Orientador: Marco Antonio Teixeira / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-10T12:10:27Z (GMT). No. of bitstreams: 1 Roberto_LuciAnyFrancisco_D.pdf: 627926 bytes, checksum: 0c8cb4e26df805282fa716847859d82f (MD5) Previous issue date: 2008 / Resumo: Neste trabalho estudamos sistemas dinâmicos possuindo estruturas Hamiltonianas e reversíveis( / Abstract: In this work we study dynamical systems possessing Hamiltonian and time-reversible structures. The reversibility concept is de¯ned in terms of an involution. Initially we discuss the dynamics of Hamiltonian vector ¯elds with 2 and 3 degrees of freedom around an elliptic equilibrium in the presence of an involution which preserves the symplectic structure. The main results discuss the existence of one-parameter families of reversible periodic solutions terminating at the equilibrium. The main techniques that are used in the proofs are Belitskii and Birkho® normal forms and the Liapunov-Schmidt Reduction. Next we consider a case of the 3-body restricted problem in rotating coordinates. In this case the two primaries are oving in an elliptic collision orbit. By the continuation method of Poincare we characterize that the periodic circular orbits and the symmetric periodic elliptic orbits from the Kepler problem which can be prolonged to pseudo periodic orbits of the planar restricted 3{body problem in rotating coordinates with the two primaries moving in an elliptic collision orbit / Doutorado / Topologia e Geometria / Doutor em Matemática

Orientador: Marco Antonio Teixeira / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-05T21:55:10Z (GMT). No. of bitstreams: 1 Lima_MauricioFirminoSilva_D.pdf: 1170094 bytes, checksum: 090e81187787a7a96591621e58ae7742 (MD5) Previous issue date: 2006 / Resumo: Neste trabalho tratamos de famílias a um-parâmetro de campos vetoriais R-reversíveis definidos em uma vizinhança de um ponto de equilíbrio ressonante em R2n. Focalizamos a atenção às 0:p:q-ressonâncias. Inicialmente estudamos a existência/bifurcação de órbitas periódicas simétricas para tais sistemas. A existência e rigidez de famílias de órbitas homoclínicas também são discutidas. Além disso, também analisamos, para n = 3, a rigidez de famílias de Cantor¿ de dois-toros invariantes por meio da Teoria KAM / Abstract: In this work we deal with one parameter families of R-reversible vector fields defined around a resonant equilibrium point in R2n. We focus our attention to 0:p:q resonances. First of all we study the existence/bifurcation of symmetric periodic orbits for such systems. The existence and rigidity of families of homoclinic orbits are also discussed. We also analyze for n = 3 the rigidity of ¿Cantor families¿ of invariant two-torus by means of KAM Theory / Doutorado / Sistemas Dinamicos / Doutor em Matemática

Estudamos o comportamento dinâmico de séries temporais experimentais obtidas de um circuito de Chua quando dois parâmetros de controle, $Delta R_1$ e $Delta R_2$, são variados.Investigamos os comportamentos caótico e periódico, analisando as séries temporais ao redor e no interior de duas janelas periódicas presentes no espaço de parâmetros $(Delta R_1,Delta R_2)$ do circuito. Na vizinhança da janela de período três, analisamos como a dinâmica simbólica se altera quando construída em diferentes seções de Poincaré de um mesmo atrator, e investigamos a dimensão dos mapas de retorno, uni ou bidimensional, para diferentes atratores caóticos presentes nessa região do espaço de parâmetros. Ainda nessa vizinhança, empregamos técnicas de caracterização topológica para confirmar a existência de fibras caóticas, que são curvas de codimensão um no espaço de parâmetros onde as propriedades caóticas dos atratores são preservadas.Ao redor da janela de período quatro, investigamos a transição entre os três comportamentos caóticos para os quais construímos os respectivos moldes topológicos. Propusemos também um molde topológico para o regime caótico após a crise por fusão ocorrer no circuito. Finalizando, investigamos as bifurcações e a estrutura topológica das órbitas periódicas que formam as janelas de período três e de período quatro, construindo um espaço de parâmetros topológico, baseado em um mapa bi-modal, para descrever as duas janela periódicas. / We have studied the dynamical behavior of experimental time series obtained from a Chua's circuit by variation of two parameter control, $Delta R_1$ and $Delta R_2$. We investigated the chaotic and periodic behaviors of the circuit, analyzing temporal series around and inside of two periodic windows in the two-parameter space $(Delta R_1,Delta R_2)$. In the period-three window neighborhood, we analyzed how the symbolic dynamics changes when it is built by different Poincaré sections of an attractor, and we studied the dimension of return map, one- or two-dimensional, for many chaotic attractors in this region of the parameter space. In this neighborhood, we also applied topological techniques to confirm the existence of chaotic fibers: codimension one curves where the chaotic properties of the attractors remain unchanged in the two-parameter space.Around the period-four window, we investigated, by template analysis, the transition between three chaotic attractors found in the Chua's circuit. We proposed a template for chaotic regime of the circuit after merge-crisis. Finally, we investigated the bifurcations and topological structure of periodic orbits in period-three and period-four windows and also proposed a topological parameter space, based in a bimodal map model, that describe these two periodic windows.

