Made available in DSpace on 2014-06-11T19:26:56Z (GMT). No. of bitstreams: 0 Previous issue date: 2006-03-03Bitstream added on 2014-06-13T20:27:28Z : No. of bitstreams: 1 bronzi_ma_me_sjrp.pdf: 904425 bytes, checksum: 2344eb35a112034c2f1741b2e229f1ec (MD5) / Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) / Estudamos intersecções homoclínicas de variedades estável e instável de pontos peródicos. Toda intersecção homoclínica produz um comportamento curioso na dinâmiôa. Nosso modelo de tal fenômeno é a famosa ferradura de Smale, a qual é um conjunto hiperbólico para um difeomorfismo. Além disso, estudamos dinâmica não hiperbólica cuja perda de hiperbolicidade é divido à tangências homoclínicas. Elas tem um papel central na teoria de sistemas dinâmicos. O desdobramento de uma tangência homoclínica produz dinâmicas muito interessantes. Neste trabalho estudamos a criação de cascatas de bifurcações de duplicação de período e um esquema de renormalização para uma tangência homoclínica. / We study homoclinic intersection of stable and unstable manifolds of periodic points. Every homoclinic intersection produce a intricate behavior of the dynamics. Our model of such phenomena is the so called Smalesþs horseshoe, which is a hyperbolic set for a di eomorphism. We also study non hyperbolic dynamics whose lack of hyperbolicity is due to homoclinic tangencies. They play a central role in the theory of dynamical systems. The unfolding of a homoclinic tangency produce many interesting dynamics. In this work we study creation of cascade of period doubling bifurcations and a renormalization scheme for a homoclinic tangency.

We describe some classical results on global injectivity of local dieomorphism on Euclidian spaces. This is not exhaustive, and it does not purport to be a complete history, it simply describes some useful results in injectivity. The first part describes some results related to the Qualitative Theory of Diferential Equations, and presents a characterization of global injectivity on planar applications by using the existence of an isochronous global center. The Global Asymptotic Stability Problem is also described. The second part describes the so called Palais-Smale condition. / Se dan algunos teoremas que garantizan la inyectividad global de los difeomorsmos locales en espacios euclidanos. De momento el trabajo no es aun exaustivo, ni pretende serlo, simplemente se describe algunos resultados utiles en la teoría del estudio de las aplicaciones inyectivas. La primera parte describe algunos resultados relacionados con la teoría cualitativa de las ecuaciones diferenciales, y presenta una caracterización de la inyectividad global de aplicaciones en el plano por medio de la existencia de un centro global isócrono. También se presenta el problema de la estabilidada sintóotica global. La segunda parte describe la "condicion de Palais Smale".

Neste trabalho estudamos problemas de reação difusão em domínios unidimensionais que surgem de materiais compostos e obtemos resultados comparando os fluxos do problema original e do problema limite quando a difusão fica muito grande em partes do domínio. Provamos que os autovalores e autofunções do operador linear ilimitado associado à equação limite têm a propriedade de Sturm Liouville e provamos que as soluções do problema de reação difusão têm a propriedade do decrescimento do número de zeros ao longo do tempo. Estes resultados são usados para provar que as variedades instável e estável de pontos de equilíbrios são genericamente transversais e que o fluxo no atrator para o problema de reação difusão é genericamente estruturalmente estável. Estes fatos permitem obter a equivalência topológica dos fluxos restritos aos atratores dos problemas original e seu problema limite. / In this work we study some reaction-difusion problems in one dimensional domains that arise from composite materials. We obtain some results comparing the flux of the original problem and the flux of the limit problem when the difusion becomes large on parts of the physical domain. We prove that the eigenvalues and eigenfunctions of the linear unbounded operator associated with the equation have the Sturm Liouville property and also that the solutions of the reaction difusion equation have the property that the zeros do not increase with time. These results are used to obtain that the stable and unstable manifolds of equilibrium points are generically transversal and that the flux on the attractor for the reaction difusion problem is generically structurally stable. Using this we are able to prove the topological equivalence of the fluxs restricted to the attractors of the original and the limit problem.

