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Doctoral Thesis
DOI
10.11606/T.43.1996.tde-13122007-093342
Document
Author
Full name
Murilo da Silva Baptista
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Paulo, 1996
Supervisor
Committee
Caldas, Ibere Luiz (President)
Aguiar, Marcus Aloizio Martinez de
Ragazzo, Clodoaldo Grotta
Sartorelli, Jose Carlos
Viana, Ricardo Luiz
Title in Portuguese
Perturbando Sistemas Não-Lineares, uma Abordagem do Controle de Caos
Keywords in Portuguese
1. Sistemas Dinâmicos 2. Estudo de Comportamento Caótico 3. Controle de Caos 4. Rotas para o Caos
Abstract in Portuguese
Inicialmente, consideramos o mapa Logístico com os vários fenômenos nele presentes, para depois, ao perturbarmos esse mapa, adicionando periodicamente um termo de amplitude constante, identificarmos os novos fenômenos e as alterações que a introdução da perturbação faz aparecer. Apresentamos o circuito eletrônico de Matsumoto e, em seguida, o consideramos em um regime caótico perturbado por uma tensão elétrica senoidal externa. A introdução desta perturbação faz o circuito permanecer caótico, tornar-se periódico ou quasi-periódico no toro de duas frequências. Aplicamos diversos métodos de controle de caos a três sistemas (mapa Logístico, mapa de Hénon e circuito de Matsumoto). Para a estabilização de uma órbita periódica, consideramos os métodos de Ott-Grebogi-Yorke (OGY), de Romeiras, de Pyragas, de Sinha, de Singer e de H¨ubbler. Para o direcionamento da trajetória para um ponto de equilíbrio, usamos o método de Sinha. Para a transferência da trajetória para um dos atratores coexistentes no sistema de Matsumoto, usamos o método de Jackson-H¨ubbler (OPCL). Usando um conjunto de pertubações constantes em um parâmetro previamente escolhido, mostramos como é possével dirigir rapidamente uma trajetória, de qualquer um dos três sistemas considerados nesta tese, para um determinado alvo. Além disso, é mostrado como esse método pode ser aplicado experimentalmente.
Title in English
Perturbing non-linear systems, an approach to the control of chaos.
Keywords in English
Chaos control
Chua\'s circuit.
homoclinic bifurcation
periodic windows
route to chaos
Abstract in English
Initially, we consider the Logistic map with its many non-linear phenomena. Then, we use this knowledge to discern new phenomena that shall appear when the map is perturbed, that is the Logistic map perturbed by a periodic and constant term. The Matsumoto's circuit is presented and, after we set this circuit to behave chaotically, we perturb it with a sinoidal wave, characterized by its frequency and amplitude. This perturbation is responsible for the appearence of a quasi-periodic and periodic oscillations, or the maintenance of chaos. We presented and applied many methods for controlling chaotic oscillations in three systems (the Logistic and Henon maps, and the Matsumoto's circuit), showing many ways for stabilizing a periodic orbit, using the methods of Ott-Grebogi-York (OGY), Romeiras, Singer, Sinhas and Huebbler. For targeting the trajectory to a equilibrium point, the Sinha's method was used. To transfer the system trajectory from one to another of the coexisting attractors presented in the Matsumoto's circuit, we use the Jackson-Huebbler (OPCL) method. Using a set of constant perturbations, in a previously chosen parameter, we showed how we can rapidly direct a trajectory of any of the considered three systems to a aimed target. Besides, it is shown how this method can be experimentally applied.
 
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BIBLIOGRAPHY.pdf (77.42 Kbytes)
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Publishing Date
2008-02-14
 
WARNING: The material described below relates to works resulting from this thesis or dissertation. The contents of these works are the author's responsibility.
  • BAPTISTA, M.s., et al. Periodic driving of plasma turbulence [doi:10.1063/1.1561612]. Physics of Plasmas [online], 2003, vol. 10, p. 1283.
  • BAPTISTA, M.s., et al. Phase Synchronization in the perturbed Chua Curcuit [doi:10.1103/PhysRevE.67.056212]. Physical Review E. (Cessou em 2000. Cont. 1539-3755 Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics) [online], 2003, vol. 67, p. 05621.
  • RODRIGUES, A, et al. Conditional Targeting for Communication [doi:10.1016/j.chaos.2003.12.048]. Chaos, Solitons and Fractals [online], 2004, vol. 21, p. 1271-1280.
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