Topics in Dynamical Systems

Hook, JL (2012) Topics in Dynamical Systems. Doctoral thesis, Manchester Institute for Mathematical Sciences, The University of Manchester.

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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 sys- tems 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 net- work 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.

Item Type: Thesis (Doctoral)
Uncontrolled Keywords: dynamical systems, max-plus algebra, max-plus linear systems, max-plus stochastic, stochastic, queuing theory, mrna, asynchronous iteration, graph laplacian, chaotic relaxation, non-smooth mechanics, regularization, time-series filtering, signal processing
Subjects: MSC 2010, the AMS's Mathematics Subject Classification > 37 Dynamical systems and ergodic theory
Depositing User: Mr James Hook
Date Deposited: 23 Jul 2013
Last Modified: 20 Oct 2017 14:13

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