Dissertation
CLUSTER SOLUTIONS IN NETWORKS OF WEAKLY COUPLED OSCILLATORS ON A 2D TORUS
Washington State University
Doctor of Philosophy (PhD), Washington State University
01/2021
DOI:
https://doi.org/10.7273/000005432
Handle:
https://hdl.handle.net/2376/119115
Abstract
The present dissertation is devoted to the analysis of the existence and stability of cluster solutions in networks of weakly coupled oscillators on a 2D square torus with von Neumann neighbor coupling. We use the phase reduction technique to analysis the phase difference model for the oscillators in our network. We then use the phase model to study the existence of cluster solutions on the 2D torus network topology. Additionally, the stability of the cluster solutions will be done using spectrum analysis techniques.
We consider an N by N lattice of weakly coupled oscillators with periodic boundary conditions, or equivalently a 2D square torus, for our network topology. We begin with nearest neighbor coupling and study the existence and stability of cluster solutions of the network with heterogenous coupling. We continue with a similar analysis for the second nearest neighbor case, and then, we consider the case for a general von Neumann nearest neighbor coupling. The analysis of homogeneous coupling is also presented. We also discuss the generality of distance dependent coupling and directionally independent phase differences. The analysis of the stability of the cluster solutions is made possible by the use of the Kronecker product and circulant matrix theory in the construction of the adjacency matrix for our 2D network.
In our numerical results section, we conduct various simulations to validate the above analytical results. We consider systems of identical Morris-Lecar neurons with inhibitory coupling, various network sizes, and couplings. Within these simulations we examine the effect of varying coupling values on the robustness of the stability of our cluster solutions. Specifically, we examine how varying coupling from the homogenous to heterogeneous case can cause a change in stability. We also conduct parameter analysis to explore the effects of varying parameters on the stability of our cluster solutions.
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Details
- Title
- CLUSTER SOLUTIONS IN NETWORKS OF WEAKLY COUPLED OSCILLATORS ON A 2D TORUS
- Creators
- Jordan Michael Culp
- Contributors
- Xueying Wang (Advisor)Robert Dillon (Committee Member)Mark Schumaker (Committee Member)Nairanjana Dasgupta (Committee Member)
- Awarding Institution
- Washington State University
- Academic Unit
- Department of Mathematics and Statistics
- Theses and Dissertations
- Doctor of Philosophy (PhD), Washington State University
- Publisher
- Washington State University
- Number of pages
- 160
- Identifiers
- 99900591956501842
- Language
- English
- Resource Type
- Dissertation