Le but de cette thèse est d'étudier les structures symplectiques dans la catégorie des variétés linéaires par morceaux (PL). La question centrale est de déterminer si toute variété symplectique lisse $(M,omega)$ peut être triangulée de manière symplectique, au sens où il existe une variété linéaire par morceaux $K$ et une triangulation $h :K -> M$ telle que $h^*omega$ est une forme symplectique constante par morceaux. Nous étudions d'abord un problème plus simple, qui consiste à trianguler les formes volumes lisses. Étant donnée une variété lisse $M$ munie d'une forme volume $Omega$, nous montrons qu'il existe une triangulation lisse $h :K -> M$ telle que $h^*Omega$ est une forme volume constante par morceaux. En particulier, les variétés symplectiques lisses de dimension 2 admettent donc des triangulations symplectiques. Étant donnée une variété symplectique fermée $(M,omega)$, nous montrons ensuite que pour certaines triangulations lisses $h :K -> M$, on peut, par une modification arbitrairement petite du complexe $K$, supposer que la forme $h^*omega$ est de rang maximal le long de tous les simplexes de $K$. Ce résultat permet d'approximer arbitrairement bien toute variété symplectique fermée par une variété symplectique PL. Nous nous intéressons finalement au cas d'une sous-variété symplectique $M$ d'un espace ambiant qui admet lui-même une triangulation symplectique. Nous montrons qu'il est possible de construire un cobordisme entre la sous-variété $M$ considérée et une approximation lisse par morceaux de celle-ci, triangulée par un complexe symplectique. / In this thesis, we study symplectic structures in a piecewise linear (PL) setting. The central question is to determine whether a smooth symplectic manifold can be triangulated symplectically, in the sense that there exists a triangulation $h :K -> M$ such that $h^*omega$ is a piecewise constant symplectic form on $K$. We first focus on a simpler related problem, and show that any smooth volume form $Omega$ on $M$ can be triangulated. This means that there always exists a triangulation $h :K -> M$ such that $h^*Omega$ is a piecewise constant volume form. In particular, symplectic surfaces admit symplectic triangulations. Given a closed symplectic manifold $(M,omega)$, we then prove that there exists triangulations $h :K -> M$ for which the piecewise smooth form $h^*omega$ has maximal rank along all the simplices of $K$. This result allows to approximate arbitrarily closely any closed symplectic manifold by a PL one. Finally, we investigate the case of a symplectic submanifold $M$ of an ambient space which is itself symplectically triangulated, and give the construction of a cobordism between $M$ and a piecewise smooth approximation of $M$, triangulated by a symplectic complex. / Doctorat en Sciences / info:eu-repo/semantics/nonPublished

After introducing concepts from convex analysis, we study how to continuously transform one convex function into another. A natural choice is the arithmetic average, as it is pointwise continuous; however, this choice fails to average functions with different domains. On the contrary, the proximal average is not only continuous (in the epi-topology) but can actually average functions with disjoint domains. In fact, the proximal average not only inherits strict convexity (like the arithmetic average) but also inherits smoothness and differentiability (unlike the arithmetic average). Then we introduce a computational framework for computer-aided convex analysis. Motivated by the proximal average, we notice that the class of piecewise linear-quadratic (PLQ) functions is closed under (positive) scalar multiplication, addition, Fenchel conjugation, and Moreau envelope. As a result, the PLQ framework gives rise to linear-time and linear-space algorithms for convex PLQ functions. We extend this framework to nonconvex PLQ functions and present an explicit convex hull algorithm. Finally, we discuss a method to find primal-dual symmetric antiderivatives from cyclically monotone operators. As these antiderivatives depend on the minimal and maximal Rockafellar functions [5, Theorem 3.5, Corollary 3.10], it turns out that the minimal and maximal function in [12, p.132,p.136] are indeed the same functions. Algorithms used to compute these antiderivatives can be formulated as shortest path problems.

