The main goal of this work is to analyse coupled systems of ordinary differential equations on networks, that is, where the coupling is described by a graph. The aim is to give conditions on the networks, and on the different parameters appearing in the equations, in order to be able to infer the existence of a globally stable equilibrium for the complete system. For this we have mainly followed the work [14]. At each vertex of the graph there is odes system with its own dynamics, and the interaction between vertices is represented by edges. This interaction can be nonlinear and dependent on the variables of both vertices. The main objective will be to construct global Lyapunov functions that depend on both the variables and the properties of the graphs. The existence of such a function will in many cases imply the existence of a globally stable equilibrium. This general result will be applied to different concrete problems: oscillators, population dynamics and epidemic problems.

Universidad de Sevilla. Grado en Matemáticas