Different Approaches to the Global Periodicity Problem

Abstract

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.

Discrete Melnikov functions

“…This result is a natural extension of a similar property that holds for Tperiodic Abel differential equations, proved in the pioneering work [14]. We will prove for (7) the above result by studying the first order discrete Melnikov function M 1 associated to the particular family…”

“…In this section, as an application of Theorem 1, we reproduce some results of [3]. We prove that when the five sequences defining (7) are N -periodic there are examples of this dynamical system having at least N − 1 isolated N -periodic orbits. As a consequence, in contrast of what happens for the Riccati case, there is no upper bound for the number of isolated periodic sequences generated by general discrete dynamical systems of the form (7).…”

“…We prove that when the five sequences defining (7) are N -periodic there are examples of this dynamical system having at least N − 1 isolated N -periodic orbits. As a consequence, in contrast of what happens for the Riccati case, there is no upper bound for the number of isolated periodic sequences generated by general discrete dynamical systems of the form (7). This result is a natural extension of a similar property that holds for Tperiodic Abel differential equations, proved in the pioneering work [14].…”

Pointwise periodic maps with quantized first integrals

“…It is known that periodicity issues are related with integrability since most continuous periodic maps are completely integrable, see [10] and [11]. In this work we revisit the examples and results of Chang et al in the above cited references under the light of their properties as integrable systems.…”

Some open problems in low dimensional dynamical systems

“…For instance, when k = 1 there are periodic Möbius maps with all the periods. To see more information about related problems, see [35] and its references.…”