Studying chaotic behavior in nonlinear systems requires numerous computations in order to simulate the behavior of such systems. The Standard Map Machine was designed and implemented as a special computer for performing these intensive computations with high-speed and high-precision. Its impressive performance is due to its simple architecture specialized to the numerical computations required of nonlinear systems. This report discusses the design and implementation of the Standard Map Machine and its use in the study of nonlinear mappings; in particular, the study of the standard map.

This thesis presents two dimensional discrete dynamical system, the extended standard family of maps, which approximates homoclinic bifurcations of continuous dissipative systems. The main subject of study is the problem of separatrix splitting which was first discovered by Poincaré in the context of the n-body problem. Separatrix splitting leads to chaotic behaviour of the system on exponentially small region in parameter space. To estimate the size of the region the dissipative map is extended to complex variables and approximated by differential equation on a specific domain. This approach was proposed by Lazutkin to study separatrix splitting for Chirikov’s standard map. Furthermore the complex nearly periodic function is used to estimate the width of the exponentially small region where chaos prevails and the map is related to the semistandard map. Numerical computations require solving complex differential equation and provide the constants involved in the asymptotic formula for the size of the region. Another problem studied in this thesis is the prevalence of resonance for the dissipative standard map on a specific invariant set, which for one dimensional map corresponds to a circle. The regions in parameter space where periodic behaviour occurs on the invariant set is known as Arnold tongues. The width of Arnold tongue is studied and numerical results obtained by iterating the map and solving differential equation are related to the semistandard map.

Made available in DSpace on 2016-12-12T20:15:51Z (GMT). No. of bitstreams: 1 Ana Carolina.pdf: 30877153 bytes, checksum: df1e32c0dad640cd008d62b73fa92072 (MD5) Previous issue date: 2014-07-30 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / In this work/study, have showed the results of the investigation about dynamics standard map in the forms; conservative, dissipative and dissipative relativistic, through a numerical and analytical approach for the last case. For the three abovementioned cases, have been build and inspected the respective phase spaces, parameter spaces and spaces of periods, by which has been found the coexistence of chaotic and periodic domains. It has still been characterized and discussed the influence of the parameters responsibles for the dissipation and damping or by a relativistic component in the dynamics of a particle subjected to periodic pulses, as well as the influence of the parameter responsible for noise in the system. The results showed the existence of periodic structures that are organized from the smallest to highest period immersed in chaotic regions. Finally, it has showed that the self-similar structures were gradually disturbed by the noise, as well as the less stable regions, where there are bifurcations, even were hit and porteriormente are totally destroyed and replaced by a chaotic structure. / Neste trabalho apresentaram-se os resultados obtidos da investigação da dinâmica do mapa padrão nas formas; conservativa, dissipativa e dissipativa relativística, através de uma abordagem numérica e analítica para o último caso. Para os três casos supracitados, construíram-se e inspecionaram-se os respectivos espaços de fases, espaços de parâmetros e espaços de períodos, pelos quais constatou-se a coexistência de domínios caóticos e periódicos. Caracterizou-se e discutiu-se ainda a influência dos parâmetros responsáveis pelo amortecimento ou dissipação e por uma componente relativística na dinâmica de uma partícula submetida a pulsos periódicos, assim como a influência do parâmetro responsável pelo ruído no sistema. Os resultados evidenciaram a existência de estruturas periódicas que se organizam do menor para o maior período imersas em regiões caóticas. Finalmente , mostrou-se que as estruturas auto similares foram gradativamente perturbadas pelo ruído, assim como as regiões menos estáveis onde há bifurcações, também foram atingidas e porteriormente são totalmente destruídas e substituídas por uma estrutura caótica.

Submitted by LARYSSA KIMI KATO (laryssakimi@hotmail.com) on 2018-04-27T19:27:34Z No. of bitstreams: 1 revisada.pdf: 3145745 bytes, checksum: 4e2c58e36b72087e4c2b3fa95a58d289 (MD5) / Approved for entry into archive by Ana Paula Santulo Custódio de Medeiros null (asantulo@rc.unesp.br) on 2018-04-27T20:10:17Z (GMT) No. of bitstreams: 1 kato_lk_me_rcla.pdf: 3143588 bytes, checksum: 16c151552f1c1e1e340ef2037cac4d8d (MD5) / Made available in DSpace on 2018-04-27T20:10:17Z (GMT). No. of bitstreams: 1 kato_lk_me_rcla.pdf: 3143588 bytes, checksum: 16c151552f1c1e1e340ef2037cac4d8d (MD5) Previous issue date: 2018-01-30 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / De maneira geral, o comportamento dinâmico de sistemas não lineares é caracterizado pela imprevisibilidade e extrema sensibilidade às condições iniciais e aos parâmetros do sistema. A sensibilidade dessas condições pode ser analisada a partir dos expoentes de Lyapunov, quando são consideradas órbitas infinitesimalmente próximas. O mapa escolhido para análise é o modelo denominado “Mapa padrão não - twist dissipativo labiríntico”, que apresenta as chamadas curvas shearless. O estudo desenvolvido analisa esse sistema com a introdução de dissipação e com parâmetros de perturbação variáveis na presença de três curvas shearless. O objetivo é compreender a evolução da dinâmica destas curvas no espaço de fase e no diagrama de Lyapunov a fim de caracterizar qual shearless é mais robusta frente á variação dos parâmetros de dissipação e perturbação. / In general, the dynamical behavior of non-linear systems is characterized by unpredictability and extreme sensibility to the initial conditions and to the parameters of the system. The sensitivity of these conditions can be analyzed from the Lyapunov exponents, when infinitesimally close orbits are considered. The map we have chosen for analysis is the model denoted as "Labyrinthic non-twist standard map", which presents the so-called "shearless" curves. The present study analyzes this system with the introduction of dissipation and with changeable parameters of perturbation in the presence of three shearless curves. The objective is to understand the evolution of the dynamics of the curves in the phase space and in the diagram of Lyapunov in order to characterize which shearless is more robust under the variation of both parameters, dissipation and perturbation.