Transporte, escape de partículas e propriedades dinâmicas de mapeamentos não lineares [thesis]

Diogo Ricardo da Costa
disponibilizar o laboratório e acesso computacional. Esta pesquisa tornou-se possível graças aos recursos computacionais disponibilizados pelo Núcleo de Computação Científica (NCC/GridUNESP) da Universidade Estadual Paulista (UNESP) e ao cluster computacional do grupo de Estudos de Sistemas Complexos e Dinâmica Não Linear, adquirido pelos projetos FAPESP 2012/23688-5, 2008/57528-9 e 2005/56253-8. A todos os Profs. das disciplinas de Pós-Graduação que cursei ao longo do doutoramento. Aos
more » ... amento. Aos pesquisadores Mário R. Silva, Carl P. Dettmann, Juliano A. de Oliveira e Diego F. M. de Oliveira, com o qual tive oportunidade de publicar e submeter alguns trabalhos. Aos colegas de Pós-Graduação, que de alguma forma colaboraram no desenvolvimento deste trabalho. A CAPES, CNPq e FAPESP (2010/52709-5) pelo apoio financeiro durante a realização deste projeto. Ao grupo do prof. Carl P. Dettmann (Bristol -Inglaterra) pela visita de 01/02/2013à 31/07/2013 com bolsa BEPE -FAPESP de doutorado sanduíche (2012/18962-0). Aos demais funcionários e docentes da Universidade de São Paulo. Muito obrigado! Diogo Ricardo da Costa ii Palavras-chave: caos, sistemas dinâmicos, bilhares, propriedades de escape, Aceleração de Fermi iii ABSTRACT Abstract We investigate some dynamical and transport properties for a set of non-interacting classical particles. The systems here described, for the most part, present mixed structure in the phase space in the sense that invariant spanning curves, chaotic seas and periodic islands are present. The dynamics of each model is described by using nonlinear mappings. We show all the details to construct the mappings and discuss some of their dynamical properties including fixed points stability among others. Lyapunov exponents will be obtained to characterize the chaotic dynamics observed in the phase space. Moreover some scaling hypotheses are used to prove that certain observables, including the average energy, are scaling invariant. We consider also that when a particle or an ensemble of them reach a certain portion of the phase space, they can escape. When studying the escape, we see that the histogram for the number of particles that reach certain height (or energy) h in the phase space for the iteration n, for which we observe to be scaling invariant, grows quickly until reaching a maximum and then goes towards zero for large enough n. When changing the height h proportionally to the position of the first invariant spanning curve, we can confirm the scaling invariance. The same happens for the survival probability for a particle in the chaotic dynamics. In this way, we will discuss the following problems: (1) A corrugated waveguide; (2) A family of two-dimensional Hamiltonian mappings which can reproduce different scaling exponents; (3) Particles confined to bounce in the interior of a time-dependent potential well; (4) We will analyse a rotating oval billiard, where for certain conditions we observed that this system does not present the unbounded energy growth (Fermi acceleration), in this way it is a possible counterexample of the LRA conjecture. This thesis is as summary of eight papers already published.
doi:10.11606/t.43.2014.tde-29092014-151049 fatcat:jg4xfakzo5emzgqnz3grjgbfpa