Wo, Weifeng. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2011. / Includes bibliographical references (leaves 99-102). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstract also in Chinese.

The purpose of this work is to illuminate some of the non-smooth phenomena found in piecewise-smooth continuous and discrete dynamical systems, which do not occur in smooth systems. We will explain how such non-smooth phenomena arise in applications which experience impact, such as impact oscillators, and a type of rotating machine, called magnetic bearing systems. The study of their dynamics and sensitivity to parameter variation gives not just insights into the critical motion found in these applications, but also into the complexity and beauty in their own right. This work comprises two parts. The first part studies a general one-dimensional discontinuous power law map which can arise from impact oscillators with a repelling wall. Parameter variation and the influence of the exponent on the existence and stability of periodic orbits is presented. In the second part we analyse two coupled oscillators that model rotating machines colliding with a circular boundary under friction. The study of the dynamics of rigid bodies impacting with and without friction is approached in two ways. On the one hand existence and stability conditions for non-impacting and impacting invariant sets are derived using local and global methods. On the other hand the analysis of parameter variation reveals new non-smooth bifurcations. Extensive numerical studies confirm these results and reveal further phenomena not attainable otherwise.

Orientador: Edson Denis Leonel / Banca: Juliano Antônio de Oliveira / Banca: Renê Orlando Medrado Torricos / Resumo: Neste trabalho consideramos o modelo do poço de potencial dependente do tempo e construimos de forma detalhada o mapeamento discreto bidimensional nas variáveis energia e fase que descreve a dinâmica do sistema. Mostramos que o espaço de fases é do tipo misto, contendo mares de caos, curvas invariantes e ilhas de estabilidade. Encontramos a matriz Jacobiana para o mapeamento assim como seu determinante, confirmando a propriedade de preservação de área. Estudamos a evolução no tempo da energia quadrática média e discutimos leis de escala para o comportamento dessa evolução. Por fim demos início à resolução da equação da difusão a fim de encontrarmos uma equação analitíca para energia quadrática média / Abstract: In this work we consider the model of the time-dependent potential well and we construct in detail the two-dimensional discrete mapping in the energy and phase variables that describes the dynamics of the system. We show that the phase space is of the mixed type, containing chaotic seas, invariant curves and stability islands. We obtain the Jacobian matrix for the mapping as well as its determinant, confirming the area preservation property. We study the evolution in time of the average squared energy and discuss scaling laws for the behavior of this evolution. Finally we started the resolution of the diffusion equation in order to find an analytical equation for mean quadratic energy / Mestre

Goal-directed behavior requires flexible selection among action plans and updating behavioral strategies when they fail to achieve desired goals. Lateral prefrontal cortex (LPFC) is implicated in the execution of behavior-guiding rule-based cognitive control while anterior cingulate cortex (ACC) is implicated in monitoring processes and updating rules. Rule-based cognitive control requires selective processing while process monitoring benefits from combinatorial processing. I used a combination of computational and experimental methods to investigate how network oscillations and neuronal heterogeneity contribute to cognitive control through their effects on selective versus combinatorial processing modes in LPFC and ACC. First, I adapted an existing LPFC model to explore input frequency- and coherence-based output selection mechanisms for flexible routing of rate-coded signals. I show that the oscillatory states of input encoding populations can exhibit a stronger influence over downstream competition than their activity levels. This enables an output driven by a weaker resonant input signal to suppress lower-frequency competing responses to stronger, less resonant (though possibly higher-frequency) input signals. While signals are encoded in population firing rates, output selection and signal routing can be governed independently by the frequency and coherence of oscillatory inputs and their correspondence with output resonant properties. Flexible response selection and gating can be achieved by oscillatory state control mechanisms operating on input encoding populations. These dynamic mechanisms enable experimentally-observed LPFC beta and gamma oscillations to flexibly govern the selection and gating of rate-coded signals for downstream read-out. Furthermore, I demonstrate how differential drives to distinct interneuron populations can switch working memory representations between asynchronous and oscillatory states that support rule-based selection. Next, I analyzed physiological data from the LeBeau laboratory and built a de novo model constrained by the biological data. Experimental data demonstrated that fast network oscillations at both the beta- and gamma frequency bands could be elicited in vitro in ACC and neurons exhibited a wide range of intrinsic properties. Computational modeling of the ACC network revealed that the frequency of network oscillation generated was dependent upon the time course of inhibition. Principal cell heterogeneity broadened the range of frequencies generated by the model network. In addition, with different frequency inputs to two neuronal assemblies, heterogeneity decreased competition and increased spike coherence between the networks thus conferring a combinatorial advantage to the network. These findings suggest that oscillating neuronal populations can support either response selection (routing), or combination, depending on the interplay between the kinetics of synaptic inhibition and the degree of heterogeneity of principal cell intrinsic conductances. Such differences may support functional differences between the roles of LPFC and ACC in cognitive control.

by Guan-xin Li. / "January 1999." / Thesis (Ph.D.)--Chinese University of Hong Kong, 1999. / Includes bibliographical references (p. 223-225). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Mode of access: World Wide Web. / Abstracts in English and Chinese.

