Some properties of the<i>k</i>-dimensional Lyness' map

Cima, Anna; Gasull, Armengol; Mañosa, Vı́ctor

doi:10.1088/1751-8113/41/28/285205

Cited by 13 publications

(13 citation statements)

References 32 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…If F [5] = F e,d,c,b,a has a meromorphic first integral, the same holds for all the other maps Of course, all the known rationally integrable cases, like F [5] = F 5 a , F [10] = F 5 ba , F [15] = F 3 c,b,a and F [20] In fact we prove:…”

Section: Rational Integrability and Associated Dynamicssupporting

confidence: 54%

Integrability and non-integrability of periodic non-autonomous Lyness recurrences

2013

Self Cite

View full text Add to dashboard Cite

This paper studies non-autonomous Lyness type recurrences of the form x n+2 = (a n + x n )/x n+1 , where {a n } n is a k-periodic sequence of positive numbers with prime period k. We show that for the cases k ∈ {1, 2, 3, 6} the behavior of the sequence {x n } n is simple(integrable) while for the remaining cases satisfying k =5 this behavior can be much more complicated(chaotic). The cases k =5 are studied separately.

show abstract

Section: Rational Integrability and Associated Dynamicssupporting

confidence: 54%

Integrability and non-integrability of periodic non-autonomous Lyness recurrences

2013

Self Cite

View full text Add to dashboard Cite

show abstract

“…is globally 4-periodic because it corresponds to G • G, where G is the Todd's map given in Example 1; see also [12]. In the first octant it has the Lie symmetry given by…”

Section: Example 9 the Mapmentioning

confidence: 99%

Different Approaches to the Global Periodicity Problem

Cima

Gasull

Mañosa

et al. 2016

Springer Proceedings in Mathematics &Amp; Statistics

Self Cite

View full text Add to dashboard Cite

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.

show abstract

“…It is worth to comment that for a = 1 it is well known that the map (16) has effectively 3 functionally independent rational first integrals. Two of them exist for any a ∈ C. They are see for instance [6] and its references, and a third one can be seen in [4] and it is found by using the tools introduced in that paper. It exists because this map, for a = 1, corresponds to the celebrated 3 rd order Todd's difference equation x n+3 = (1 + x n+2 + x n )/x n which is globally 8-periodic, that is x n+8 = x n for all n, whenever x k is well defined.…”

Section: Higher Dimensional Examplesmentioning

confidence: 95%

Meromorphic first integrals of analytic diffeomorphisms

Ferragut¹,

Gasull²,

Xiang³

2020

Preprint

Self Cite

View full text Add to dashboard Cite

We give an upper bound for the number of functionally independent meromorphic first integrals that a discrete dynamical system generated by an analytic map f can have in a neighborhood of one of its fixed points. This bound is obtained in terms of the resonances among the eigenvalues of the differential of f at this point. Our approach is inspired on similar Poincaré type results for ordinary differential equations. We also apply our results to several examples, some of them motivated by the study of several difference equations.

show abstract

Some properties of thek-dimensional Lyness' map

Abstract: This paper is devoted to study some properties of the k-dimensional Lyness' map

Cited by 13 publications

References 32 publications

Integrability and non-integrability of periodic non-autonomous Lyness recurrences

Integrability and non-integrability of periodic non-autonomous Lyness recurrences

Different Approaches to the Global Periodicity Problem

Meromorphic first integrals of analytic diffeomorphisms

Contact Info

Product

Resources

About