Rigorous numerics of tubular, conic, star-shaped neighborhoods of slow manifolds for fast-slow systems
This work provides a systematic, rigorous approach to constructing neighborhoods of slow manifolds, which is important for analyzing dynamics in singularly perturbed systems.
The authors develop a rigorous numerical method to validate tubular neighborhoods of slow manifolds in fast-slow systems, providing explicit radii and smoothness guarantees for a range of ε values.
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 $\{λ_i(y;ε), u_i(y;ε)\}_{i=1}^n$ of $f_x(h_ε(y),y,ε)$ over the slow manifold $S_ε= \{x = h_ε(y)\}$ 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 $S_ε$ 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 $[0,ε_0]$ of $ε$ with rigorous numerics.