Global periodicity conditions for maps and recurrences via normal forms
Rights accessOpen Access
All rights reserved. This work is protected by the corresponding intellectual and industrial property rights. Without prejudice to any existing legal exemptions, reproduction, distribution, public communication or transformation of this work are prohibited without permission of the copyright holder
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.
CitationCima, A.; Gasull, A.; Mañosa, V. Global periodicity conditions for maps and recurrences via normal forms. "International journal of bifurcation and chaos", Novembre 2013, vol. 23, núm. 11, p. 1350182-1-1350182-18.