Complex oscillations in the delayed Fitzhugh-Nagumo equation

Motivated by the dynamics of neuronal responses, we analyze the dynamics of the Fitzhugh-Nagumo slow-fast system with delayed self-coupling. This system provides a canonical example of a canard explosion for sufficiently small delays. Beyond this regime, delays significantly enrich the dynamics, leading to mixed-mode oscillations, bursting and chaos. These behaviors emerge from a delay-induced subcritical Bogdanov-Takens instability arising at the fold points of the S-shaped critical manifold. Underlying the transition from canard-induced to delay-induced dynamics is an abrupt switch in the nature of the Hopf bifurcation

Abelian Integral Method and its Application

Oscillation is a common natural phenomenon in real world problems. The most efficient mathematical models to describe these cyclic phenomena are based on dynamical systems. Exploring the periodic solutions is an important task in theoretical and practical studies of dynamical systems. Abelian integral is an integral of a polynomial differential 1-form over the real ovals of a polynomial Hamiltonian, which is a basic tool in complex algebraic geometry. In dynamical system theory, it is generalized to be a continuous function as a tool to study the periodic solutions in planar dynamical systems. The zeros of Abelian integral and their distributions provide the number of limit cycles and their locations. In this thesis, we apply the Abelian integral method to study the limit cycles bifurcating from the periodic annuli for some hyperelliptic Hamiltonian systems. For two kinds of quartic hyperelliptic Hamiltonian systems, the periodic annulus is bounded by either a homoclinic loop connecting a nilpotent saddle, or a heteroclinic loop connecting a nilpotent cusp to a hyperbolic saddle. For a quintic hyperelliptic Hamiltonian system, the periodic annulus is bounded by a more degenerate heteroclinic loop, which connects a nilpotent saddle to a hyperbolic saddle. We bound the number of zeros of the three associated Abelian integrals constructed on the periodic structure by employing the combination technique developed in this thesis and Chebyshev criteria. The exact bound for each system is obtained, which is three. Our results give answers to the open questions whether the sharp bound is three or four. We also study a quintic hyperelliptic Hamiltonian system with two periodic annuli bounded by a double homoclinic loop to a hyperbolic saddle, one of the periodic annuli surrounds a nilpotent center. On this type periodic annulus, the exact number of limit cycles via Poincar{\\u27e} bifurcation, which is one, is obtained by analyzing the monotonicity of the related Abelian integral ratios with the help of techniques in polynomial boundary theory. Our results give positive answers to the conjecture in a previous work. We also extend the methods of Abelian integrals to study the traveling waves in two weakly dissipative partial differential equations, which are a perturbed, generalized BBM equation and a cubic-quintic nonlinear, dissipative Schr\ {o}dinger equation. The dissipative PDEs are reduced to singularly perturbed ODE systems. On the associated critical manifold, the Abelian integrals are constructed globally on the periodic structure of the related Hamiltonians. The existence of solitary, kink and periodic waves and their coexistence are established by tracking the vanishment of the Abelian integrals along the homoclinic loop, heteroclinic loop and periodic orbits. Our method is novel and easily applied to solve real problems compared to the variational analysis

Qualitative Analysis of Polycycles in Filippov Systems

In this paper, we are concerned about the qualitative behaviour of planar Filippov systems around some typical minimal sets, namely, polycycles. In the smooth context, a polycycle is a simple closed curve composed by a collection of singularities and regular orbits, inducing a first return map. Here, this concept is extended to Filippov systems by allowing typical singularities lying on the switching manifold. Our main goal consists in developing a method to investigate the unfolding of polycycles in Filippov systems. In addition, we applied this method to describe bifurcation diagrams of Filippov systems around certain polycycles

Global dynamics of a SD oscillator

In this paper we derive the global bifurcation diagrams of a SD oscillator which exhibits both smooth and discontinuous dynamics depending on the value of a parameter a. We research all possible bifurcations of this system, including Pitchfork bifurcation, degenerate Hopf bifurcation, Homoclinic bifurcation, Double limit cycle bifurcation, Bautin bifurcation and Bogdanov-Takens bifurcation. Besides we prove that the system has at most five limit cycles. At last, we give all numerical phase portraits to illustrate our results

Sobre ciclos degenerados em campos vetoriais descontínuos e o problema de Dulac

Orientadores: Marco Antonio Teixeira, Ricardo Miranda Martins, Mike R. JeffreyTese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação CientíficaResumo: Neste trabalho, estuda-se ciclos que ocorrem tipicamente em campos vetoriais descontínuos, planares definidos em duas zonas, Z=(X,Y), com variedade de descontinuidade dada pela imagem inversa do 0 por uma função suave h, definida no plano e assumindo valores reais, para a qual 0 é um valor regular. Primeiramente, mostra-se que, se X e Y são campos vetoriais analíticos e C é um policiclo de Z, então, genericamente, não existem ciclos limite se acumulando em C. Depois disso, o objetivo é estudar bifurcações de ciclos típicos contendo um ponto do tipo sela-regular. Mais especificamente, considera-se ciclos compostos por um segmento de órbita regular de Z, que cruza a variedade de descontinuidade transversalmente, e um ponto do tipo sela-regular resultando numa conexão quase-homoclínica. São apresentados diagramas de bifurcação para o caso onde o raio de hiperbolicidade do ponto de sela é um número irracional, o caso onde o raio de hiperbolicidade da sela é um número racional é ilustrado com alguns modelos. Finalmente, dois modelos comuns em aplicações e que apresentam tal ciclo são estudados por meio de cálculos numéricosAbstract: In this work, a study is performed on cycles occurring typically in planar discontinuous vector fields in two zones, Z=(X,Y), with switching manifold being the inverse image of 0 by a smooth function h, defined on the plane and assuming real values, for which 0 is a regular value. Firstly, it is shown that if X and Y are analytic vector fields and C is a polycycle of Z, then, generically, C cannot have limit cycles accumulating onto it. After that, the objective is to study the bifurcations of typical cycles through a saddle-regular point. More specifically, we consider a cycle composed by one segment of a regular orbit of Z, which crosses the switching manifold transversally, and a saddle-regular point, resulting in a homoclinic-like connection. Bifurcation diagrams are presented for the case where the hyperbolicity ratio of the saddle point is a irrational number, the case where hyperbolicity ratio is a rational number is illustrated with models. Finally, two application models presenting cycles through saddle-regular points are studied by means of numeric calculationsDoutoradoMatematicaDoutora em Matemática2013/07523-9FAPESPCAPE

