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

Abstract

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 \begin{equation*} \begin{cases} x' = f(x,y,ε), and y' =εg(x,y,ε). & \end{cases} \end{equation*} Our main focus is the 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 slow manifold $S_{\epsilon }=\{x=h_{\epsilon }(y)\}$ 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 $S_{\epsilon }$ 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 $[0,{\epsilon }_0]$ of $\epsilon$ with rigorous numerics.

45 pages, 7 figures. Put the validation (raw) code at [28]. In the latest version, we have re-computed all validation results replacing CAPD library ( http://capd.ii.uj.edu.pl ) by kv library ( http://verifiedby.me/kv/ )

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