The categorical limit of a sequence of dynamical systems

TL;DR

This paper introduces a category-theoretic approach to defining limits of sequences of dynamical systems, resolving ambiguity issues present in topological methods and enabling better approximation of continuous dynamics.

Contribution

It formalizes limits of dynamical system sequences using category theory, explicitly modeling behavior relations and proving the existence of projective limits.

Findings

All projective limits exist in the defined category.

The approach resolves ambiguities present in topological limit definitions.

Enables approximation of continuous dynamics through discrete sequences.

Abstract

Modeling a sequence of design steps, or a sequence of parameter settings, yields a sequence of dynamical systems. In many cases, such a sequence is intended to approximate a certain limit case. However, formally defining that limit turns out to be subject to ambiguity. Depending on the interpretation of the sequence, i.e. depending on how the behaviors of the systems in the sequence are related, it may vary what the limit should be. Topologies, and in particular metrics, define limits uniquely, if they exist. Thus they select one interpretation implicitly and leave no room for other interpretations. In this paper, we define limits using category theory, and use the mentioned relations between system behaviors explicitly. This resolves the problem of ambiguity in a more controlled way. We introduce a category of prefix orders on executions and partial history preserving maps between them to describe both discrete and continuous branching time dynamics. We prove that in this category all projective limits exist, and illustrate how ambiguity in the definition of limits is resolved using an example. Moreover, we show how various problems with known topological approaches are now resolved, and how the construction of projective limits enables us to approximate continuous time dynamics as a sequence of discrete time systems.