On invariant manifolds in singularly perturbed systems (Q1972812)

The author deals with a slow-fast system \[ \dot x=f(x,y, \varepsilon), \quad \dot y=\varepsilon g(x,y, \varepsilon), \quad \dot\varepsilon=0 \] where \(x\in\mathbb{R}^\ell\), \(y\in\mathbb{R}^m\), \(\varepsilon\in \mathbb{R}\). Assuming that the fast system \(\dot x=f(x,y,0)\), \(\dot y=0\) has a compact smooth invariant manifold with boundary \(M_0=M_0(y)\) smoothly depending on \(y\), where \(y\) runs over some ball \(B\subset \mathbb{R}^m\), the author proves that for a sufficiently small \(\varepsilon\), the whole system has an invariant manifold close to \(\bigcup_{y\in B}M_0(g) \times\{y\}\). The degree of its smoothness is analyzed.