Orientador: Ketty Abaroa de Rezende / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-24T17:12:31Z (GMT). No. of bitstreams: 1 EspirituLedesma_GuidoGerson_D.pdf: 1229937 bytes, checksum: 00f2d538b5b2a2c4147d828351f4ef16 (MD5) Previous issue date: 2014 / Resumo: Neste trabalho, usamos os grafos de Lyapunov como uma ferramenta combinat{\'o}ria para obter classifica\c{c}{\~o}es completas de fluxos Smale sobre $\ss$ e fluxos Morse-Novikov sobre superf{\'i}cies orient{\'a}veis e n{\~a}o orient{\'a}veis. Esta classifica\c{c}{\~a}o consiste em obter condi\c{c}{\~o}es necess{\'a}rias e suficientes que devem ser satisfeitas por um grafo de Lyapunov abstrato de forma a ser associado a um fluxo Smale sobre $\ss$ ou um fluxo Morse-Novikov sobre uma superf{\'i}cie respectivamente. Assim nesta tese de doutorado obtemos os seguintes resultados: \begin{enumerate} \item As condições locais que devem ser satisfeitas por cada vértice do grafo de Lyapunov, assim como as condições globais que devem ser satisfeitas pelos grafos para estarem associados a um fluxo Smale sobre $\ss$ ou a um fluxo Morse-Novikov sobre uma superfície s{\~a}o determinadas. \item A realização destes grafos abstratos sujeita {\'a}s condições determinadas acima, como fluxos Smale sobre $\ss$ ou fluxos Morse-Novikov sobre superfícies respectivamente, são obtidas. \end{enumerate} / Abstract: In this work Lyapunov graphs are used as a combinatorial tool in order to obtain a complete classification of Smale flows on $\ss$ and Morse-Novikov flows on orientable and non-orientable surfaces. This classification consists in determining necessary and sufficient conditions that must be satisfied by an abstract Lyapunov graph so that it is associated to a Smale flow on $\ss$ or to a Morse-Novikov flow on a surface respectively.\\ In summary in this doctoral thesis we obtain the following results: \begin{enumerate} \item The local conditions that must be satisfied by each vertex on a Lyapunov graph is determinated as well as the global conditions on the graph in order for it to be associated to a Smale flow on $\ss$ or a Morse-Novikov flow on a surface. \item The realization of these graphs subject to the conditions found above as Smale flows on $\ss$ or as Morse-Novikov flows on surfaces respectively is obtained. \end{enumerate} / Doutorado / Matematica / Doutor em Matemática

Cette thèse porte sur deux aspects de la théorie de Morse: théorie de Morse discrète de Forman (cas de la dimension finie) et homologie de Heegaard-Floer (cas de la dimension infinie).<br />Dans une première partie, on s'intéresse au problème de relèvement de signe pour l'homologie de Heegaard-Floer combinatoire. On montre que la construction originale faite par Manolescu, Ozsváth, Szabó et D. Thurston peut être refaite de manière plus conceptuelle. On donne ensuite le lien entre ces deux constructions puis finalement on décrit un algorithme qui permet de calculer les signes.<br />La seconde partie porte sur la théorie de Morse discrète définie par Forman. Après avoir fait le lien entre l'algèbre sur les complexes de chaînes et la théorie de Morse discrète, on montre que le complexe de Thom-Smale donné par une fonction de Morse lisse sur variété lisse close peut être réalisé par une triangulation et une fonction de Morse discrète sur celle-ci. On utilise cela pour obtenir une représentation particulière sous forme de couplage complet de toute structure d'Euler sur une variété de dimension 3 close orientée.

We study the hyperbolic dynamical systems known as Smale spaces. More specifically we investigate the C*-algebras constructed from these systems. The K group of one of these algebras has a natural ring structure arising from an asymptotically abelian property. The K groups of the other algebras are then modules over this ring. In the case of a shift of finite type we compute these structures explicitly and show that the stable and unstable algebras exhibit a certain type of duality as modules. We also investigate the Bowen measure and its stable and unstable components with respect to resolving factor maps, and prove several results about the traces that arise as integration against these measures. Specifically we show that the trace is a ring/module homomorphism into R and prove a result relating these integration traces to an asymptotic of the usual trace of an operator on a Hilbert space.

We study aspects of noncommutative geometry on hyperbolic dynamical systems known as Smale spaces. In particular, there are two C*-algebras, defined on the stable and unstable groupoids arising from the hyperbolic dynamics. These give rise to two additional crossed product C*-algebras known as the stable and unstable Ruelle algebras. We show that the Ruelle algebras exhibit noncommutative Poincaré duality. As a consequence we obtain isomorphisms between the K-theory and K-homology groups of the stable and unstable Ruelle algebras. A second result defines spectral triples on these C*-algebras and we show that the spectral dimension recovers the topological entropy of the Smale space itself. Finally we define a natural Fredholm module on the Ruelle algebras in the special case that the Smale space is a shift of finite type. Using unitary operators arising from the Pimsner-Voiculescu sequence we compute the index pairing with our Fredholm module for specific examples.

