Averaging and rates of averaging for uniform families of deterministic fast-slow skew product systems

Alexey Korepanov¹ ; Zemer Kosloff² ; Ian Melbourne¹

¹ Mathematics Institute University of Warwick Coventry, CV4 7AL, UK
² Mathematics Institute University of Warwick Coventry, CV4 7AL, UK and Permanent address: Einstein Institute of Mathematics The Hebrew University Edmond J. Safra Campus (Givat Ram) Jerusalem 91904, Israel

Studia Mathematica, Tome 238 (2017) no. 1, pp. 59-89

Résumé

We consider families of fast-slow skew product maps of the form $$ x_{n+1} = x_n+{\epsilon }a(x_n,y_n,{\epsilon }),\ \hskip 1em y_{n+1} = T_{\epsilon }y_n, $$ where $T_{\epsilon }$ is a family of nonuniformly expanding maps, and prove averaging and rates of averaging for the slow variables $x$ as ${\epsilon }\to 0$. Similar results are obtained also for continuous time systems $$ \dot x = {\epsilon }a(x,y,{\epsilon }),\ \hskip 1em \dot y = g_{\epsilon }(y). $$ Our results include cases where the family of fast dynamical systems consists of intermittent maps, unimodal maps (along the Collet–Eckmann parameters) and Viana maps.

Détail

DOI : 10.4064/sm8540-1-2017

Keywords: consider families fast slow skew product maps form epsilon y epsilon hskip epsilon where epsilon family nonuniformly expanding maps prove averaging rates averaging slow variables nbsp epsilon similar results obtained continuous time systems dot epsilon epsilon hskip dot epsilon results include cases where family fast dynamical systems consists intermittent maps unimodal maps along collet eckmann parameters viana maps