Topics in Dynamical Systems

Abstract

In this thesis I explore three new topics in Dynamical Systems. In Chapters 2 and 3 I investigate the dynamics of a family of asynchronous linear systems. These systems are of interest as models for asynchronous processes in economics and computer science and as novel ways to solve linear equations. I find a tight sandwich of bounds relating the Lyapunov exponents of these asynchronous systems to the eigenvalue of their synchronous counterparts. Using ideas from the theory of IFSs I show how the random behavior of these systems can be quickly sampled and go some way to characterizing the associated probability measures.In Chapter 4 I consider another family of random linear dynamical system but this time over the Max-plus semi-ring. These models provide a linear way to model essentially non-linear queueing systems. I show how the topology of the queue network impacts on the dynamics, in particular I relate an eigenvalue of the adjacency matrix to the throughput of the queue.In Chapter 5 I consider non-smooth systems which can be used to model a wide variety of physical systems in engineering as well as systems in control and computer science. I introduce the Moving Average Transformation which allows us to systematically 'smooth' these systems enabling us to apply standard techniques that rely on some smoothness, for example computing Lyapunov exponents from time series data.

Date of Award	1 Aug 2012
Original language	English
Awarding Institution	The University of Manchester
Supervisor	David Broomhead (Supervisor) & Paul Glendinning (Supervisor)