Dynamical Systems as Functorial Realisations of Abstract Evolution Shapes

We develop a categorical framework for closed dynamical systems in which the abstract pattern of admissible evolutions is separated from its concrete realisation.

A closed dynamical system is formulated as a functor $X\colon S\to C$ from a small category $S$, viewed as an abstract evolution shape, to a coefficient category $C$.

By varying $S$ and $C$, this single definition encompasses many important examples including autonomous, non-autonomous, switched, hybrid, and stochastic systems.

Within this framework, we introduce invariant subsystems, equilibria, and orbits in functorial terms.

We then formulate convergence by combining a cosieve-based intrinsic notion of eventuality on the evolution shape with neighbourhood filters of invariant subsystems.

Finally, we establish a categorical Lyapunov principle based on categorical sublevel neighbourhoods.

This yields abstract stability and convergence criteria that recover the classical Lyapunov method in standard examples.