let Z ∈ ΛF satisfying condition (C) and X1 · Y1(0) < 0. Let V 0 be
the neighborhood given in Theorem 2.15. Then any smooth curve γ : α ∈ (−α0, α0) 7→ Zα ∈ V0
which is transverse to ΛF at γ(0) = Z leads to the same behaviors in V+
0 and in V − 0 .
This behavior depends of the sign of (det Z)x(0):
1. If (det Z)x(0) > 0:
• Every Z ∈ V−
0 has two invisible fold points and there exists a crossing region
between them. The sliding vector field has a pseudo-saddle in the sliding region situated on “left” of both folds.
• Every Z ∈ V+
0 has two invisible fold points and there exists a crossing region
between them. The sliding vector field has a pseudo-saddle in the escaping region situated on the “right” of both folds.
2. If (det Z)x(0) < 0:
• Every Z ∈ V−
0 has two invisible fold points and there exists a crossing region
be-tween them. The sliding vector field has a unstable pseudo-node in the escaping region situated on the “right” of both folds.
• Every Z ∈ V+
0 has two invisible fold points and there exists a crossing region
between them. The sliding vector field has a stable pseudo-node in the sliding region situated on “left” of both folds.
And therefore, there exists a weak equivalence between any two unfoldings of Z. Proof. We are going to focus on the case X1(0) > 0. Assume that Zα ∈ V−
0 if α < 0 and
Zα ∈ V+
0 if α > 0.
For α = 0, since Z0 = Z by hypothesis X1· Y1(0) < 0 then the discontinuity
curve is decomposed as Σ = Σs∪Σe.
For α 6= 0, it is easy to see that for all points (x, 0) between the folds Tα X and
TYα the vector field Zα satisfies Xα2· Yα2(x, 0) > 0. Therefore, a piece of crossing region
appears between the folds for α 6= 0. The discontinuity curve becomes Σ = Σe∪Σc∪Σs,
where Σs= {x ∈ Σ : x < min{Tα X, TYα}}, Σc= {x ∈ Σ : x ∈ (min{Tα X, T α Y},max{T α X, T α Y})}, Σe= {x ∈ Σ : x > max{Tα X, T α Y}}. (2.35)
Since the generic condition for the visible-invisible case is γ 6= 0 (see (2.8)) which implies (det Z)x(0) 6= 0, the Corolary 2.10 guarantees the existence of a unique
point Q(α) ∈ Σ such that det Zα(Q(α), 0) = 0. To check if Q(α) belongs to Σs,e and
therefore it is a pseudo-equilibrium one must analyze separately the cases (det Z)x(0) > 0
and (det Z)x(0) < 0.
Suppose that (det Z)x(0) > 0 and so for |α| << 1 we have (det Z)αx(x, 0) > 0
