Impact of network connectivity on the synchronization and global dynamics of coupled systems of differential equations

• We investigated a general class of coupled dynamical systems on networks.
• We established the global dynamics when the network is not strongly connected.
• We investigated the impact of network connectivity on the global dynamics.
• When vertex systems are gradient, network connectivity completely determines the synchronization.
• We applied the theory to two significant applications in epidemiology and ecology.

The global dynamics of coupled systems of differential equations defined on an interaction network are investigated. Local dynamics at each vertex, when interactions are absent, are assumed to be simple: solutions to each vertex system are assumed to converge to an equilibrium, either on the boundary or in the interior of the feasible region. The interest is to investigate the collective behaviours of the coupled system when interactions among vertex systems are present. It was shown in Li and Shuai (2010) that, if the interaction network is strongly connected, then solutions to the coupled system synchronize at a single equilibrium. We focus on the case when the underlying network is not strongly connected and the coupled system may have mixed equilibria whose coordinates are in the interior at some vertices while on the boundary at others. We show that solutions on a strongly connected component of the network will synchronize. Considering a condensed digraph by collapsing each strongly connected component, we are able to introduce a partial order on the set PP of all equilibria, and show that all solutions of the coupled system converge to a unique equilibrium P∗P∗ that is the maximizer in PP. We further establish that behaviours of the coupled system at minimal elements of the condensed digraph determine whether the global limit P∗P∗ is a mixed equilibrium. The theory are applied to mathematical models from epidemiology and spatial ecology.