Publication Date

2009-04-25

Availability

Open access

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PHD)

Department

Mathematics (Arts and Sciences)

Date of Defense

2009-04-15

First Committee Member

Lev Kapitanski - Committee Chair

Second Committee Member

Gregory J. Galloway - Committee Member

Third Committee Member

Subramanian Ramakrishnan - Committee Member

Fourth Committee Member

Hüseyin Koçak - Outside Committee Member

Abstract

Dynamics with choice is a generalization of discrete-time dynamics where instead of the same evolution operator at every time step there is a choice of operators to transform the current state of the system. Many real life processes studied in chemical physics, engineering, biology and medicine, from autocatalytic reaction systems to switched systems to cellular biochemical processes to malaria transmission in urban environments, exhibit the properties described by dynamics with choice. We study the long-term behavior in dynamics with choice. We prove very general results on the existence and properties of global compact attractors in dynamics with choice. In addition, we study the dynamics with restricted choice when the allowed sequences of operators correspond to subshifts of the full shift. One of practical consequences of our results is that when the parameters of a discrete-time system are not known exactly and/or are subject to change due to internal instability, or a strategy, or Nature's intervention, the long term behavior of the system may not be correctly described by a system with "averaged" values for the parameters. There may be a Gestalt effect.

Keywords

Symbolic Dynamics; Discrete Dynamical System; Attractor; Dynamics With Choice

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