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School of Mathematics

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College of Sciences

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Now showing 1 - 10 of 11

Physical Billiards and Open Dynamical Systems

(Georgia Institute of Technology, 2021-07-14) Attarchi, Hassan

This thesis consists of four works in dynamical systems with a focus on billiards. In the first part, we consider open dynamical systems, where there exists at least a hole of positive measure in the phase space which some portion of points in phase space escapes through that hole at each iterate of the dynamical system map. Here, we study the escape rate (a quantity that presents at what rate points in phase space escape through the hole) and various estimations of the escape rate of an open dynamical system. We uncover a reason why the escape rate is faster than expected, which is the convexity of the function defining escape rate. Moreover, exact computations of escape rate and its estimations are present for the skewed tent map and Arnold’s cat map. In the second part of the thesis, we study physical billiards where the moving particle has a finite nonzero size. In contrast to mathematical billiards where a trajectory is excluded when it hits a corner point of the boundary, in physical billiards reflection of the physical particle (a ball) off a visible corner point is well-defined. Initially, we study properties of such reflections in a physical billiards. Our results confirm that the reflection considered in the literature about physical billiards are indeed no-slip friction-free (elastic) collisions. In the third part of the thesis, we study physical Ehrenfests' wind-tree models, where we show that physical wind-tree models are dynamically richer than the well-known Lorentz gas model. More precisely, when we replace the point particle by a physical one (a ball), the wind-tree models show a new superdiffusive regimes that never been observed in any other model such as Lorentz gas. Finally, we prove that typical physical polygonal billiard is hyperbolic at least on a subset of positive measure and therefore has a positive Kolmogorov-Sinai entropy for any positive radius of the moving particle.
Topics in dynamical systems

(Georgia Institute of Technology, 2019-06-14) Shu, Longmei

The thesis consists of two parts. the first one is dealing with isosspectral transformations and the second one with the phenomenon of local immunodeficiency. Isospectral transformations (IT) of matrices and networks allow for compression of either object while keeping all the information about their eigenvalues and eigenvectors. Chapter 1 analyzes what happens to generalized eigenvectors under isospectral transformations and to what extent the initial network can be reconstructed from its compressed image under IT. We also generalize and essentially simplify the proof that eigenvectors are invariant under isospectral transformations and generalize and clarify the notion of spectral equivalence of networks. In the recently developed theory of isospectral transformations of networks isospectral compressions are performed with respect to some chosen characteristics (attributes) of the network's nodes (edges). Each isospectral compression (when a certain characteristic is fixed) defines a dynamical system on the space of all networks. Chapter 2 shows that any orbit of this dynamical system which starts at any finite network (as the initial point of this orbit) converges to an attractor. This attractor is a smaller network where the chosen characteristic has the same value for all nodes (or edges). We demonstrate that isospectral compressions of one and the same network defined by different characteristics of nodes (or edges) may converge to the same as well as to different attractors. It is also shown that a collection of networks may be spectrally equivalent with respect to some network characteristic but nonequivalent with respect to another. These results suggest a new constructive approach which allows to analyze and compare the topologies of different networks. Some basic aspects of the recently discovered phenomenon of local immunodeficiency generated by antigenic cooperation in cross-immunoreactivity (CR) networks are investigated in chapter 3. We prove that stable with respect to perturbations local immunodeficiency (LI) already occurs in very small networks and under general conditions on their parameters. Therefore our results are applicable not only to Hepatitis C where CR networks are known to be large, but also to other diseases with CR. A major necessary feature of such networks is the non-homogeneity of their topology. It is also shown that one can construct larger CR networks with stable LI by using small networks with stable LI as their building blocks. Our results imply that stable LI occurs in networks with quite general topologies. In particular, the scale-free property of a CR network, assumed previously, is not required.
Topics in dynamics: First passage probabilities and chaotic properties of the physical wind-tree model

(Georgia Institute of Technology, 2018-04-06) Bolding, Mark M.

We prove that the evolution of the "most chaotic" dynamical systems consists of three different stages. Consider a finite Markov partition (coarse graining) X of the phase space of a system. In the first short time interval there is a hierarchy with respect to the values of the first passage probabilities for the elements of X and therefore predictions can be made about which element an orbit will most likely hit. In the third infinitely long time interval there is an opposite hierarchy of the elements of X and therefore analogous finite time predictions can be made. Our results demonstrate that finite time predictions for the evolution of strongly chaotic dynamical systems are possible. We then show that mathematical billiards generated by the motion of a point particle do not adequately describe the dynamics of real physical particles within the same domain (billiard table), and introduce a new model for diffusion intended to revitalize the Ehrenfests’ Wind-Tree model.
Isospectral graph reductions, estimates of matrices' spectra, and eventually negative Schwarzian systems

