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Relationships between network connectivity and global dynamics of complex dynamical systems Open Access


Other title
interaction network
complex system
clustered behavior
coupled dynamical system
strongly connected component
Type of item
Degree grantor
University of Alberta
Author or creator
Supervisor and department
Michael Y. Li (Department of Mathematical and Statistical Sciences)
Examining committee member and department
Hao Wang (Department of Mathematical and Statistical Sciences)
Anthony T-M Lau (Department of Mathematical and Statistical Sciences)
Bin Han (Department of Mathematical and Statistical Sciences)
Xingfu Zou (Department of Applied Mathematics, University of Western Ontario)
James Muldowney (Department of Mathematical and Statistical Sciences)
Department of Mathematical and Statistical Sciences
Applied Mathematics
Date accepted
Graduation date
Doctor of Philosophy
Degree level
The global dynamics of complex systems is investigated in this thesis, using the framework of coupled dynamical systems. For a coupled dynamical system on an interaction network, we show the impact of the connectivity of the interaction network on its dynamical behavior. We lay particular emphasis on non-strongly connected interaction networks, and clustered behavior of coupled dynamical systems. Two typical kinds of coupled dynamical systems are studied in the thesis: coupled gradient systems and coupled oscillators. We present a general approach to investigating the dynamical behaviors of coupled gradient systems. The approach is demonstrated through two multi-group epidemic models: one ordinary differential equation model and one functional differential equation model with distributed delay. We show disease either persists in all groups of one strongly connected component or dies out in all groups of one strongly connected component. Moreover, we present a threshold value that determines whether disease persists or dies out in one strongly connected component. We study both coupled linear and nonlinear oscillators in the thesis. For systems of coupled linear oscillators, we show its dynamical behavior under arbitrary interaction networks. When the interaction network is strongly connected, synchronization occurs; otherwise, clustered behavior may occur. In the case of clustered behavior, we show the frequency of oscillators in the same strongly connected components are the same. For systems of coupled nonlinear oscillators, synchronization occurs when its interaction network is strongly connected; otherwise, we show synchronization can occur when the coupling strength between any two strongly connected components is sufficiently large. For coupled gradient systems and coupled oscillators, our analysis shows synchronization occurs under strongly connected interaction networks; while non-strongly connected interaction networks give rise to clustered behavior. In the case of clustered behavior, local systems in one strongly connected components are in the same dynamical cluster.
This thesis is made available by the University of Alberta Libraries with permission of the copyright owner solely for the purpose of private, scholarly or scientific research. This thesis, or any portion thereof, may not otherwise be copied or reproduced without the written consent of the copyright owner, except to the extent permitted by Canadian copyright law.
Citation for previous publication
P. Du and M. Y. Li, {Impact of network connectivity on the synchronization and global dynamics of coupled systems of differential equations}, {\it Physica D}, \textbf{286-287} (2014), 32--42.

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