Discretization in the method of averaging

Abstract:

Let $f:R \times {R^{\overline m }} \times R \to {R^{\overline m }},f = f(\varepsilon ,x,t)$ be a ${C^2}$-mapping $1$-periodic in $t$ having the form $f(0,x,t) = Ax + o(|x|)$ as $x \to 0$ where $A \in \mathcal {L}({R^{\overline m }})$ has no eigenvalues with zero real parts. We study the relation between local stable manifolds of the equation \[ x’ = \varepsilon f(\varepsilon ,x,t),\varepsilon > 0{\text {is}}\;{\text {small}}\] and of its discretization \[ {x_{n + 1}} = {x_n} + (\varepsilon /m)f(\varepsilon ,{x_n},{t_n}),{t_{n + 1}} = {t_n} + 1/m,\] where $m \in \{ 1,2, \ldots \} = \mathcal {N}$. We show behavior of these manifolds of the discretization for the following cases: (a) $m \to \infty ,\varepsilon \to \overline \varepsilon > 0$, (b) $m \to \infty ,\varepsilon \to 0$, (c) $m \to k \in \mathcal {N},\varepsilon \to 0$.