We consider a general class of finite dimensional deterministic dynamical systems with finitely many local attractors each of which supports a unique ergodic probability measure, which includes in particular the class of Morse–Smale systems in any finite dimension. The dynamical system is perturbed by a multiplicative non-Gaussian heavytailed Lévy type noise of small intensity ε > 0. Specifically we consider perturbations leading to a Itô, Stratonovich and canonical (Marcus) stochastic differential equation. The respective asymptotic first exit time and location problem from each of the domains of attractions in case of inward pointing vector fields in the limit of ε-> 0 has been investigated by the authors. We extend these results to domains with characteristic boundaries and show that the perturbed system exhibits a metastable behavior in the sense that there exits a unique ε-dependent time scale on which the random system converges to a continuous time Markov chain switching between the invariant measures. As examples we consider α-stable perturbations of the Duffing equation and a chemical system exhibiting a birhythmic behavior.

Submitted by Santos Davilene (davilenes@ufba.br) on 2017-06-01T20:01:50Z No. of bitstreams: 1 Tese - Wescley.pdf: 2339393 bytes, checksum: 22bfa3dae7b30d963d35d1f54414a3df (MD5) / Approved for entry into archive by Vanessa Reis (vanessa.jamile@ufba.br) on 2017-06-07T11:19:46Z (GMT) No. of bitstreams: 1 Tese - Wescley.pdf: 2339393 bytes, checksum: 22bfa3dae7b30d963d35d1f54414a3df (MD5) / Made available in DSpace on 2017-06-07T11:19:46Z (GMT). No. of bitstreams: 1 Tese - Wescley.pdf: 2339393 bytes, checksum: 22bfa3dae7b30d963d35d1f54414a3df (MD5) / O conteúdo desta Tese está relacionado a versão da conjectura de Smale sobre a trivialidade do centralizador para certas classes de fluxos e ações de Rn. A conjectura de Smale estabelece que a maioria dos sistemas dinâmicos tem centralizador trivial, significando que toda a dinâmica que comuta com a original é um reescalonamento temporal da mesma. Neste trabalho, mostramos a trivialidade do centralizador para as seguintes classes de sistemas dinâmicos: (i) conjunto aberto de campos de classe C1 com singularidades hiperbólicas não-ressonantes e que satisfazem a Komuro-expansividade, os quais contém o atrator de Lorenz clássico como caso particular; (ii) conjunto Baire residual de campos conservativos de classe C1; (iii) conjunto Baire residual de campos hamiltonianos de classe C1. Além disso, provamos que o conjunto das ações de Rd localmente livres, expansivas e de classe C1 têm centralizador quase-trivial. Em particular, obtivemos os seguintes: (i) Rd-ações Anosov transitivas em variedades compactas têm centralizador quase-trivial; (ii) caracterização de sub-ações expansivas de ações de Rd.

We model hyperbolic toral automorphisms by two types of Smale spaces; shifts of finite type and substitution tilings spaces. Smale spaces are dynamical systems with local hyperbolic product structure. In 1970, Bowen showed that an irreducible Smale space is a factor of a shift of finite type by showing that it has Markov partitions. Putnam extended Bowen's theorem by showing that every irreducible Smale space has a factor map that can be split into a s-bijective and u-bijective map; thereby better modelling a Smale space on its characterizing expanding and contracting spaces separately. In this thesis, we define two new constructions of Markov partitions for hyperbolic toral automorphisms inspired by the work of Adler, Weiss, and Praggastis. With one of the constructions, we investigate when a factor map from a shift of finite type to a hyperbolic toral automorphism can be written as a composition of a s-bijective and u-bijective map and we show that if such a splitting exists then the Markov partition must satisfy a Border Continuity condition. The second construction can be thought of as an explicit example of Putnam's theorem for the case of hyperbolic toral automorphisms whose defining matrix is in dimension 2 and has positive entries. We define a full splitting for all such hyperbolic toral automorphisms with one exception; the Arnold Cat map. / Graduate