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.

In this thesis, we study the normally elliptic singular perturbation problems including both finite and infinite dimensional cases, which could also be non-autonomous. In particular, we establish the existence and smoothness of O(1) local invariant manifolds and provide various estimates which are independent of small singular parameters. We also use our results on local invariant manifolds to study the persistence of homoclinic solutions under weakly dissipative and conservative per- turbations. We apply Semi-group Theory and Lyapunov-Perron Integral Equations with some careful estimates to handle the O(1) driving force in the system so that we can approximate the full system through some simpler limiting system. In the investigation of homoclinics, a diagonalization procedure and some normal form transformation should be first carried out. Such diagonalization procedure is not trivial at all. We discuss this issue in the appendix. We use Melnikov type analysis to study the weakly dissipative case, while the conservative case is based on some energy methods. As a concrete example, we have shown rigrously the persistence of homoclinic solutions of an elastic pendulum model which may be affected by damping, external forcing and other potential fields.

Made available in DSpace on 2014-06-11T19:30:22Z (GMT). No. of bitstreams: 0 Previous issue date: 2006Bitstream added on 2014-06-13T20:07:43Z : No. of bitstreams: 1 carnevarollojr_rp_me_sjrp.pdf: 404426 bytes, checksum: 4ea777e4e28a9a139f9b0bcb3f7b0f7c (MD5) / O objetivo principal desse trabalho é provar, sob certas hipóteses de transversalidade e sobre os autovalores, que se uma família a um-parâmetro de equações diferenciais possuindo, para um determinado valor do parâmetro, um laço homoclínico conectado a um ponto de equilíbrio do tipo sela, então existe uma variedade central invariante, de dimensão dois, que contém o laçco homoclínico, que contém todas as trajetórias que permanecem numa vizinhança do laço homoclínico e ainda é tangente ao autoespaço gerado por autovetores associados aos autovalores que determinam o laço homoclínico. / The main goal of this work is to prove, under certain hypothesis of transversality and about the eigenvalues, that if a one-parameter family of ordinary differential equations possess, for a determined value of the parameter, a homoclinic loop connected to an equilibrium point of type saddle, then there exists an invariant center manifold, of dimension two, that contains the homoclinic loop, that contains all trajectories which stay in a small neighborhood of the homoclinic loop and that is tangent to the eigenspace spanned by the eigenvectors associated to the eigenvalues that determine the homoclinic loop.

[pt] Uma classe homoclínica de um campo vetorial é o fecho do conjunto de pontos homoclínicos transversais associados a uma órbita periódica hiperbólica. Provamos as propriedades seguintes. 1. As classes homoclínicas de campos vetoriais C¹ genéricos em variedades de dimensão n são conjuntos transitivos maximais, saturados, e isolados se e somente se omega-isolados. 2. Os campos vetorias C¹ genéricos não têm ciclos formados por classes homoclínicas. 3. As singularidades de codimensão 1, i.e., com um único autovalor positivo ou um único autovalor negativo, de campos vetoriais C¹ genéricos estão contidas em conjuntos transitivos maximais. 4. Os campos vetoriais C¹ genéricos com finitas classes homoclínicas têm finitos atratores cujas bacias formam um aberto denso da variedade. 5. Existem conjuntos localmente residuais de campos vetoriais C¹ em uma variedade de dimensão 5 exibindo finitos atratores e repulsores, porém infinitas classes homoclínicas. Conseguimos também uma condição suficiente para que um conjunto atrativo (at-tracting set) seja C 1 fracamente robusto. Observamos que esses resultados generalizam propriedades conhecidas dos campos vetoriais Axioma A. / [en] A homoclinic class of a vector field is the closure of the set of transverse homoclinic points associated to a hyperbolic periodic orbit.We prove the following properties. 1.The homoclinic classes of generic C¹ vector fields on n-manifolds are maximal transitive sets, they are satured sets and isolated if and only if (omega)-isolated . 2. Generic C¹ vector fields do not exhibit cycles associated to homoclinic classes. 3.Codimension 1 singularities, i.e. with a unique positive or negative eigenvalue, of generic C¹ vector fields are contained in maximal transitive sets. 4.Generic C¹ vector fields with finitely many homoclinic classes have finitely many attractors the union of the basins of which form an open dense set of the manifold. 5. There are locally residual sets of C¹ vector fields on a 5-manifold exhibitinf finitely many attractors and repellers but infinitely many homoclinic classes. We also show a sufficient condition for an attracting set to be C¹ weakly robust. Let us observe that these results generalize well Known properties of Axiom a vector fields. / [es] Una clase homoclínica de un campo vectorial es la clausura del conjunto de puntos homoclínicos transversales asociados a una órbita periódica hiperbólica. Fueron provadas las propriedades siguientes. 1. Las clases homoclínicas de campos vetoriales C¹ genéricos en variedades de dimensión n son conjuntos transitivos maximales, saturados, e aislados si y solo si son omega-aislados. 2. Los campos vetoriales C¹ genéricos no tienen ciclos formados por clases homoclínicas. 3. Las singularidades de codimensión 1, i.e., con un único valor propio positivo o un único valor propio negativo, de campos vectoriales C¹ genéricos están contenidas en conjuntos transitivos maximales. 4. Los campos vectoriales C¹ genéricos con finitas clases homoclínicas tienen finitos atractores cuyas bacias forman un abierto denso de la variedad. 5. Existen conjuntos localmente residuales de campos vetoriales C¹ en una variedad de dimensión 5 que exhiben finitos atratores y repulsores, no obstante infinitas clases homoclínicas. Conseguimos também una condición suficiente para que un conjunto atrativo (at-tracting set) sea C1 débilmente robusto. Observamos que esos resultados generalizan propriedades conocidas de los campos vetoriales Axioma A.

