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|Title:||Synchronisation on non-compact manifolds|
|School/Discipline:||School of Physical Sciences|
|Abstract:||The Kuramoto model is a widely studied model of phase synchronisation which continues to display novel and interesting emergent behaviours with its many extensions and generalisations. We formulate a non-compact version of the Kuramoto model by replacing the compact symmetry group SO(2) with the non-compact group SO(1, 1). The N equations are similar to the Kuramoto equations, except that the trigonometric terms are replaced with hyperbolic functions. Solution trajectories are generally unbounded, lying on the unit hyperbola, but synchronisation occurs for any positive coupling constants and arbitrary driving parameters. We show solutions can develop singularities for negative coupling constants. We also develop a vector model of synchronisation on non-compact manifolds. By choosing the group SO(1, 2) trajectories are confined to a one-sheeted hyperboloid. This model also displays unbounded trajectories but can synchronise for negative couplings and restricted initial conditions. Singularities can occur for positive couplings or if the initial conditions are too widely distributed. We describe a physical interpretation of the SO(1, 1) model as a system of interacting relativistic particles in 1 + 1 spacetime dimensions.|
|Dissertation Note:||Thesis (MPhil.) -- University of Adelaide, School of Physical Sciences, 2018|
|Provenance:||This electronic version is made publicly available by the University of Adelaide in accordance with its open access policy for student theses. Copyright in this thesis remains with the author. This thesis may incorporate third party material which has been used by the author pursuant to Fair Dealing exceptions. If you are the owner of any included third party copyright material you wish to be removed from this electronic version, please complete the take down form located at: http://www.adelaide.edu.au/legals|
|Appears in Collections:||Research Theses|
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