http://dspace.cityu.edu.hk:80/handle/2031/718 Tue, 30 Apr 2013 10:35:56 GMT 2013-04-30T10:35:56Z http://dspace.cityu.edu.hk:80/handle/2031/6640 Title: Dynamics of one-dimensional inelastic particle systems Authors: Yang, Rong ( 楊榮) Abstract: In this thesis, we investigate the dynamics of one-dimensional inelastic particle systems composed of rigid, frictionless, inelastic particles. Collisions between particles are assumed to be inelastic with constant coefficient of restitution, and between collisions the particles move with constant velocity. We consider two different models and in each case consider the dynamics of an arbitrary number of particles of arbitrary mass. First, we consider a system bounded by two walls, with external forcing from one of the walls. Second, we consider a system with periodic boundary conditions, that can also be thought of as a set of particles on a ring. We show that both systems exhibit surprising behavior that is completely absent in equivalent elastic systems. In the first case, we investigate continuous transitions between different periodic orbits. We show that continuous transitions that occur when adding or subtracting a single collision are, generically, of co-dimension 2. We give a full mechanical description of the system and explain why this is the case. Surprisingly, we also show that there are an infinite set of degenerate transitions of co-dimension 1. We provide a theoretical analysis that gives a simple criteria to classify which transitions are degenerate purely using the discrete set of collisions that occur in the orbits. Our analysis allows us to understand the nature of the degeneracy. We also show that higher degrees of degeneracy can occur, and provide an explanation. In the second case, we consider the dynamics of sequences of collisions that are self-similar in the sense that the relative positions return to their original relative positions after the collision sequence, while the relative velocities are reduced by a constant factor. For a given collision sequence, we develop the analytic machinery to determine the particle velocities and the locations of collisions, and show that the problem of determining self-similar orbits reduces to solving an eigenvalue problem to obtain the particle velocities and solving a linear system to obtain the locations of inter-particle collisions. For inelastic systems, we show that the collision locations can always be uniquely determined. We also show that this is in sharp contrast to the case of elastic systems in which infinite families of self-similar orbits can co-exist. Notes: CityU Call Number: QA851 .Y36 2011; v, 97 leaves 30 cm.; Thesis (Ph.D.)--City University of Hong Kong, 2011.; Includes bibliographical references (leaves [92]-97) Sat, 01 Jan 2011 00:00:00 GMT http://dspace.cityu.edu.hk:80/handle/2031/6640 2011-01-01T00:00:00Z http://dspace.cityu.edu.hk:80/handle/2031/6639 Title: Synchronization and consensus analysis of complex networks Authors: Xiong, Wenjun ( 熊文軍) Abstract: The last decade has witnessed the birth of a new movement of interest and research in the study of complex networks, i.e. networks whose structure is irregular, complex and dynamically evolving in time. The main focus has moved from the analysis of small networks to that of systems with thousands or millions of nodes, with a renewed attention to the properties of networks of dynamical units. This flurry of activity, triggered by two seminal papers, that by Watts and Strogatz on small-world networks, appeared in Nature in 1998, and that by Barabási and Albert on scale-free networks appeared one year later in Science, has seen the physics' community among the principal actors, and has been certainly induced by the increased computing powers and by the possibility to study the properties of a plenty of large databases of real networks. The research on complex networks begun with the effort of defining new concepts and measures to characterize the topology of real networks. The main result has been the identification of a series of unifying principles and statistical properties common to most of the real networks considered. Over the past decade, much of the interesting dynamical behavior of complex dynamical networks, such as synchronization, consensus and spatio-temporal chaos, has recently attracted increasing attention from researchers in different areas. Among these, synchronization and consensus techniques have been applied in many fields, such as secure communication, harmonic oscillation generation and parallel image processing. Hence, synchronization and consensus analysis for complex networks has important real-life application. The major work of this thesis is to analyze synchronization and consensus of complex networks. In Chapter 2, due to the fact that we usually have to consider some algebraic constraints of complex networks in modeling the real world problems, we aim to build singular hybrid coupled systems to describe complex networks with a special class of constraints. Based on a reference state and the Lyapunov stability, a sufficient condition is obtained ensuring the globally asymptotical synchronization of a class of singular hybrid coupled networks with finite-varying nonlinear perturbation. In Chapter 3, pinning synchronization of a directed network with Markovian jump and nonlinear perturbations is considered. By analyzing the structure of the network, a detailed pinning scheme is given to ensure the synchronization of all nodes in a directed network. This pinning scheme can overcome the difficulties of deciding which nodes needs to be pinned. This scheme can also identify the exact least number of pinned nodes for a directed network model. In addition, the time-varying polytopic directed network with Markovian jump is discussed. In Chapter 4, the consensus problem of multiagent nonlinear directed networks is discussed in the case that a multiagent nonlinear directed network does not have a spanning tree to reach the consensus of all nodes. By using the Lie algebra theory, a linear node-and-node pinning method is proposed to achieve a consensus for all nonlinear functions satisfying a given set of conditions. Based on some optimal algorithms, large-size networks are aggregated to small-size ones. Subsequently, by applying the principle minor theory to the small-size networks, a sufficient condition is given to reduce the number of controlled nodes. In Chapter 5, the consensus problem of a multi-agent directed network with nonlinear perturbations is investigated. Based on a reduced-order transformation, it is shown that the discussed multi-agent model cannot reach a consensus under a Hypoth esis even though the discussed network has a spanning tree. An impulsive approach is then introduced and a simple criterion is presented to guarantee the consensus of all agents in the multi-agent model. Notes: CityU Call Number: TK5105.5 .X56 2010; ix, 140 leaves 30 cm.; Thesis (Ph.D.)