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

TL;DR

This paper introduces a rigorous numerical method to validate tubular neighborhoods of slow manifolds in fast-slow systems, providing explicit radii and smoothness properties using interval Newton-like methods.

Contribution

The authors develop a systematic approach combining rigorous numerics and eigenpair validation to construct explicit, smooth tubular neighborhoods of slow manifolds in fast-slow dynamical systems.

Findings

Validated families of eigenvectors generate vector bundles over slow manifolds.

Constructed explicit tubular neighborhoods with proven smoothness and diffeomorphic properties.

Extended neighborhoods to conic and star-shaped forms with rigorous bounds.

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,\epsilon), and y' =\epsilon g(x,y,\epsilon). & \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.