<div>Over the past twenty years, ridesharing opportunities for smallsats, i.e., secondary payloads, has increased with the introduction of Evolved Expendable Launch Vehicle (EELV) Secondary Payload Adapter (ESPA) rings. However, the orbits available for these secondary payloads is limited to Low Earth Orbits (LEO) or Geostationary Orbits (GEO). By incorporating a propulsion system, propulsive ESPA rings offer the capability to transport a secondary payload, or a collection of payloads, to regions beyond GEO. In this investigation, the ridesharing scenario includes a secondary payload in a dropped-off Geosynchronous Transfer Orbit (GTO) and the region of interest is the vicinity near the Sun-Earth Lagrange points. However, mission design for secondary payloads faces certain challenges. A significant mission constraint for a secondary payload is the drop-off orbit orientation, as it is dependent on the primary mission. To address this mission constraint, strategies leveraging dynamical structures within the Circular Restricted Three-Body Problem (CRTBP) are implemented to construct efficient and flexible transfers from GTO to orbits near Sun-Earth Lagrange points. First, single-maneuver ballistic transfers are constructed from a range of GTO departure orientations. The ballistic transfer utilize trajectories within the stable manifold structure associated with periodic and quasi-periodic orbits near the Sun-Earth L1 and L2 points. Numerical differential corrections and continuation methods are leveraged to create families of ballistic transfers. A collection of direct ballistic transfers are generated that correspond to a region of GTO departure locations. Additional communications constraints, based on the Solar Exclusion Zone and the Earth’s penumbra shadow region, are included in the catalog of ballistic transfers. An integral-type path condition is derived and included throughout the differential corrections process to maintain transfers outside the required communications restrictions. The ballistic transfers computed in the CRTBP are easily transitioned to the higher-fidelity ephemeris model and validated, i.e., their geometries persist in the ephemeris model. To construct transfers to specific orbits near Sun-Earth L1 or L2, families of two-maneuver transfers are generated over a range of GTO departure locations. The two-maneuver transfers consist of a maneuver at the GTO departure location and a Deep Space Maneuver (DSM) along the trajectory. Families of two-maneuver transfers are created via a multiple- shooting differential corrections method and a continuation process. The generated families of transfers aid in the rapid generation of initial guesses for optimized transfer solutions over a range of GTO departure locations. Optimized multiple-maneuver transfers into halo and Lissajous orbits near Sun-Earth L1 and L2 are included in this analysis in both the CRTBP model and the higher-fidelity ephemeris model. Furthermore, the two-maneuver transfer strategy employed in this analysis are easily extended to other Three-Body systems. </div>

