Inversible Max-Plus algebras and integrable systems

Let F be a real or complex n-dimensional map. It is said that F is globally periodic if there exists some p ∈ N + such that F p (x) = x for all x, whereThe minimal p satisfying this property is called the period of F. Given a m-dimensional parametric family of maps, say F λ , a problem of current interest is to determine all the values of λ such that F λ is globally periodic, together with their corresponding periods. The aim of this paper is to show some techniques that we use to face this question, as well as some recent results that we have obtained. We will focus on proving the equivalence of the problem with the complete integrability of the dynamical system induced by the map F, and related issues; on the use of the local linearization given by the Bochner Theorem; and on the use the Normal Form theory. We also present some open questions in this setting.

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“…This recurrence is the ultradiscrete version of the Lyness recurrence; see [46] for further details on ultradiscrete systems.…”

Section: Introductionmentioning

confidence: 99%

Different Approaches to the Global Periodicity Problem

Cima

Gasull

Mañosa

et al. 2016

Springer Proceedings in Mathematics &Amp; Statistics

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“…Nowadays, discrete-time integrable systems is also an area of a deep research activity, see for instance Refs. [6,9,11,12,15,16,25,22,29]. Recall that a first integral of the DDS generated by F is a non-constant K-valued function H which is constant on the orbits of the DDS.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Also notice that, according to Ref. [29], equation (9) is the ultradiscrete version of the Lyness recurrence.…”

Section: Proposition 17mentioning

confidence: 98%

Global periodicity and complete integrability of discrete dynamical systems

Cima¹,

Gasull²,

Mañosa³

2006

Journal of Difference Equations and Applications

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Consider the discrete dynamical system generated by a map F. It is said that it is globally periodic if there exists a natural number p such that F p (x)Zx for all x in the phase space. On the other hand, it is called completely integrable if it has as many functionally independent first integrals as the dimension of the phase space. In this paper, we relate both concepts. We also give a large list of globally periodic dynamical systems together with a complete set of their first integrals, emphasizing the ones coming from difference equations.

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“…There have been many attempts at reconciling the role of subtraction in the ultradiscretization procedure, such as the so-called inversible max-plus algebra [22], the s-ultradiscretization [12] and more analytic approaches [18]. The approach that is closest to our framework is that of Kasman and Lafortune [18], yet they do not make any tropical geometric connections in their work.…”

Section: Introductionmentioning

confidence: 99%

Tropical geometric interpretation of ultradiscrete singularity confinement

Ormerod¹

2013

J. Phys. A: Math. Theor.

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Abstract. Using the interpretation of the ultradiscretization procedure as a non-Archimedean valuation, we use results of tropical geometry to show how roots and poles manifest themselves in piece-wise linear systems as points of non-differentiability. This will allow us to demonstrate a correspondence between singularity confinement for discrete integrable systems and ultradiscrete singularity confinement for ultradiscrete integrable systems.

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Inversible Max-Plus algebras and integrable systems

Cited by 13 publications

References 10 publications

Different Approaches to the Global Periodicity Problem

Different Approaches to the Global Periodicity Problem

Global periodicity and complete integrability of discrete dynamical systems

Tropical geometric interpretation of ultradiscrete singularity confinement

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