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Different approaches to the global periodicity problem
Cima Mollet, Anna; Gasull Embid, Armengol; Mañosa Fernández, Víctor; Mañosas, Francesc
Universitat Politècnica de Catalunya. Departament de Matemàtica Aplicada III; Universitat Politècnica de Catalunya. CODALAB - Control, Dinàmica i Aplicacions
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t 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 Fp(x) = x for all x, where F k = F ◦ F k−1, k ≥ 2. The minimal p satisfying this property is called the period of F. Given a m-dimensional parametric family of maps, say Fλ, a problem of current interest is to determine all the values of λ such that Fλ 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.
Àrees temàtiques de la UPC::Matemàtiques i estadística::Equacions diferencials i integrals
Differential equations
Differentiable dynamical systems
Globally periodic maps. Periodic orbits.
Equacions diferencials
Sistemes dinàmics diferenciables
Classificació AMS::37 Dynamical systems and ergodic theory::37G Local and nonlocal bifurcation theory
Classificació AMS::37 Dynamical systems and ergodic theory::37C Smooth dynamical systems: general theory
Classificació AMS::39 Difference and functional equations::39A Difference equations
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