Asymptotic Approximations and Extension of Time-Scales

Abstract

It is shown how to obtain $O(\varepsilon )$ (or any higher order)-approximations to the solutions of the differential equation \[\begin{gathered} \dot \phi = 1 + \sum\limits_{p = 1}^p {\varepsilon ^p X^p (\phi ,X),\quad \phi \in S^1, } \hfill \dot x = \sum_{p = 1}^p {\varepsilon ^p Y^p (\phi ,X),\quad \begin{array}{*{20}c} {\varepsilon \in (0,\varepsilon _0 ],} {x \in D_0 \subset \mathbb{R}^n,} \end{array} } \hfill \end{gathered} \] in such a way that they are valid on the interval $0 \leqq \varepsilon ^{\tilde N} t \leqq L$, with $\tilde N \in \mathbb{N}$ arbitrary and L an $\varepsilon $-independent constant under the condition that the averaged equation has an attracting nondegenerate limit-cycle.The proof uses higher order averaging techniques and the Sanchez–Palencia contraction argument, together with Gronwall-estimates. In fact, for the x-component the approximation is uniformly valid on $[0,\infty )$.An application to the Van der Pol-oscillator is given, extending the usual interval of...