In this thesis we employ two recent analytical approaches to investigate the possible classes of traveling wave solutions of some members of recently derived integrable family of generalized Camassa-Holm (GCH) equations. In the first part, a novel application of phase-plane analysis is employed to analyze the singular traveling wave equations of four GCH equations, i.e. the possible non-smooth peakon, cuspon and compacton solutions. Two of the GCH equations do no support singular traveling waves. We generalize an existing theorem to establish the existence of peakon solutions of the third GCH equation. This equation is found to also support four segmented, non-smooth M-wave solutions. While the fourth supports both solitary (peakon) and periodic (cuspon) cusp waves in different parameter regimes. In the second part of the thesis, smooth traveling waves of the four GCH equations are considered. Here, we use a recent technique to derive convergent multi-infinite series solutions for the homoclinic and heteroclinic orbits of their traveling-wave equations, corresponding to pulse and front (kink or shock) solutions respectively of the original PDEs. Unlike the majority of unaccelerated convergent series, high accuracy is attained with relatively few terms. Of course, the convergence rate is not comparable to typical asymptotic series. However, asymptotic solutions for global behavior along a full homoclinic/heteroclinic orbit are currently not available.

Provamos que \'C POT. 1\' genericamente difeomorfismos simpléticos ou são Anosov ou possuem entropia topológica limitada por baixo pelo supremo sobre o menor expoente de Lyapunov positivo dos pontos periódicos hiperbólicos. Usando isto exibimos exemplos de difeomorfismos conservativos sobre superfícies que não são pontos de semicontinuidade superior para a entropia topológica. Provamos também que \'C POT. 1\' genericamente difeomorfismos simpléticos não Anosov não admitem extensões simbólicas. Mudando de assunto, Hayashi estendeu um resultado de Mañé, provando que todo difeomorfismo f que possui uma \'C POT. 1\' vizinhança U, onde todos os pontos periódicos de qualquer g \'PERTENCE A\' U são hiperbólicos, é de fato um difeomorfismo Axioma A. Aqui, provamos o resultado análogo a este no caso conservativo, e a partir deste é possível exibir uma demonstração de um fato \"folclore\", a conjectura de Palis no caso conservativo / We prove that a \'C POT.1\' generic symplectic diffeomorphism is either Anosov or the topological entropy is bounded from below by the supremum over the smallest positive Lyapunov exponent of the periodic points. By means of that we give examples of area preserving diffeomorphisms which are not point of upper semicontinuity of entropy function in \'C POT. 1\' topology. We also prove that \'C POT. 1\'- generic symplectic diffeomorphisms outside the Anosov ones do not admit symbolic extension. Changing of subject, Hayashi has extended a result of Mañé, proving that every diffeomorphism f which has a \'C POT. 1\'-neighborhood U, where all periodic points of any g \'IT BELONGS\' U are hyperbolic, it is an Axiom A diffeomorphism. Here, we prove the analogous result in the volume preserving scenario, and using it we prove a \"folklore\" fact, the Palis conjecture in this context

