2026/07/20 by Bangxin Wang
Mathematics · #math.CT #math.DS
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→ 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.