Boolean networks, neural networks and reaction-diffusion automata share a common structure which we identify as a special class of a new notion of dynamical systems on graphs for which we present a Lyapunov function type concept which implies phase-locking of the dynamics.  For dynamical systems with 'time' being a monoid instead of the integers or the reals, we define a notion of chaos which extends Devaney's classical chaos notion and we prove a theorem that sensitive dependence of initial conditions is a consequence of the two other properties in the definition. The common theme of the two topics is the intention to push the limits of dynamical systems theory in order to investigate how coupling or feedback motifs influence macroscopic behavior and discuss the role of time being a line.