Provably convergent implementations of the subdivision algorithm for the computation of invariant objects

TL;DR

This paper proves that practical, slightly modified versions of the subdivision algorithm for invariant set computation are convergent under mild assumptions, without requiring exact dynamics or fine sampling.

Contribution

It demonstrates the convergence of implementable subdivision algorithms for invariant sets without the need for precise dynamics or dense sampling.

Findings

Convergence holds under mild assumptions.

Exact dynamics and fine sampling are unnecessary.

Modified algorithms are practically implementable.

Abstract

The subdivision algorithm by Dellnitz and Hohmann for the computation of invariant sets of dynamical systems decomposes the relevant region of the state space into boxes and analyzes the induced box dynamics. Its convergence is proved in an idealized setting, assuming that the exact time evolution of these boxes can be computed. In the present article, we show that slightly modified, directly implementable versions of the original algorithm are convergent under very mild assumptions on the dynamical system. In particular, we demonastrate that neither a fine net of sample points nor very accurate approximations of the precise dynamics are necessary to guarantee convergence of the overall scheme.