Sobre condições de estabilidade para sistemas de Filippov e sistemas hamiltonianos

Orientadores: Marco Antonio Teixeira, Maria Teresa Martinez-Seara Alonso e Marcel Guardia MunarrizTese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação CientíficaResumo: Neste trabalho, abordamos aspectos qualitativos de vários fenômenos em sistemas de Filippov e em sistemas Hamiltonianos. No contexto de sistemas dinâmicos suaves por partes, concentramos nossa atenção em problemas em dimensões 2 e 3. No caso planar, desenvolvemos um mecanismo para analisar o desdobramento de policiclos que passam por certas singularidades de sistemas de Filippov (conhecidas como ?-singularidades) em uma configuração típica, e o utilizamos para descrever completamente o diagrama de bifurcação de sistemas de Filippov em torno de alguns policiclos elementares. No caso tridimensional, obtivemos uma caracterização completa dos sistemas que são localmente estruturalmente estáveis em um ponto ???? da variedade de descontinuidade. Mais ainda, caracterizamos completamente os sistemas de Filippov robustos em uma vizinhança da variedade de descontinuidade, os quais são chamados de sistemas semi-localmente estruturalmente estáveis. Além disso, estudamos alguns fenômenos globais em sistemas de Filippov 3????. Primeiramente, descrevemos o diagrama de bifurcação de um sistema em torno de um laço ("loop") do tipo homoclínico de codimensão um em uma singularidade genérica denominada dobra-regular, o qual não possui contrapartida no contexto suave. Em seguida, analisamos uma classe de sistemas que apresenta conexões robustas entre certas singularidades típicas, conhecidas como ????-singularidades, as quais garantiram a existência de um comportamento caótico nas folheações associadas a tais sistemas de Filippov. Em relação aos sistemas Hamiltonianos, estudamos alguns problemas que apresentam fenômenos exponencialmente pequenos. Mais especificamente, consideramos um modelo de interação kink-defect dado por um Hamiltoniano singularmente perturbado ???????? (???? ? 0 representa o parâmetro perturbativo) com dois graus de liberdade, e determinamos condições sobre a energia do sistema para a existência de certas conexões heteroclínicas que surgem da quebra (???? > 0) de uma órbita heteroclínica contida no nível de energia zero do sistema limite ????0. Finalmente, investigamos a existência de soluções breather de equações diferenciais parciais reversíveis do tipo Klein-Gordon, as quais podem ser vistas como órbitas homoclínicas de um sistema Hamiltoniano de dimensão infinitaAbstract: In this work, we discussed qualitative aspects of several phenomena in Filippov and Hamiltonian systems. In the context of piecewise smooth dynamical systems, we have focused on problems in dimensions 2 and 3. In the planar case, we have provided a mechanism to analyze the unfolding of polycycles passing through certain singularities of Filippov systems (known as ?-singularities) in a typical scenario and we have used it to completely describe the bifurcation diagram of Filippov systems around some elementary polycycles. In the three-dimensional case, we have obtained a complete characterization of the systems which are locally structurally stable at a point ???? in the switching manifold ?. Moreover, we have completely characterized the Filippov systems which are robust in a neighborhood of the whole switching manifold, named semi-local structurally stable systems. In addition, we have studied some global phenomena in 3???? Filippov systems. First we described the bifurcation diagram of a system around a codimension one homoclinic-like loop at a generic singularity named fold-regular singularity, which has no counterpart in the smooth context. Second, we analyzed a class of systems presenting robust connections between certain typical singularities, known as ????-singularities, which have lead us to the existence of a chaotic behavior in the foliations associated to such Filippov systems. Concerning to Hamiltonian Systems, we have studied some problems exhibiting exponentially small phenomena. More specifically, we considered a model of kink-defect interaction given by a singularly perturbed 2-dof Hamiltonian ???????? (???? ? 0 stands for the perturbation parameter) and we have provided conditions on the energy of the system for the existence of certain heteroclinic connections arising from the breakdown (???? > 0) of a heteroclinic orbit lying in the zero energy level of the limit system ????0. Finally, we have investigated the existence of breathers of reversible Klein-Gordon partial differential equations, which can be seen as homoclinic orbits of an infinite-dimensional Hamiltonian systemDoutoradoMatematicaDoutor em Matemática2015/22762-5CAPESFAPES