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We provide a rigorous numerical computation method to validate tubularneighborhoods of normally hyperbolic slow manifolds with the explicit radii forthe fast-slow system \begin{equation*} \begin{cases} x' = f(x,y,\epsilon), andy' =\epsilon g(x,y,\epsilon). & \end{cases} \end{equation*} Our main focus isthe validation of the continuous family of eigenpairs $\{\lambda_i(y;\epsilon),u_i(y;\epsilon)\}_{i=1}^n$ of $f_x(h_\epsilon(y),y,\epsilon)$ over the slowmanifold $S_\epsilon = \{x = h_\epsilon(y)\}$ admitting the graphrepresentation. In order to obtain such a family, we apply the intervalNewton-like method with rigorous numerics. The validated family of eigenvectorsgenerates a vector bundle over $S_\epsilon$ determining normally hyperboliceigendirections rigorously. The generated vector bundle enables us to constructa tubular neighborhood centered at slow manifolds with explicit radii.Combining rate conditions for providing smoothness of center-(un)stablemanifolds, we can validate smooth tubular neighborhoods with diffeomorphicfamily of affine change of coordinates, as well as several extensions such asconic and star-shaped neighborhoods. Our procedure provides a systematicconstruction of smooth neighborhoods of slow manifolds in an explicit range$[0,\epsilon_0]$ of $\epsilon$ with rigorous numerics.

Rigorous numerics of tubular, conic, star-shaped neighborhoods of slow manifolds for fast-slow systems