Discretization in the method of averaging

Abstract

Let f : R × R m ¯ × R → R m ¯ , f = f ( ε , x , t ) f:R \times {R^{\overline m }} \times R \to {R^{\overline m }},f = f(\varepsilon ,x,t) be a C 2 {C^2} -mapping 1 1 -periodic in t t having the form f ( 0 , x , t ) = A x + o ( | x | ) f(0,x,t) = Ax + o(|x|) as x → 0 x \to 0 where A ∈ L ( R m ¯ ) 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 \[ x ′ = ε f ( ε , x , t ) , ε > 0 is small x’ = \varepsilon f(\varepsilon ,x,t),\varepsilon > 0{\text {is}}\;{\text {small}} \] and of its discretization \[ x n + 1 = x n + ( ε / m ) f ( ε , x n , t n ) , t n + 1 = t n + 1 / m , {x_{n + 1}} = {x_n} + (\varepsilon /m)f(\varepsilon ,{x_n},{t_n}),{t_{n + 1}} = {t_n} + 1/m, \] where m ∈ { 1 , 2 , … } = N m \in \{ 1,2, \ldots \} = \mathcal {N} . We show behavior of these manifolds of the discretization for the following cases: (a) m → ∞ , ε → ε ¯ > 0 m \to \infty ,\varepsilon \to \overline \varepsilon > 0 , (b) m → ∞ , ε → 0 m \to \infty ,\varepsilon \to 0 , (c) m → k ∈ N , ε → 0 m \to k \in \mathcal {N},\varepsilon \to 0 .