Three-dimensional gravity-capillary steady waves on water of finite-depth, which are uniformly translating in a horizontal propagation direction and periodic in a transverse direction, are considered. The exact Euler equations are formulated as a spatial dynamic system in which the variable used for the propagating direction is the time-like variable. The existence of the solutions of the system is determined by two non-dimensional constants: the Bond number b and λ (the inverse of the square of the Froude number). The property of Sobolev spaces and the spectral analysis show that the spectrum of the linear part consists of isolated eigenvalues of finite algebraic multiplicity and the number of purely imaginary eigenvalues are finite. The distribution of eigenvalues is described by b and λ. Assume that C₁ is the curve in (b,λ)-plane on which the first two eigenvalues for three-dimensional waves collide at the imaginary axis, and that the intersection point of the curve C₁ with the line λ=1 is (b₀,1) where b₀>0. Two cases (b₀,1) and (b,λ) â C₁ where 0< b< b₀ are investigated. A center-manifold reduction technique and a normal form analysis are applied to show that for each case the dynamical system can be reduced to a system of ordinary differential equations with finite dimensions. The dominant system for the case (b₀,1) is coupled Schrödinger-KdV equations while it is a Schrödinger equation for another case (b,λ) â C₁. Then, from the existence of the homoclinic orbit connecting to the two-dimensional periodic solution (called generalized solitary wave) for the dominant system, it is obtained that such generalized solitary wave solution persists for the original system by using the perturbation method and adjusting some appropriate constants. / Ph. D.

Neste trabalho, apresentamos um método computacional rigoroso para a demonstração de existência de órbitas periódicas de alguns sistemas de equações diferenciais ordinárias com campo autônomo do tipo polinomial. Mostraremos que o problema de encontrar órbitas periódicas para esses sistemas de equações é equivalente a buscar por raízes de certas funções definidas no espaço de Banach das sequências com decaimento algébrico. O método pode ser dividido em duas etapas. Na primeira, buscamos numericamente por soluções periódicas aproximadas. Na segunda, mostraremos a existência de uma órbita periódica numa vizinhança da curva encontrada numericamente. O rigor das verificações computacionais é garantido pelo uso de aritimética intervalar. / In this work, we present a rigorous computational method for proving the existence of periodic orbits of some systems of ordinary differential equations with autonomous vector field of polynomial type. We show that the problem of finding periodic orbits for these systems is equivalent to check for roots of certain functions defined in the Banach space of sequences with algebraic decay. The method can be divided into two steps. First, we seek, numerically, to approximated periodic solutions. Then, we show the existence of a periodic orbit in a neighborhood of the curve numerically found in the previous stage. The accuracy of the computational verifications is guaranteed by the use of interval arithmetic.

On étudie les résonances semi-excitées pour un Opérateur h-Pseudo-différentiel (h-PDO)H(x, hDx) sur L2(M) induites par une orbite périodique de type hyperbolique à l’énergie E = 0. Par exemple M = Rn et H(x, hDx; h) est l’opérateur de Schrödinger avec effet Stark, ouH(x, hDx; h) est le flot géodesique sur une variété axi-symétrique M, généralisant l’exemplede Poincaré de systèmes Lagrangiens à 2 degrés de liberté. On étend le formalisme de Gérard and Sjöstrand, au sens où on autorise des valeurs propres hyperboliques et elliptiques del’application de Poincaré, et où l’on considère des résonances dont la partie imaginaire est del’ordre de hs, pour 0 < s < 1.On établit une règle de quantification de type Bohr-Sommerfeld au premier ordre en fonction des nombres quantiques longitudinaux (réels) et transverses (complexes), incluantl’intégrale d’action le long de l’orbite, la 1-forme sous-principale, et l’indice de Conley-Zehnder. / In this Thesis we consider semi-excited resonances for a h-Pseudo-Differential Operator (h-PDO for short) H(x, hDx; h) on L2(M) induced by a periodic orbit of hyperbolic type at energy E = 0, as arises when M = Rn and H(x, hDx; h) is Schrödinger operator withAC Stark effect, or H(x, hDx; h) is the geodesic flow on an axially symmetric manifold M,extending Poincaré example of Lagrangian systems with 2 degree of freedom. We generalizethe framework of Gérard and Sjöstrand, in the sense that we allow for hyperbolic and ellipticeigenvalues of Poincaré map, and look for (excited) resonances with imaginary part of magnitude hs, with 0 < s < 1,It is known that these resonances are given by the zeroes of a determinant associatedwith Poincaré map. We make here this result more precise, in providing a first order asymptoticsof Bohr-Sommerfeld quantization rule in terms of the (real) longitudinal and (complex)transverse quantum numbers, including the action integral, the sub-principal 1-form and Gelfand-Lidskii index.