Inference of regulatory relations in dynamical systems is a promising active research area. Recently, most of the investigations in this field have been stimulated by the researches in functional genomics. In this thesis, the inferential modeling problem for switching hybrid systems is studied. The hybrid systems refers to dynamical systems in which discrete and continuous variables regulate each other, in other words the jumps and flows are interrelated. In this study, piecewise linear approximations are used for modeling purposes and it is shown that piecewise linear models are capable of displaying the evolutionary characteristics of switching hybrid systems approxi- mately. For the mentioned systems, detection of switching instances and inference of locally linear parameters from empirical data provides a solid understanding about the system dynamics. Thus, the inference methodology is based on these issues. The primary difference of the inference algorithm is the idea of transforming the switch- ing detection problem into a jump detection problem by derivative estimation from discrete data. The jump detection problem has been studied extensively in signal processing literature. So, related techniques in the literature has been analyzed care- fully and suitable ones adopted in this thesis. The primary advantage of proposed method would be its robustness in switching detection and derivative estimation. The theoretical background of this robustness claim and the importance of robustness for real world applications are explained in detail.

Many dynamical systems in nature and technology involve delays in the interaction of variables forming the system. Furthermore, many of such systems involve external inputs or perturbations which might force the system to have arbitrary initial function. The conventional way to model these systems is using delay differential equations (DDE). However, DDEs with arbitrary initial functions has serious problems for finding analytical and computational solutions. This fact is a strong motivation for considering abstractions and approximations for dynamical systems involving delay. In this thesis, the piecewise linear systems with delay on piecewise constant part which is a useful subclass of hybrid dynamical systems is studied. We introduced various representations of these systems and studied the state transition conditions. We showed that there exists fixed point and periodic stable solutions. We modelled the genomic regulation of fission yeast cell cycle. We discussed various potential uses including approximating the DDEs and finally we concluded.

Regulatory processes and history dependent behavior appear in many dynamical systems in nature and technology. For modeling regulatory processes, hybrid systems offer several advances. From this point of view, to observe the capability of hybrid systems in a history dependent system is a strong motivation. In this thesis, we developed functional hybrid systems which exhibit memory dependent behavior such that the dynamics of the system is determined by both the location of the state vector and the memory. This property was explained by various examples. We used the hybrid system with memory in modeling the gene regulatory network of human immune response to Influenza A virus infection. We investigated the sensitivity of the piecewise linear model with memory. We introduced how the model can be developed in future.

After introducing concepts from convex analysis, we study how to continuously transform one convex function into another. A natural choice is the arithmetic average, as it is pointwise continuous; however, this choice fails to average functions with different domains. On the contrary, the proximal average is not only continuous (in the epi-topology) but can actually average functions with disjoint domains. In fact, the proximal average not only inherits strict convexity (like the arithmetic average) but also inherits smoothness and differentiability (unlike the arithmetic average). Then we introduce a computational framework for computer-aided convex analysis. Motivated by the proximal average, we notice that the class of piecewise linear-quadratic (PLQ) functions is closed under (positive) scalar multiplication, addition, Fenchel conjugation, and Moreau envelope. As a result, the PLQ framework gives rise to linear-time and linear-space algorithms for convex PLQ functions. We extend this framework to nonconvex PLQ functions and present an explicit convex hull algorithm. Finally, we discuss a method to find primal-dual symmetric antiderivatives from cyclically monotone operators. As these antiderivatives depend on the minimal and maximal Rockafellar functions [5, Theorem 3.5, Corollary 3.10], it turns out that the minimal and maximal function in [12, p.132,p.136] are indeed the same functions. Algorithms used to compute these antiderivatives can be formulated as shortest path problems.

Orientador: Paulo Ricardo da Silva / Banca: Weber Flavio Pereira / Banca: Marcelo Messias / Resumo: Consideramos dois casos principais de bifurcação de órbitas periódicas não hiperbólicas que dão origem a ciclos limite. Nosso estudo é feito para sistemas lineares por partes com três zonas em sua fórmula mais geral, que inclui situações sem simetria. Obtemos estimativas tanto para a amplitude como para o período do ciclo limite e apresentamos uma aplicação de interesse em engenharia: sistemas de controle. / Abstract: We consider two main cases of bifurcation of non hyperbolic periodic orbits that give rise to limit cycles. Our study is done concerning piecewise linear systems with three zones in the more general formula that includes situations without symmetry. We obtain estimates for both the amplitude and the period of limit cycles and we present a applications of interest in engineering: control systems. / Mestre