--City University of Hong Kong, 2010.; Includes bibliographical references (leaves [122]-140) Fri, 01 Jan 2010 00:00:00 GMT http://dspace.cityu.edu.hk:80/handle/2031/6639 2010-01-01T00:00:00Z http://dspace.cityu.edu.hk:80/handle/2031/6638 Title: Some mathematical theories on the Cauchy problem and boundary layer problem for the Boltzmann equation Authors: Sun, Jie ( 孫傑) Abstract: This thesis focuses on the mathematical theories on the Cauchy problem and the boundary layer problem of the Boltzmann equation. The Cauchy problem is well-known. The boundary layer problem arises in the physical consideration of the condensation-evaporation problem, and is the first order approximation of the Boltzmann equation for small Knudsen number near a plane. The thesis is mainly divided into two parts. In the first part, the Cauchy problems of the Boltzmann equation with potential force in the whole space and in torus are investigated. In the whole space, we consider the Cauchy problem with potential force with some less restrictive assumptions compared to the previous works. We obtain the well-posedness theory and the optimal convergence rate of the solution to the Boltzmann equation even for the hard potential case by energy method, when the initial data is sufficiently close to a steady state. In torus, global existence and stability of solutions to the Cauchy problem of the Boltzmann equation with potential forces for hard potentials are considered. We prove the stationary state is asymptotically stable with exponential rate in time for any initially smooth, periodic, origin symmetric small perturbation which preserves the same total mass, momentum and mechanical energy as the natural steady state and any origin symmetric small potential force. In the second part, the boundary layer solutions to the Boltzmann equation with mixed boundary condition for the inverse power law are discussed. The existence of boundary layer solutions to the Boltzmann equation with mixed boundary condition, that is Dirichlet boundary condition weakly perturbed by diffuse reflection boundary condition at the wall, is considered. The boundary condition is imposed on the incoming particles, and the solution is supposed to approach to a global Maxwellian in the far field. Like the problem with Dirichlet boundary condition, the existence of a solution depends on the Mach number of the far field Maxwellian. Furthermore, an implicit solvability condition on the boundary data which shows the codimension of the boundary data is related to the number of positive characteristic speeds is also given. Notes: CityU Call Number: QC175.2 .S85 2011; v, 115 leaves 30 cm.; Thesis (Ph.D.)--City University of Hong Kong, 2011.; Includes bibliographical references (leaves [111]-115) Sat, 01 Jan 2011 00:00:00 GMT http://dspace.cityu.edu.hk:80/handle/2031/6638 2011-01-01T00:00:00Z http://dspace.cityu.edu.hk:80/handle/2031/6637 Title: Modeling diffraction gratings by modal methods and Chebyshev collocation Dirichlet-to-Neumann map method Authors: Song, Dawei ( 宋大偉) Abstract: Diffraction gratings are periodic structures with many practical applications, such as monochromators, spectrometers, lasers, wavelength division multiplexing devices, optical pulse compressing devices and other optical instruments. Numerical methods are essential in the design, analysis and optimization of grating structures. In principle, when the problem is formulated on one period of the structure, it can be solved by standard numerical methods, such as the finite element method (FEM). However, these general methods give rise to large, complex, indefinite linear systems that are relatively expensive to solve. Less general methods that take advantage of available geometric features are often more efficient. Existing methods for diffraction gratings include the analytic modal method, the Fourier modal method (FMM), the finite difference modal method, the differential method and the integral equation method, etc. All modal methods require that the structure consists of uniform layers, so that the wave field can be expanded in eigenmodes in each layer. Computing the eigenmodes in each layer is usually the most expensive part of the method. In this thesis, we first review the analytic modal method and the Fourier modal method. FMM calculates the eigenmodes based on Fourier series expansions. Since it is relatively easy to implement, FMM is extremely popular. Next, we derive a fourth order finite difference modal method. All modal methods need to solve the eigenvalue problems. Since the eigenvalue problems are relatively expensive to solve, we develop a Dirichlet-to-Neumann (DtN) map method for diffraction gratings with uniform layers. Instead of computing the eigenmodes in each layer, we calculate an operator that maps the wave field to its normal derivative at the boundaries of the layer. In practice, this operator, the so-called DtN map, is approximated by a matrix, and it is efficiently calculated using a highly accurate Chebyshev collocation method and a fourth order finite difference method to discretize the uniform and periodic directions, respectively. The DtN formalism has been previously used to analyze periodic arrays of cylinders and piecewise uniform waveguides. For circular cylinders, the DtN maps are constructed from cylindrical harmonics. For uniform waveguide segments, the Chebyshev collocation method was used with a second order finite difference method in the transverse direction to approximate the DtN maps. In our work, the fourth order finite difference scheme is used to discretize the periodic direction. As illustrated in numerical examples, our new method is more accurate than FMM, when the same degrees of freedom are used in the discretization, and it is also more efficient than FMM, since the time consuming eigenvalue decomposition is avoided and the DtN map can be calculated efficiently. Finally, the Chebyshev collocation DtN map method is extended to diffraction gratings in conical mounting. Notes: CityU Call Number: QC417 .S66 2010; vi, 92 leaves : ill. 30 cm.; Thesis (Ph.D.)--City University of Hong Kong, 2010.; Includes bibliographical references (leaves [89]-92) Fri, 01 Jan 2010 00:00:00 GMT http://dspace.cityu.edu.hk:80/handle/2031/6637 2010-01-01T00:00:00Z