Studying discrete dynamical systems trough differential equations

Abstract

In this paper we consider dynamical systems generated by a diffeomorphism F defined on U an open subset of R^n, and give conditions over F which imply that their dynamics can be understood by studying the flow of an associated differential equation, $\dot{x}=X(x),$ also defined on U. In particular the case where F has n-1 functionally independent first integrals is considered. In this case X is constructed by imposing that it shares with $F$ the same set of first integrals and that the functional equation $\mu (F(x))=\det ((DF(x))\mu (x),$ for x in U has some non-zero solution. Several examples for n=2,3 are presented, most of them coming from several well-known difference equations.

22 pages; 3 Figures

References in corpus (2)

• Dynamics of the third order Lyness' difference equation
• Third-order integrable difference equations generated by a pair of second-order equations

Cited by in corpus (11)

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• Non-integrability of measure preserving maps via Lie symmetries
• On two and three periodic Lyness difference equations
• On Poncelet's maps
• Lie Symmetries of Birational Maps Preserving Genus 0 Fibrations
• Meromorphic first integrals of analytic diffeomorphisms
• Kahan-Hirota-Kimura maps preserving original cubic hamiltonians