Orientador: Luci Any Francisco Roberto / Coorientador: Claudio Aguinaldo Buzzi / Banca: Ana Cristina Mereu / Banca: Claudio Gomes Pessoa / Resumo: Neste trabalho temos como objetivo estudar o número e a distribuição de ciclos limites em sistemas diferenciais lineares por partes. Em particular estudamos o número de ciclos limites do sistema diferencial linear por partes planar ˙x = −y − ε φ ( x) , ˙y = x, onde ε 6= 0 é um parâmetro pequeno e φ é uma função periódica linear por partes ímpar de período 4 . Provamos que dado um inteiro arbitário positivo n, o sistema acima possui exatamente n ciclos limites na faixa |x| ≤ 2 (n + 1 ). Consequentemente, existem sistemas diferenciais lineares por partes contendo uma infinidade de ciclos limites no plano real. Inicialmente obtemos uma quota inferior par a o número destes ciclos limites na faixa | x| ≤ 2 (n + 1 ) via Teoria do Averaging . Em seguida , utilizando a Teoria de Campos de Vetores Rodados, verificamos que o sistema acima tem exatamente n ciclos limites na faixa | x| ≤ 2 (n + 1 ) / Abstract: The main goal of this work aim to study the number and distribution of limit cycles in piecewise linear differential systems. In particular we consider the planar piecewise linear differential system ˙x = −y − ε φ ( x) , ˙y = x, where ε 6= 0 is a small parameter and φ is an odd piecewise linear periodic function of period 4 . We prove that given an arbitrary positive integer n, the system above has exactly n limit cycles in the strip | x| ≤ 2 (n + 1 ) . Consequently, there are piecewise differential systems containing an infinite number of limit cycles in the real plane. First we get a lower bound on the number of limit cycles in the strip |x| ≤ 2 (n + 1 ) via Averaging Theory. In the following , using the Theory of Rotated Vector Fields, we see that above system has exactly n limit cycles in the strip | x| ≤ 2 (n + 1 ) / Mestre

Estudamos uma classe de campos de vetores seccionalmente lineares no plano denotada por X. Tais campos aparecem frequentemente em modelos matemáticos aplicados à engenharia. Baseados no trabalho de J. Sotomayor e R. Garcia [SG03], impondo condições sobre as singularidades, órbitas periódicas e separatrizes, definimos um conjunto de campos de vetores que são estruturalmente estáveis em X. Provamos que esse conjunto é aberto, denso e tem medida de Lebesgue total em X, o qual é um espaço vetorial de dimensão finita. / We study a class of piecewise-linear vector fields in the plane denoted by X. These vector fields appear often in mathematical models applied to Engineering. Based on Jorge Sotomayor and Ronaldo Garcia paper [SG03], we impose conditions on singularities, periodic orbits and separatrices, to define a set of vector fields structurally stable in X. We give a proof that this set is open, dense and has full Lebesgue measure in X, that is a finite dimensional vector space.

After introducing concepts from convex analysis, we study how to continuously transform one convex function into another. A natural choice is the arithmetic average, as it is pointwise continuous; however, this choice fails to average functions with different domains. On the contrary, the proximal average is not only continuous (in the epi-topology) but can actually average functions with disjoint domains. In fact, the proximal average not only inherits strict convexity (like the arithmetic average) but also inherits smoothness and differentiability (unlike the arithmetic average). Then we introduce a computational framework for computer-aided convex analysis. Motivated by the proximal average, we notice that the class of piecewise linear-quadratic (PLQ) functions is closed under (positive) scalar multiplication, addition, Fenchel conjugation, and Moreau envelope. As a result, the PLQ framework gives rise to linear-time and linear-space algorithms for convex PLQ functions. We extend this framework to nonconvex PLQ functions and present an explicit convex hull algorithm. Finally, we discuss a method to find primal-dual symmetric antiderivatives from cyclically monotone operators. As these antiderivatives depend on the minimal and maximal Rockafellar functions [5, Theorem 3.5, Corollary 3.10], it turns out that the minimal and maximal function in [12, p.132,p.136] are indeed the same functions. Algorithms used to compute these antiderivatives can be formulated as shortest path problems. / Graduate Studies, College of (Okanagan) / Graduate