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

We provide a rigorous numerical computation method to validate tubular neighborhoods of normally hyperbolic slow manifolds with the explicit radii for the fast-slow system { x′ = f(x, y, ), y′ = g(x, y, ). Our main focus is the validation of the continuous family of eigenpairs {λi(y; ), ui(y; )}i=1 of fx(h (y), y, ) over the slow manifold S = {x = h (y)} admitting the graph representation. In order to obtain such a family, we apply the interval Newton-like method with rigorous numerics. The validated family of eigenvectors generates a vector bundle over S determining normally hyperbolic eigendirections rigorously. The generated vector bundle enables us to construct a tubular neighborhood centered at slow manifolds with explicit radii. Combining rate conditions for providing smoothness of center-(un)stable manifolds, we can validate smooth tubular neighborhoods with diffeomorphic family of affine change of coordinates, as well as several extensions such as conic and star-shaped neighborhoods. Our procedure provides a systematic construction of smooth neighborhoods of slow manifolds in an explicit range [0, 0] of with rigorous numerics.