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Title: Bifurcations in lattice dynamical systems
Author: Johnson, M. E.
Awarding Body: University of Cambridge
Current Institution: University of Cambridge
Date of Award: 1997
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In this thesis I consider bifurcations in lattice dynamical systems, primarily coupled map lattices and lattice differential equations. First I explain how certain coupled map lattices can be considered as cellular automata, on all or part of each of state- or parameter-space, and outline what can be gained from such a consideration. Then I consider bifurcations of fixed points in locally bistable coupled map lattices, introducing various analytical techniques for the study of piecewise-linear lattice dynamical systems, and showing how numerical techniques allow smooth systems to be investigated also. In particular, bifurcation diagrams and bifurcation sets for "kink" and "bump" fixed points are constructed, and these reveal that there will be small regions of parameter space in which no stable bump fixed points exist, though many such fixed points exist on each side of these regions. Strange behaviour occurs in such regions, with very long transients being observed. Bifurcations from homogenous fixed points of lattice dynamical systems are then investigated using the methods of equivalent bifurcation theory. As well as locating such bifurcations, I obtain information regarding branching directions, numbers of branches, and branch stabilities. The effect of the local dynamical units having odd symmetry is discussed. Some aspects of the theory of infinite-dimensional lattice dynamical systems are then considered. I discuss the connection between fixed points of these systems and orbits of a certain area-preserving map of the plane, and extend some earlier results regarding the symbolic dynamics of this map. I then introduce a shadowing-based technique which allows us to infer the existence of certain fixed points on the infinite lattice using our knowledge of similar fixed points on finite lattices.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available