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Master's Dissertation
DOI
Document
Author
Full name
Wayner de Souza Klen
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Paulo, 2019
Supervisor
Committee
Mendes, Carlos Molina (President)
Caldas, Ibere Luiz
Hase, Masayuki Oka
Saa, Alberto Vazquez
Title in Portuguese
Dinâmica relativística de partículas em torno de objetos ultracompactos
Keywords in Portuguese
Bifurcação Bogdanov-Takens
Buracos Negros
Estabilidade de Jacobi
Estabilidade de Lyapunov
Sistemas Dinâmicos
Abstract in Portuguese
Nesta dissertação de mestrado o problema da estabilidade de geodésicas do tipo luz e do tipo tempo é estudado sobre o ponto de vista do formalismo de sistemas dinâmicos. Uma breve revisão bibliográfica sobre aspectos importantes de sistemas dinâmicos contínuos no tempo é realizada, bem como uma sucinta revisão de tópicos de interesse em relatividade geral. As equações de movimento para as geodésicas são deduzidas para geometrias com simetria esférica, e o caso Schwarzschild é inicialmente analisado. Em seguida, analisamos o caso das geometrias proposta por Casadio, Fabbri e Mazzacurati e um caso de buraco de minhoca assintoticamente de Sitter. A caracterização dos pontos fixos dos sistemas de interesse é feita, e a sua estabilidade é analisada sob a ótica dos métodos de Lyapunov e Jacobi, assim como bifurcações foram mapeadas. A fotosfera é caracterizada como um ciclo limite, sendo um ponto fixo estritamente instável no espaço de estados de buracos negros. A análise dos buracos de minhoca revelam a existência de uma fotosfera estável em determinadas regiões do espaço de parâmetros do sistema
Title in English
Relativistic dynamics of particles around ultracompact objects
Keywords in English
Black Hole
Bogdanov-Takens Bifurcation
Dynamical Systems
Jacobi Stability
Lyapunov Stability
Abstract in English
In this dissertation, the problem associated with the stability of timelike and null geodesics is studied from the dynamical system point of view. A succinct bibliographical review covering important aspects of time-continuous dynamical systems is made, and a short review about some topics of interest of general relativity is also presented. The geodesic equations of motion are shown for geometries with spherical symmetry, and the Schwarzschild case is first analyzed. In the following, we analyze the geometries proposed by Casadio, Fabbri, and Mazzacurati and an asymptotically de Sitter wormhole case. The characterization of the fixed points of the system is performed, and their stability is studied from the perspective of the Lyapunov and Jacobi methods, as well as the bifurcation analysis. The photon sphere is characterized as a limit cycle, being a strictly unstable fixed point in the state space of the system. The wormhole analysis reveals the existence of a stable photon sphere in certain regions of the parameter space of the system
 
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Publishing Date
2019-09-05
 
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