(Georgia Institute of Technology, 2011-03-18) Webb, Benjamin Zachary

This dissertation can be essentially divided into two parts. The first, consisting of Chapters I, II, and III, studies the graph theoretic nature of complex systems. This includes the spectral properties of such systems and in particular their influence on the systems dynamics. In the second part of this dissertation, or Chapter IV, we consider a new class of one-dimensional dynamical systems or functions with an eventual negative Schwarzian derivative motivated by some maps arising in neuroscience. To aid in understanding the interplay between the graph structure of a network and its dynamics we first introduce the concept of an isospectral graph reduction in Chapter I. Mathematically, an isospectral graph transformation is a graph operation (equivalently matrix operation) that modifies the structure of a graph while preserving the eigenvalues of the graphs weighted adjacency matrix. Because of their properties such reductions can be used to study graphs (networks) modulo any specific graph structure e.g. cycles of length n, cliques of size k, nodes of minimal/maximal degree, centrality, betweenness, etc. The theory of isospectral graph reductions has also lead to improvements in the general theory of eigenvalue approximation. Specifically, such reductions can be used to improved the classical eigenvalue estimates of Gershgorin, Brauer, Brualdi, and Varga for a complex valued matrix. The details of these specific results are found in Chapter II. The theory of isospectral graph transformations is then used in Chapter III to study time-delayed dynamical systems and develop the notion of a dynamical network expansion and reduction which can be used to determine whether a network of interacting dynamical systems has a unique global attractor. In Chapter IV we consider one-dimensional dynamical systems of an interval. In the study of such systems it is often assumed that the functions involved have a negative Schwarzian derivative. Here we consider a generalization of this condition. Specifically, we consider the functions which have some iterate with a negative Schwarzian derivative and show that many known results generalize to this larger class of functions. This includes both systems with regular as well as chaotic dynamic properties.
Billiards and statistical mechanics

(Georgia Institute of Technology, 2009-05-18) Grigo, Alexander

In this thesis we consider mathematical problems related to different aspects of hard sphere systems. In the first part we study planar billiards, which arise in the context of hard sphere systems when only one or two spheres are present. In particular we investigate the possibility of elliptic periodic orbits in the general construction of hyperbolic billiards. We show that if non-absolutely focusing components are present there can be elliptic periodic orbits with arbitrarily long free paths. Furthermore, we show that smooth stadium like billiards have elliptic periodic orbits for a large range of separation distances. In the second part we consider hard sphere systems with a large number of particles, which we model by the Boltzmann equation. We develop a new approach to derive hydrodynamic limits, which is based on classical methods of geometric singular perturbation theory of ordinary differential equations. This method provides new geometric and dynamical interpretations of hydrodynamic limits, in particular, for the of the dissipative Boltzmann equation.
Some problems in the theory of open dynamical systems and deterministic walks in random environments

(Georgia Institute of Technology, 2008-11-11) Yurchenko, Aleksey

The first part of this work deals with open dynamical systems. A natural question of how the survival probability depends upon a position of a hole was seemingly never addresses in the theory of open dynamical systems. We found that this dependency could be very essential. The main results are related to the holes with equal sizes (measure) in the phase space of strongly chaotic maps. Take in each hole a periodic point of minimal period. Then the faster escape occurs through the hole where this minimal period assumes its maximal value. The results are valid for all finite times (starting with the minimal period), which is unusual in dynamical systems theory where typically statements are asymptotic when time tends to infinity. It seems obvious that the bigger the hole is the bigger is the escape through that hole. Our results demonstrate that generally it is not true, and that specific features of the dynamics may play a role comparable to the size of the hole. In the second part we consider some classes of cellular automata called Deterministic Walks in Random Environments on Z^1. At first we deal with the system with constant rigidity and Markovian distribution of scatterers on Z^1. It is shown that these systems have essentially the same properties as DWRE on Z^1 with constant rigidity and independently distributed scatterers. Lastly, we consider a system with non-constant rigidity (so called process of aging) and independent distribution of scatterers. Asymptotic laws for the dynamics of perturbations propagating in such environments with aging are obtained.
Lorentz Lattice Gases on Graphs

(Georgia Institute of Technology, 2003-11-26) Kreslavskiy, Dmitry Michael

The present work consists of three parts. In the first part (chapters III and IV), the dynamics of Lorentz lattice gases (LLG) on graphs is analyzed. We study the fixed scatterer model on finite graphs. A tight bound is established on the size of the orbit for arbitrary graphs, and the model is shown to perform a depth-first search on trees. Rigidity models on trees are also considered, and the size of the resulting orbit is established. In the second part (chapter V), we give a complete description of dynamics for LLG on the one-dimensional integer lattice, with a particular interest in showing that these models are not capable of universal computation. Some statistical properties of these models are also analyzed. In the third part (chapter VI) we attempt to partition a pool of workers into teams that will function as independent TSS lines. Such partitioning may be aimed to make sure that all groups work at approximately the same rate. Alternatively, we may seek to maximize the rate of convergence of the corresponding dynamical systems to their fixed points with optimal production at the fastest rate. The first problem is shown to be NP-hard. For the second problem, a solution for splitting into pairs is given, and it is also shown that this solution is not valid for partitioning into teams composed of more than two workers.
Dynamical and statistical properties of Lorentz lattice gases

(Georgia Institute of Technology, 2003-05) Khlabystova, Milena
Dynamics of billiards

(Georgia Institute of Technology, 2002-08) Del Magno Gianluigi
The peak-crossing bifurcation in lattice dynamical systems

(Georgia Institute of Technology, 1996-08) Venkatagiri, Shankar C.