Abstract
A singularly perturbed system with a small parameter ε at the velocity of the slow variable y and with the fast variable x is considered. The main hypothesis is that for all y from some bounded domain D, the fast subsystem has a stable invariant or overflowing manifold M 0(y) and that the motions in this system going in the directions transversal to M 0(y) are more fast than the mutual approaching of trajectories on M 0(y) (a precise statement is given in terms of appropriate Lyapunov-type characteristic numbers). It is proved that for a sufficiently small ε, the whole system has an invariant manifold close to \(\bigcup\limits_{y \in D} {M_0 (y) \times \{ y\}}\) the degree of its smoothness is specifed.
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Anosova, O.D. On Invariant Manifolds in Singularly Perturbed Systems. Journal of Dynamical and Control Systems 5, 501–507 (1999). https://doi.org/10.1023/A:1021739205527
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DOI: https://doi.org/10.1023/A:1021739205527