Spatial decay analysis in interconnected dynamical systems using vector Lyapunov functions

In this paper we analyze the propagation of input signals in a large-scale network of dynamical systems. Using vector Lyapunov functions, the individual multidimensional subsystems are first reduced to an approximating scalar representation in the form of the evolution of their weighted norm. The norm simultaneously qualifies as a local Lyapunov function for the isolated subsystem. Employing properties of M-matrices, we then derive linear system dynamics which provide an upper bound for the evolution of the original system, and use them to investigate the decay between hops from subsystem to subsystem of the steady state magnitude. Two input signals are considered: Constant input, and a sinusoidal input. The results are demonstrated using numerical simulations.