Investigou-se a gênese de atividade irregular em neurônios de centros geradores de padrões através de modelos eletrofisiologicamente realistas. Para tanto, foram adotadas abordagens paralelas. Primeiramente, desenvolveram-se técnicas para determinar quais os mecanismos biofísicos subjacentes aos processos de codificação de informação nestas células. Também foi proposta uma nova metodologia híbrida (baseada em continuação numérica e em varreduras força bruta) para análise de bancos de dados de modelos neuronais, permitindo estendê-los e revelar instâncias de multiestabilidade entre regimes oscilatórios e quiescentes. Além disto, a fim de determinar a origem de comportamento complexo em modelos neuronais simplificados, empregaram-se métodos geométricos da teoria de sistemas dinâmicos. A partir da análise de mapas unidimensionais perturbados por ruído, foram discutidos possíveis cenários para o surgimento de caos em sistemas dinâmicos aleatórios. Finalmente mostrou-se que, levando em conta o ruído, uma classe de modelos de condutâncias reproduz padrões de disparo observados in vivo. Estas pertubações revelam a riqueza da dinâmica transiente, levando o sistema a visitar um arcabouço determinista complexo preexistente -- sem recorrer a ajustes finos de parâmetros ou a construções ad hoc para induzir comportamento caótico. / We investigated the origin of irregularities in the dynamics of central pattern generator neurons, through analyzing electrophysiologically realistic models. A number of parallel approaches were adopted for that purpose. Initially, we studied information coding processes in these cells and proposed a technique to determine the underlying biophysical mechanisms. We also developed a novel hybrid method (based on numerical continuation and brute force sweeps) to analyze neuronal model databases, extending them and unveiling instances of multistability between oscillatory and resting regimes. Furthermore, in order to determine the origin of irregular dynamics in simplified neuronal models, we employed geometrical methods from the theory of dynamical systems. The analysis of stochastically perturbed maps allowed us to discuss possible scenarios for the generation of chaotic behaviour in random dynamical systems. Finally we showed that, by taking noise into account, a class of conductance based models gives rise to firing patterns akin to the ones observed \\emph{in vivo}. These perturbations unveil the richness of the transient dynamics, inducing the system to populate a preexistent complex deterministic scaffolding -- without resorting to parameter fine-tuning or ad hoc constructions to induce chaotic activity.

Ma, Man Shun. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2011. / Includes bibliographical references (leaves 75-76). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.6 / Chapter 2 --- Basic notions in Riemannian geometry --- p.10 / Chapter 2.1 --- Basic manifold theory --- p.11 / Chapter 2.2 --- "Connection, curvature" --- p.19 / Chapter 2.3 --- Submanifold theory --- p.29 / Chapter 3 --- Basic facts in symplectic and complex geometry --- p.33 / Chapter 3.1 --- "Symplectic manifolds, Lagrangian submanifolds" --- p.34 / Chapter 3.2 --- Kahler and Calabi-Yau manifolds --- p.39 / Chapter 3.3 --- Calibration --- p.49 / Chapter 4 --- Mean curvature flow --- p.52 / Chapter 4.1 --- Basic equations in Lagrangian immersions --- p.53 / Chapter 4.2 --- Evolution equation for --- p.57 / Chapter 4.3 --- Evolution equations for H and θ --- p.62 / Chapter 5 --- Lagrangian angle of a foliation --- p.67 / Chapter 5.1 --- "Proof of equation (5.1), (5.2)" --- p.68 / Chapter 5.2 --- Main theorem --- p.70 / Chapter 5.3 --- Examples of invariant solution --- p.73 / Bibliography --- p.75

