Relationships between network connectivity and global dynamics of complex dynamical systems

Abstract

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.