Neste trabalho, apresentamos um método computacional rigoroso para a demonstração de existência de órbitas periódicas de alguns sistemas de equações diferenciais ordinárias com campo autônomo do tipo polinomial. Mostraremos que o problema de encontrar órbitas periódicas para esses sistemas de equações é equivalente a buscar por raízes de certas funções definidas no espaço de Banach das sequências com decaimento algébrico. O método pode ser dividido em duas etapas. Na primeira, buscamos numericamente por soluções periódicas aproximadas. Na segunda, mostraremos a existência de uma órbita periódica numa vizinhança da curva encontrada numericamente. O rigor das verificações computacionais é garantido pelo uso de aritimética intervalar. / In this work, we present a rigorous computational method for proving the existence of periodic orbits of some systems of ordinary differential equations with autonomous vector field of polynomial type. We show that the problem of finding periodic orbits for these systems is equivalent to check for roots of certain functions defined in the Banach space of sequences with algebraic decay. The method can be divided into two steps. First, we seek, numerically, to approximated periodic solutions. Then, we show the existence of a periodic orbit in a neighborhood of the curve numerically found in the previous stage. The accuracy of the computational verifications is guaranteed by the use of interval arithmetic.

Το κύριο στοιχείο της διατριβής είναι οι Εξελικτικοί Αλγόριθμοι. Στο πρώτο μέρος παρουσιάζονται οι Μιμιδικοί Αλγόριθμοι. Οι Μιμιδικοί Αλγόριθμοι είναι υβριδικά σχήματα που συνδυάζουν τους Εξελιτκικούς Αλγορίθμους με μεθόδους τοπικής αναζήτησης. Οι Μιμιδικοί Αλγόριθμοι συγκρίθηκαν με τους Εξελικτικούς Αλγορίθμους σε πληθώρα προβλημάτων ολικής βελτιστοποίησης και είχαν καλύτερα αποτελέσματα. Στο δεύτερο μέρος μελετήθηκαν προβλήματα μη γραμμικής δυναμικής. Αυτά ήταν η εκτίμηση της περιοχής ευστάθειας διατηρητικών απεικονίσεων, η ανίχνευση συντονισμών και ο υπολογισμός περιοδικών τροχιών. Τα αποτελέσματα ήταν ικανοποιητικά. / The main objective of the thesis was the study of Evolutionary Algorithms. At the first part, Memetic Algorithms were introduced. Memetic Algorithms are hybrid schemes that combine Evolutionary Algorithms and local search methods. Memetic Algorithms were compared to Evolutionary Algorithms in various problems of global optimization and they had better performance. At the second part, problems from nonlinear dynamics were studied. These were the estimation of the stability region of conservative maps, the detection of resonances and the computation of periodic orbits. The results were satisfactory.

In this work some existence and stability results of periodic orbits of van der Pol-Mathieu Equation are studied. By using the Averaging Theorem we are able to prove, under mild conditions, the existence of two asymptotically stable periodic orbits of this equation. Moreover, the existence of invariant quadrics can be settled in plane phase of this equation. / Neste trabalho alguns resultados sobre existência e estabilidade de soluções periódicas da equação de van der Pol-Mathieu são estudados. Por meio do Teorema da Média é provado, sob condições adequadas, que esta equação possui duas órbitas periódicas assintóticamente estáveis. Além disso é obtida a existência de cônicas invariantes no plano de fase desta equação. / Mestre em Matemática