Invariant Manifolds in Singularly Perturbed Systems

A singularly perturbed system is considered with a small parameter $\varepsilon$ in the velocity of a slow variable $y$ and with a fast variable $x$. It is assumed that, for any $y$ from a certain bounded domain $D$, the fast subsystem has a manifold $M_0(y)$ that is compact stable invariant or overflowing (in another variant, it is hyperbolic bilaterally invariant) and that the motions in this system in the direction transverse to $M_0(y)$ are faster than the mutual approaching of trajectories on $M_0(y)$ (the precise formulation is given in terms of the generalized Lyapunov characteristic numbers). It is proved that, for sufficiently small $\varepsilon$, the full system has an invariant manifold close to $\bigcup _{y\in D}M_0(y)\times \{y\}$; its degree of smoothness is refined. In the stable case, this manifold attracts close trajectories. In the hyperbolic case, the behavior of trajectories near this manifold is hyperbolic (in the direction transverse to the manifold).