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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Discretization in the method of averaging


Author: Michal Fečkan
Journal: Proc. Amer. Math. Soc. 113 (1991), 1105-1113
MSC: Primary 34C29; Secondary 34A45
DOI: https://doi.org/10.1090/S0002-9939-1991-1068119-7
MathSciNet review: 1068119
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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(\vert x\vert)$ 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

$\displaystyle x' = \varepsilon f(\varepsilon ,x,t),\varepsilon > 0{\text{is}}\;{\text{small}}$

and of its discretization

$\displaystyle {x_{n + 1}} = {x_n} + (\varepsilon /m)f(\varepsilon ,{x_n},{t_n}),{t_{n + 1}} = {t_n} + 1/m,$

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$.

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DOI: https://doi.org/10.1090/S0002-9939-1991-1068119-7
Article copyright: © Copyright 1991 American Mathematical Society