Coupled nonlinear dynamical systems: asymptotic behavior and uncertainty propagation

We discuss asymptotic behavior and uncertainty propagation in nonlinear dynamical systems on graphs. We utilize digraph decomposition theory of the system employed here in its Markov chain theory realization, to prove several results on asymptotic dynamics of large systems that depend only on their structural properties. We discuss disturbance propagation in such systems and show that it depends on the node distance in the associated graph. We use the framework to refine the recent symmetry grupoid theory of interconnected dynamical systems.