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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Stable manifolds in the method of averaging


Author: Stephen Schecter
Journal: Trans. Amer. Math. Soc. 308 (1988), 159-176
MSC: Primary 34C29; Secondary 34C30, 58F27, 58F30
MathSciNet review: 946437
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Abstract: Consider the differential equation $ \dot z = \varepsilon f(z,\,t,\,\varepsilon )$, where $ f$ is $ T$periodic in $ t$ and $ \varepsilon > 0$ is a small parameter, and the averaged equation $ \dot z = \overline f (z): = (1/T)\,\int_0^T {\,f(z,\,t,\,0)\,dt} $. Suppose the averaged equation has a hyperbolic equilibrium at $ z = 0$ with stable manifold $ \overline W $. Let $ {\beta _\varepsilon }(t)$ denote the hyperbolic $ T$-periodic solution of $ \dot z = \varepsilon f(z,\,t,\,\varepsilon )$ near $ z \equiv 0$. We prove a result about smooth convergence of the stable manifold of $ {\beta _\varepsilon }(t)$ to $ \overline W \times {\mathbf{R}}$ as $ \varepsilon \to 0$. The proof uses ideas of Vanderbauwhede and van Gils about contractions on a scale of Banach spaces.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1988-0946437-2
PII: S 0002-9947(1988)0946437-2
Article copyright: © Copyright 1988 American Mathematical Society