On the Computational Complexity of Limit Cycles in Dynamical Systems

Christos H. Papadimitriou; Nisheeth K. Vishnoi

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arXiv · 1511.07605

On the Computational Complexity of Limit Cycles in Dynamical Systems

Christos H. Papadimitriou·Nisheeth K. Vishnoi

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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" Poincaré-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.

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BibTeX RIS

Christos H. Papadimitriou, Nisheeth K. Vishnoi. 2015-11-24. On the Computational Complexity of Limit Cycles in Dynamical Systems. https://arxiv.org/abs/1511.07605

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