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Doctoral Thesis
DOI
https://doi.org/10.11606/T.43.2021.tde-10032021-171732
Document
Author
Full name
Gabriel Díaz Iturry
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Paulo, 2021
Supervisor
Committee
Leonel, Edson Denis (President)
Barreiro, Luiz Antonio
Batista, Antonio Marcos
Oliveira, Mario Jose de
Viana, Ricardo Luiz
Title in Portuguese
Estudo do comportamento da entropia em bilhares
Keywords in Portuguese
Bilhares
Caos
Difus\\~ao
Abstract in Portuguese
Neste trabalho estudamos como usar o comportamento da entropia para medir o expoente de difus\~ao de um conjunto de condi\c c\~oes iniciais em sistemas do tipo bilhar. Os modelos considerados s\~ao o Modelo Fermi Ulam Simplificado, o Mapa Padr\~ao e o Bilhar Ov\'oide. Nos preocupamos com a difus\~ao perto da ilha principal no espa\c co de fases, onde existe o fen\^omeno de aprisionamento tempor\'ario. Calculamos o expoente de difus\~ao para diversos valores do par\^ametro de controle do Mapa Padr\~ao e o Bilhar Ov\'oide, onde para cada valor a ilha principal tinha uma forma diferente, e mostramos que as mudan\c cas de comportamento no expoente est\~ao relacionadas com mudan\c cas na \'area da ilha principal. Particularmente, mostramos que toda vez que a \'area da ilha principal se reduzia abruptamente, devido a destrui\c c\~ao de toros invariantes e a cria\c c\~ao de pontos fixos hiperb\'olicos e el\'ipticos, o expoente de difus\~ao cresce. Para investigar melhor a conex\~ao entre o expoente de difus\~ao e a cria\c c\~ao de pontos fixos hiperb\'olicos e el\'ipticos, desenvolvemos um esquema de controle apropriado no Mapa Padr\~ao, com o qual mostramos que fechando os caminhos de fuga das proximidades da ilha o expoente de difus\~ao tornou-se menor. Em seguida, relacionamos os caminhos de fuga com a variedade inst\'avel dos pontos hiperb\'olicos.
Title in English
Study of entropy behaviour in billiard systems
Keywords in English
Billiards
Chaos
Diffusion
Abstract in English
In this work we studied how to use the behaviour of the entropy to measure the diffusion exponent of a set of initial conditions in Billiard like systems. The considered models are the Simplified Fermi Ulam Model, Standard Map and the Oval Billiard. We care about the diffusion near the main island in the phase space, where exists the stickiness phenomenon. We calculated the diffusion exponent for many values of the nonlinear parameter of the Standard Map and the Oval Billiard, where for each value the main island has a different shape, then we show that the changes of behaviour in the diffusion exponent are related to changes in the area of the main island. Particularly, we show when the main island's area is abruptly reduced, due to the destruction of invariant tori and consequently creation of hyperbolic and elliptic fixed points, the diffusion exponent grows. To further investigate the connection between the diffusion exponent and the creation of hyperbolic and elliptic fixed points, we developed an appropriate control scheme in the Standard Map, with which we showed that closing paths of escape from the island shore the diffusion exponent became smaller. Then we related the paths of escape with the unstable manifold of the hyperbolic points.
 
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IturryGabrielDO.pdf (4.07 Mbytes)
Publishing Date
2021-04-06
 
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