Cheung, Chun Ngai. / "August 2011." / Thesis (M.Phil.)--Chinese University of Hong Kong, 2011. / Includes bibliographical references (leaves 55-57). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.6 / Chapter 2 --- Coding the Lorenz knots --- p.10 / Chapter 2.1 --- Lorenz words --- p.10 / Chapter 3 --- Lorenz links and positive braids --- p.15 / Chapter 3.1 --- Lorenz braids --- p.15 / Chapter 3.2 --- Properties of Lorenz links as the closure of a positive braid --- p.17 / Chapter 4 --- T-Links and the braid index --- p.28 / Chapter 4.1 --- T-links --- p.30 / Chapter 4.2 --- Symmetries --- p.35 / Chapter 4.3 --- Trip number and the braid index --- p.39 / Chapter 5 --- Modular knots --- p.47 / Chapter 5.1 --- The Modular flow --- p.47 / Chapter 5.2 --- Modular Knots --- p.49 / Bibliography --- p.55

Thesis (Ph.D.) - University of Queensland, 2004. / Includes bibliography.

Thesis (Ph.D.(Electronic engineering))--University of Pretoria, 2008. / Summary in English. Includes bibliographical references.

The theory of deterministic chaos has generated a lot of interest and continues to be one of the much-focused research areas in the field of dynamics today. This is due to its prevalence in essential parts of human lives such as electrical circuits, chemical reactions, the flow of blood through the human system, the weather, etc. This thesis presents a study of the Lorenz equations, a famous example of chaotic systems. In particular, it presents the analysis of the Lorenz equations from stability to chaos and various bifurcation scenarios with numerical and graphical interpretations. It studies concepts of non-linear dynamical systems such as equilibrium points, stability, linearization, bifurcation, Lyapunov function, etc. Finally, it discusses how the Lorenz equations serve as a model for the waterwheel (in detail), and the convection roll for fluid.

Made available in DSpace on 2014-07-29T15:07:10Z (GMT). No. of bitstreams: 1 Dissertacao nelio.pdf: 402379 bytes, checksum: 978793fda386f5ccef16f94380d52b6f (MD5) Previous issue date: 2008-04-05 / In this work we investigated the problem of Rayleigh-Bénard for a magnetic binary fluid, i.e., a magnetic fluid, which consist of magnetic nanopartilces stably dispersed in a liquid carrier. The theoretical calculations were performed based on a Lorenz-like model, which transforms a system of partial differential equations into ordinary differential ones. The analysis of the magnetic binary fluid problem used the Navier-Stokes, thermal conduction and mass diffusion equations. The magnetic body force was obtained using the Cowley- Rosensweig tensor as well as the Maxwell equations. The mass flux had included the difusive contribution, associated to Fick s law, and also the thermal diffusion term, due to the Soret effect. Our model consist of a system of eight ordinary differential equations, which were shown to mantain the same mathematical form as the ones obtained earlier by Cross for a non-magnetic binary fluid. However, as expected, our coefficients depend on the magnetic field. According to our investigation on the site www.isiknowledge.com this is the first time in the literature that those equations are obtained, which we named the Lorenz-Cross equations on Ferrohydrodynamics. The validity of our system of equations were, also, checked in the limit of a simple fluid, where our model returns to the Lorenz equations. The only difference is the existence of an effective Rayleigh number, represented by the sum of the Rayleigh number and the magnetic Rayleigh one. Finally, the efect of magnetophoresis in the system of equations had also been discussed. / Neste trabalho investigamos o problema de Rayleigh-Bénard para um fluido binário magnético, ou seja, um fluido magnético, que consiste de nanopartículas magnéticas dispersas em um líqüido carreador.Os cálculos teóricos foram baseados na construção de um modelo tipo Lorenz, pelo qual transformamos um sistema de equações diferenciais parciais em equações diferenciais ordinárias. O sistema de equações para o fluido binário utilizou as equações de Navier-Stokes, condução de calor e difusão de massa.A força magnética foi obtida, usando o tensor eletromagnético de Cowley-Rosensweig, levando em conta as equações de Maxwell. O fluxo de massa considerou o termo difusivo, associado a Lei de Fick, e a contribuição termodifusiva, devido ao efeito de Soret.Nosso modelo consiste de um sistema de 8 equações diferenciais ordinárias, e manteve a mesma forma matemática daquelas obtidas anteriormente por Cross para um sistema binário não-magnético. Entretanto, possui contribuições dependentes do campo magnético. De acordo com nosso levantamento bibliográfico essa é a primeira vez na literatura que essas equações são obtidas, as quais denominamos de equações de Lorenz-Cross na Ferrohidrodinâmica.A validade do nosso sistema de equações foi verificada, também, no limite de um fluido simples, no qual nosso sistema retorna ao modelo tradicional de Lorenz com a diferença da contribuição de um número de Rayleigh efetivo, que representa a soma do número de Rayleigh tradicional com um Rayleigh magnético. A contribuição do efeito magnetoforético para o sistema de equações também foi discutida.

<p>In 1998 the Swedish mathematician Warwick Tucker used rigorous interval arithmetic and normal form theory to prove the existence of a strange attractor in the Lorenz system. In large parts, that proof consists of computations implemented and performed on a computer. This thesis is an independent numerical verification of the result obtained by Warwick Tucker, as well as a study of a higher-dimensional system of ordinary differential equations introduced by the Swedish physicist Lennart Stenflo.</p><p>The same type of mapping data as Warwick Tucker obtained is calculated here via a combination of numerical integration, solving optimisation problems and a coordinate change that brings the system to a normal form around the stationary point in the origin. This data is collected in a graph and the problem of determining the existence of a strange attractor is translated to a few graph theoretical problems. The end result, after the numerical study, is a support for the conclusion that the attractor set of the Lorenz system is a strange attractor and also for the conclusion that the Lorenz-Stenflo system possesses a strange attractor.</p>

In 1998 the Swedish mathematician Warwick Tucker used rigorous interval arithmetic and normal form theory to prove the existence of a strange attractor in the Lorenz system. In large parts, that proof consists of computations implemented and performed on a computer. This thesis is an independent numerical verification of the result obtained by Warwick Tucker, as well as a study of a higher-dimensional system of ordinary differential equations introduced by the Swedish physicist Lennart Stenflo. The same type of mapping data as Warwick Tucker obtained is calculated here via a combination of numerical integration, solving optimisation problems and a coordinate change that brings the system to a normal form around the stationary point in the origin. This data is collected in a graph and the problem of determining the existence of a strange attractor is translated to a few graph theoretical problems. The end result, after the numerical study, is a support for the conclusion that the attractor set of the Lorenz system is a strange attractor and also for the conclusion that the Lorenz-Stenflo system possesses a strange attractor. / QC 20101007