Decomposing Dynamical Systems

Dynamical systems on monoids have been recently proposed as minimal mathematical models for the intuitive notion of deterministic dynamics. This paper shows that any dynamical system DS_L on a monoid L can be exhaustively decomposed into a family of mutually disconnected subsystems—the constituent systems of DS_L . In addition, constituent systems are themselves indecomposable, even though they may very well be complex. Finally, this work also makes clear how any dynamical system DS_L turns out to be identical to the sum of all its constituent systems. Constituent systems can thus be thought as the indecomposable, but possibly complex, building blocks to which the dynamics of an arbitrary complex system fully reduces. However, no further reduction of the constituents is possible, even if they are themselves complex.