In qualitative theory of differential equations, an important role is played by special classes of solutions, like periodic solutions or solutions to some boundary value problems. When a system of ordinary differential equations has equilibria, i.e. constant solutions, whose stability properties are known, it is significant to search for connections between them by trajectories of solutions of the given system. These are called homoclinic or heteroclinic, according to whether they describe a loop based at one single equilibrium or they "start" and "end" at two distinct equilibria. This thesis is devoted to the study of heteroclinic solutions for a specific class of ordinary differential equations related to the Extended Fisher-Kolmogorov equation and the Swift-Hohenberg equation. These are semilinear fourth order bi-stable evolution equations which appear as mathematical models for problems arising in Mechanics, Chemistry and Biology. For such equations, the set of bounded stationary solutions is of great interest. These solve an autonomous fourth order equation. In this thesis, we focus on such equations having a variational structure. In that case, the solutions are critical points of an associated action functional defined in convenient functional spaces. We then look for heteroclinic solutions as minimizers of the action functional. Our main contributions concern existence and multiplicity results of such global and local minimizers in the case where the functional is defined from sign changing Lagrangians. The underlying idea is to impose conditions which imply a lower bound on the action over all admissible functions. We then combine classical arguments of the Calculus of Variations with careful estimates on minimizing sequences to prove the existence of a minimum.

Provamos que \'C POT. 1\' genericamente difeomorfismos simpléticos ou são Anosov ou possuem entropia topológica limitada por baixo pelo supremo sobre o menor expoente de Lyapunov positivo dos pontos periódicos hiperbólicos. Usando isto exibimos exemplos de difeomorfismos conservativos sobre superfícies que não são pontos de semicontinuidade superior para a entropia topológica. Provamos também que \'C POT. 1\' genericamente difeomorfismos simpléticos não Anosov não admitem extensões simbólicas. Mudando de assunto, Hayashi estendeu um resultado de Mañé, provando que todo difeomorfismo f que possui uma \'C POT. 1\' vizinhança U, onde todos os pontos periódicos de qualquer g \'PERTENCE A\' U são hiperbólicos, é de fato um difeomorfismo Axioma A. Aqui, provamos o resultado análogo a este no caso conservativo, e a partir deste é possível exibir uma demonstração de um fato \"folclore\", a conjectura de Palis no caso conservativo / We prove that a \'C POT.1\' generic symplectic diffeomorphism is either Anosov or the topological entropy is bounded from below by the supremum over the smallest positive Lyapunov exponent of the periodic points. By means of that we give examples of area preserving diffeomorphisms which are not point of upper semicontinuity of entropy function in \'C POT. 1\' topology. We also prove that \'C POT. 1\'- generic symplectic diffeomorphisms outside the Anosov ones do not admit symbolic extension. Changing of subject, Hayashi has extended a result of Mañé, proving that every diffeomorphism f which has a \'C POT. 1\'-neighborhood U, where all periodic points of any g \'IT BELONGS\' U are hyperbolic, it is an Axiom A diffeomorphism. Here, we prove the analogous result in the volume preserving scenario, and using it we prove a \"folklore\" fact, the Palis conjecture in this context

<div> <div> <div> <p>Resonant orbits in a multi-body environment have been investigated in the past to aid the understanding of perceived chaotic behavior in the solar system. The invariant manifolds associated with resonant orbits have also been recently incorporated into the design of trajectories requiring reduced maneuver costs. Poincaré sections are now also extensively utilized in the search for novel, maneuver-free trajectories in various systems. This investigation employs dynamical systems techniques in the computation and characterization of resonant orbits in the higher-fidelity Circular Restricted Three-Body model. Differential corrections and numerical methods are widely leveraged in this analysis in the determination of orbits corresponding to different resonance ratios. The versatility of resonant orbits in the design of low cost trajectories to support exploration for several planet-moon systems is demonstrated. The efficacy of the resonant orbits is illustrated via transfer trajectory design in the Earth-Moon, Saturn-Titan, and the Mars-Deimos systems. Lastly, Poincaré sections associated with different resonance ratios are incorporated into the search for natural, maneuver-free trajectories in the Saturn-Titan system. To that end, homoclinic and heteroclinic trajectories are constructed. Additionally, chains of periodic orbits that mimic the geometries for two different resonant ratios are examined, i.e., periodic orbits that cycle between different resonances are determined. The tools and techniques demonstrated in this investigation are useful for the design of trajectories in several different systems within the CR3BP. </p> </div> </div> </div>

No description available.