Averaging and rates of averaging for uniform families of deterministic fast-slow skew product systems
Volume 238 / 2017
Studia Mathematica 238 (2017), 59-89
MSC: Primary 34C29; Secondary 37D25, 34E13.
DOI: 10.4064/sm8540-1-2017
Published online: 12 April 2017
Abstract
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.
