Resumen de Contribution to the study of perturbations of low dimensional maps

Pau Rabassa Sans

• The thesis is concerned with some two dimensional maps that can be seen as perturbations of one dimensional maps. The first map considered is the Hénon map for big values on the dissipation. We present a method to compute numerically cubic tangencies of the map in a systematic way. To explain the numerical results obtained with this systematic computation, we prove the existence of cubic tangencies accumulating in the parameter space to a previous one.

  The second case of study is the quasi-periodically Forced Logistic Map (FLM). To study this map we analyze its period doubling and reducibility loss bifurcations. First we compute numerically the bifurcations diagram for some regions of the (two dimensional) parameter space. We present different studies to explain the behavior observed numerically. Then we notice numerically certain properties of self-similarity and renormalization for the map. Finally we purpose an extension of the (doubling) renormalization operator to the case quasi-periodically forced one dimensional case. With an study of the operator we show that it can be used to predict the asymptotic behavior of the reducibility loss bifurcations close to the uncoupled case. With the study of the operator we present also a suitable explanation to the universality and self-similarity properties observed before.