Computing the exact number of periodic orbits for planar flows

In this paper, we consider the problem of determining the \emph{exact} number of periodic orbits for polynomial planar flows. This problem is a variant of Hilbert's 16th problem. Using a natural definition of computability, we show that the problem is noncomputable on the one hand and, on the o...

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Bibliographic Details
Published inarXiv.org
Main Authors Graça, Daniel S, Zhong, Ning
Format Paper Journal Article
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 28.10.2022
Subjects
Computer Science - Logic in Computer Science
Mathematics - Dynamical Systems
Mathematics - Logic
Orbits
Polynomials
Upper bounds
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Summary:In this paper, we consider the problem of determining the \emph{exact} number of periodic orbits for polynomial planar flows. This problem is a variant of Hilbert's 16th problem. Using a natural definition of computability, we show that the problem is noncomputable on the one hand and, on the other hand, computable uniformly on the set of all structurally stable systems defined on the unit disk. We also prove that there is a family of polynomial planar systems which does not have a computable sharp upper bound on the number of its periodic orbits.
ISSN:2331-8422
DOI:10.48550/arxiv.2101.07701