by Yiu Chun Chit. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1997. / Includes bibliographical references (leaves 81-82). / Chapter 1 --- Introduction --- p.7 / Chapter 1.1 --- Riemannian Manifolds --- p.7 / Chapter 1.1.1 --- Completeness --- p.8 / Chapter 1.1.2 --- Curvature tensor --- p.9 / Chapter 1.1.3 --- Holonomy --- p.11 / Chapter 1.2 --- Simply-connected Manifold of Nonpositive Sectional Curvature --- p.11 / Chapter 1.2.1 --- Topological structure --- p.12 / Chapter 1.2.2 --- Basic geometric properties --- p.13 / Chapter 1.2.3 --- Examples of nonpositively curved manifold --- p.20 / Chapter 1.2.4 --- Convexity properties --- p.23 / Chapter 1.2.5 --- Points at infinity for M --- p.27 / Chapter 2 --- Symmetric Spaces --- p.36 / Chapter 2.1 --- Symmetric Spaces of Noncompact Type --- p.36 / Chapter 2.1.1 --- Symmetric diffeomorphisms --- p.36 / Chapter 2.1.2 --- Transvections in I(M) --- p.38 / Chapter 2.1.3 --- Symmetric spaces as coset manifolds G/K --- p.39 / Chapter 2.1.4 --- Metric on TpM and the adjoint representation of Lie group --- p.41 / Chapter 2.1.5 --- Curvature tensor of M --- p.43 / Chapter 2.1.6 --- Killing form and classification of symmetric spaces --- p.44 / Chapter 2.1.7 --- Holonomy of M at p --- p.44 / Chapter 2.1.8 --- Rank of a symmetric space M --- p.45 / Chapter 2.1.9 --- Regular and singular points at infinity --- p.46 / Chapter 2.2 --- "The Symmetric Space Mn = SL(n,R)/SO(n,R)" --- p.46 / Chapter 2.2.1 --- Metric on TIMn --- p.47 / Chapter 2.2.2 --- Geodesic and symmetries of Mn --- p.48 / Chapter 2.2.3 --- Curvature of Mn --- p.48 / Chapter 2.2.4 --- Rank and flats in Mn --- p.49 / Chapter 2.2.5 --- Holonomy of Mn at I --- p.49 / Chapter 2.2.6 --- Eigenvalue-flag pair for a point in Mn(∞ ) --- p.50 / Chapter 2.2.7 --- Action of I0(Mn) on Mn(∞ ) --- p.52 / Chapter 2.2.8 --- Flags in opposition --- p.53 / Chapter 2.2.9 --- Joining points at infinity --- p.53 / Chapter 3 --- Group Action --- p.56 / Chapter 3.1 --- Action of Isometries on M(oo) --- p.56 / Chapter 3.1.1 --- Fundamental group as a group of isometries --- p.56 / Chapter 3.1.2 --- Lattices --- p.58 / Chapter 3.1.3 --- Duality condition --- p.59 / Chapter 3.1.4 --- Geodesic flows --- p.61 / Chapter 3.2 --- Action of Geodesic Symmetries on M(oo) --- p.62 / Chapter 3.3 --- Rank --- p.66 / Chapter 3.3.1 --- Rank of a manifold of nonpositive curvature --- p.66 / Chapter 3.3.2 --- Rank of the fundamental group --- p.68 / Chapter 3.4 --- Rigidity Theorems of Locally Symmetric Spaces --- p.69

Gravitational N-body problems are central in classical mathematical physics. Studying their long time behavior raises subtle questions about the interplay between regular and irregular motions and the boundary between integrable and chaotic dynamics. Over the last hundred years, concepts from the qualitative theory of dynamical systems such as stable/unstable manifolds, homoclinic and heteroclinic tangles, KAM theory, and whiskered invariant tori, have come to play an increasingly important role in the discussion. In the last fty years the study of numerical methods for computing invariant objects has matured into a thriving sub-discipline. This growth is driven at least in part by the needs of the world's space programs. Recent work on validated numerical methods has begun to unify the computational and analytical perspectives, enriching both aspects of the subject. Many of these results use computer assisted proofs, a tool which has become increasingly popular in recent years. This thesis presents a proof that the circular restricted four body problem is non-integrable. The proof of this result is obtained as an application of more general rigorous numerical methods in nonlinear analysis. / Includes bibliography. / Dissertation (Ph.D.)--Florida Atlantic University, 2017. / FAU Electronic Theses and Dissertations Collection

The theory of thermodynamic phase transitions has played a central role both in theoretical physics and in dynamical systems for several decades. One of its fundamental results is the classification of various physical models into equivalence classes with respect to the scaling behavior of solutions near the critical manifold. From that point of view, systems characterized by the same set of critical exponents are equivalent, regardless of how different the original physical models might be. For non-equilibrium phase transitions, the current theoretical framework is much less developed. In particular, an equivalent classification criterion is not available, thus requiring a specific analysis of each model individually. In this thesis, we propose a potential classification method for time-dependent dynamical systems, namely comparing the possible deformations of the original problem, and identifying dynamical systems which share the same deformation space. The specific model on which this procedure is developed is the Kuramoto model for interacting, disordered oscillators. Studied in the mean-field limit by a variety of methods, its associated synchronization phase transition appears as an appropriate model for cooperative phenomena ranging from coupled Josephson junctions to self-ordering patterns in biological and social systems. We investigate the geometric deformation of the dynamical system into the space of univalent maps of the unit disk, related to the Douady-Earle extension and the Denjoy-Wolff theory, and separately the algebraic deformation into the space of nonlinear sigma models for unitary operators. The results indicate that the Kuramoto model is representative for a large class of non-equilibrium synchronization models, with a rich phase-space diagram.