On the Computational Complexity of Limit Cycles in Dynamical Systems

TL;DR

This paper investigates the computational difficulty of identifying limit cycles in two-dimensional dynamical systems, revealing that exact problems are uncomputable or PSPACE-complete, but approximate cycles can be efficiently guaranteed in all dimensions.

Contribution

It establishes the uncomputability of exact limit cycle detection and introduces an approximate cycle concept with a PSPACE-complete computational complexity in all dimensions.

Findings

Exact limit cycle problems are uncomputable in continuous systems.

Discrete analogue problems are PSPACE-complete.

An approximate Poincaré-Bendixson theorem holds universally, with PSPACE-complete complexity.

Abstract

We study the Poincare-Bendixson theorem for two-dimensional continuous dynamical systems in compact domains from the point of view of computation, seeking algorithms for finding the limit cycle promised by this classical result. We start by considering a discrete analogue of this theorem and show that both finding a point on a limit cycle, and determining if a given point is on one, are PSPACE-complete. For the continuous version, we show that both problems are uncomputable in the real complexity sense; i.e., their complexity is arbitrarily high. Subsequently, we introduce a notion of an "approximate cycle" and prove an "approximate" Poincar\'e-Bendixson theorem guaranteeing that some orbits come very close to forming a cycle in the absence of approximate fixpoints; surprisingly, it holds for all dimensions. The corresponding computational problem defined in terms of arithmetic circuits is PSPACE-complete.