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Global periodicity conditions for maps and recurrences via normal forms
Cima Mollet, Anna; Gasull Embid, Armengol; Mañosa Fernández, Víctor
Universitat Politècnica de Catalunya. Departament de Matemàtica Aplicada III; Universitat Politècnica de Catalunya. CODALAB - Control, Dinàmica i Aplicacions
We face the problem of characterizing the periodic cases in parametric families of rational diffeomorphisms of K, where K is ℝ or ℂ, having a fixed point. Our approach relies on the Normal Form Theory, to obtain necessary conditions for the existence of a formal linearization of the map, and on the introduction of a suitable rational parametrization of the parameters of the family. Using these tools we can find a finite set of values p for which the map can be p-periodic, reducing the problem of finding the parameters for which the periodic cases appear to simple computations. We apply our results to several two- and three-dimensional classes of polynomial or rational maps. In particular, we find the global periodic cases for several Lyness-type recurrences.
Peer Reviewed
Àrees temàtiques de la UPC::Matemàtiques i estadística::Equacions diferencials i integrals
Differential equations
Differentiable dynamical systems
Periodic orbits
Normal forms
Maps
Difference Equations
Equacions diferencials
Sistemes dinàmics diferenciables
Classificació AMS::37 Dynamical systems and ergodic theory::37G Local and nonlocal bifurcation theory
Classificació AMS::39 Difference and functional equations::39A Difference equations
Classificació AMS::37 Dynamical systems and ergodic theory::37C Smooth dynamical systems: general theory
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