## Electronic Theses and Dissertations

#### Identifier

2612

#### Date

2016

#### Date of Award

4-18-2016

#### Document Type

Dissertation

#### Degree Name

Doctor of Philosophy

#### Major

Mathematical Sciences

#### Concentration

Mathematics

#### Committee Chair

Robert Kozma

#### Committee Member

George Anastassiou

#### Committee Member

Fernanda Botelho

#### Committee Member

Ebenezer George

#### Abstract

In this dissertation we consider some dynamical systems and their properties. We study the stability of a discrete system, which corresponds to an approximate dynamic programming problem. We investigate phase transition of a process on random graphs and find critical parameters. We analyze the bifurcation and attractor of a system given by generalized Lotka-Volterra equations. In particular:we study the stability of a discrete dynamical system of estimation error, which corresponds to an approximate dynamic programming (ADP) problem via Lyapunov's second method. We prove that the system is uniformly ultimately bounded;we show the necessary conditions for a phase transition in two randomly coupled probabilistic cellular automata in mean-field approximation and prove the existence of limit cycle behavior;we introduce a new random graph model G_{Z^2_N, p_d}, a discrete torus with random edges defined with respect to graph distances between vertices on the torus. We prove that the degree probability distribution is approximately Poisson and the diameter of the graph is D(G{Z^2_N, p_d}) = Theta (log N), whp;we study bootstrap percolation on G_{Z^2_N, p_d}. Sharp conditions are derived for phase transition at different values of k with a k-threshold rule in mean-field approximation. We generalize the bootstrap percolation on G_{Z^2_N, p_d} to the case of two types of vertices with a modified k-threshold rule. We derive some bounds for critical probabilities in the generalized model in mean-field approximation;we study the bifurcation of two coupled systems each described by generalized Lotka-Volterra equations with respect to coupling. Also, we study a dissipative system with an inhomogeneous heteroclinic cycle, that is, each equilibrium in the cycle is either with one or two unstable directions. We prove that there exists an asymptotically stable set consisting of unstable manifolds of the saddles. Most of the results in the dissertation has been published, which represent joint work with Svante Janson, Robert Kozma, Mikhail I. Rabinovich, Miklos Ruszinko, Ludmilla D. Werbos, and Paul Werbos.

#### Library Comment

dissertation or thesis originally submitted to the local University of Memphis Electronic Theses & dissertation (ETD) Repository.

#### Recommended Citation

Sokolov, Yury, "Dynamics of Discrete And Continuous Spatially Distributed Systems" (2016). *Electronic Theses and Dissertations*. 1358.

https://digitalcommons.memphis.edu/etd/1358

## Comments

Data is provided by the student.