Dinâmica de um bilhar composto: acoplamento de dois bilhares anulares
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2020-02-17
Autores
Baroni, Rodrigo Simile [UNESP]
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Universidade Estadual Paulista (Unesp)
Resumo
Bilhares exibem um rico comportamento dinâmico, típico de sistemas Hamiltonianos. Nessa dissertação, investigamos: (i) a dinâmica clássica de partículas no bilhar anular excêntrico, que possui espaço de fases misto, no limite em que o espalhador é pontual e (ii) o transporte de partículas entre dois bilhares anulares acoplados, conectados por um buraco permeável de modo que as partículas podem visitar ambos os bilhares. Chamamos a configuração apresentada em (i) de quase singular, em que uma única condição inicial (CI) preenche densamente o espaço de fases com linhas retas. Para caracterizar as órbitas nessa configuração, duas técnicas foram aplicadas: expoente de Lyapunov a tempo finito (FTLE) e recorrência temporal. A recorrência de órbitas no espaço de fases foi investigada através de gráficos de recorrência. Mostramos que durante a transição para o limite quase singular, uma órbita típica do bilhar exibe uma queda brusca no valor do expoente de Lyapunov, sugerindo uma mudança no comportamento dinâmico do sistema. Muitos tempos de recorrência são observados no limite quase singular, também indicando que a órbita é caótica. Os padrões no gráfico de recorrência revelam que essa órbita caótica é composta por segmentos quase periódicos. Na configuração apresentada em (ii), calculamos a probabilidade de sobrevivência para CIs dadas em uma componente dos bilhares acoplados e a estatística de recorrência (ou tempos de aprisionamento) da outra componente para mostrar que, ao aumentar a largura do buraco que conecta os dois bilhares, o transporte de partículas entre eles também aumenta. O diagrama de tempo de escape mostra a existência de canais que organizam o transporte. No caso com fronteiras internas dependentes do tempo, o mecanismo de aceleração de Fermi é observado e é otimizado quando ambas as componentes são concêntricas.
Billiards exhibit rich dynamical behavior, typical of Hamiltonian systems. In the present study, we investigate: (i) the classical dynamics of particles in the eccentric annular billiard, which has a mixed phase space, in the limit that the scatterer is point-like and (ii) the transport of particles between two coupled annular billiards connected through a tunable hole so that the particles can visit both billiards. We call the con guration presented in (i) the near singular, in which a single initial condition (IC) densely lls the phase space with straight lines. To characterize the orbits in this con guration, two techniques were applied: Finite-time Lyapunov exponent (FTLE) and time recurrence. The recurrence of orbits in the phase space was investigated through recurrence plots. We show that during the transition to the near singular limit, a typical orbit in the billiard exhibits a sharp drop in the value of the Lyapunov exponent, suggesting some change in the dynamical behavior of the system. Many di erent recurrence times are observed in the near singular limit, also indicating that the orbit is chaotic. The patterns in the recurrence plot reveal that this chaotic orbit is composed of quasi-periodic segments. In the con guration presented in (ii), we calculate the survival probabilities for ICs given in one component of the coupled billiards and the recurrence (or trapping times) statistics of the other component to show that by increasing the size of the hole that connects the two billiards, the transport of particles between them also increases. The escape time diagram shows the existence of channels that assist the transport. In the case with time dependent inner boundaries, the Fermi acceleration mechanism is observed and is optimized when both components are concentric.
Billiards exhibit rich dynamical behavior, typical of Hamiltonian systems. In the present study, we investigate: (i) the classical dynamics of particles in the eccentric annular billiard, which has a mixed phase space, in the limit that the scatterer is point-like and (ii) the transport of particles between two coupled annular billiards connected through a tunable hole so that the particles can visit both billiards. We call the con guration presented in (i) the near singular, in which a single initial condition (IC) densely lls the phase space with straight lines. To characterize the orbits in this con guration, two techniques were applied: Finite-time Lyapunov exponent (FTLE) and time recurrence. The recurrence of orbits in the phase space was investigated through recurrence plots. We show that during the transition to the near singular limit, a typical orbit in the billiard exhibits a sharp drop in the value of the Lyapunov exponent, suggesting some change in the dynamical behavior of the system. Many di erent recurrence times are observed in the near singular limit, also indicating that the orbit is chaotic. The patterns in the recurrence plot reveal that this chaotic orbit is composed of quasi-periodic segments. In the con guration presented in (ii), we calculate the survival probabilities for ICs given in one component of the coupled billiards and the recurrence (or trapping times) statistics of the other component to show that by increasing the size of the hole that connects the two billiards, the transport of particles between them also increases. The escape time diagram shows the existence of channels that assist the transport. In the case with time dependent inner boundaries, the Fermi acceleration mechanism is observed and is optimized when both components are concentric.
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Palavras-chave
Caos determinístico, Sistemas hamiltonianos, Sistemas dinâmicos, Bilhares, Deterministic chaos, Hamiltonian systems, Dynamical systems, Billiard