A Restricted Conley Index and Robust Dynamics of Coupled Oscillator Systems
Ismail, Asma Farj Alramle
Date: 10 June 2013
Thesis or dissertation
Publisher
University of Exeter
Degree Title
PhD in Mathematics
Abstract
In this thesis, we explore the robustness of heteroclinic cycles which can appear as solutions
of dynamical systems subject to certain constraints. We develop a method, using
topological notions, to inspect the dynamics of simple heteroclinic cycles; in particular,
in the first part of the thesis we develop a “restricted Conley ...
In this thesis, we explore the robustness of heteroclinic cycles which can appear as solutions
of dynamical systems subject to certain constraints. We develop a method, using
topological notions, to inspect the dynamics of simple heteroclinic cycles; in particular,
in the first part of the thesis we develop a “restricted Conley index”. This tool is defined
by restricting the general Conley index to specific invariant subspaces associated with
constraints on the vector field. The resulting restricted Conley index allows us to find
connections that are robust to perturbations that respect these constraints. In the second
part of this thesis we study the dynamical problem of designing a system of globally
coupled oscillator systems with a specific structure. We extend some known results on
conditions for stability of cluster states in these systems and, as an example, we give sufficient
stability conditions for cluster states with three non-trivial clusters, (2, 2, 2)−cluster
states, in a system of six globally coupled oscillators. We show that robust heteroclinic
cycles connecting these states can appear as a result of our investigation on dynamics of
a systems of six globally coupled oscillators, and we use the restricted Conley index to
investigate the robustness of heteroclinic cycles between three nontrivial clusters.
Doctoral Theses
Doctoral College
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