Rigorous Enclosures of a Slow Manifold

TL;DR

This paper presents a rigorous numerical method for computing enclosures of slow manifolds in slow-fast dynamical systems, enabling precise analysis of invariant structures and bifurcations.

Contribution

It introduces a novel algebraic triangulated approximation technique for invariant manifolds in systems with one fast and two slow variables.

Findings

Validated existence of invariant manifold tangencies

Provided bounds on tangency locations

Demonstrated method on singular Hopf bifurcation

Abstract

Slow-fast dynamical systems have two time scales and an explicit parameter representing the ratio of these time scales. Locally invariant slow manifolds along which motion occurs on the slow time scale are a prominent feature of slow-fast systems. This paper introduces a rigorous numerical method to compute enclosures of the slow manifold of a slow-fast system with one fast and two slow variables. A triangulated first order approximation to the two dimensional invariant manifold is computed "algebraically". Two translations of the computed manifold in the fast direction that are transverse to the vector field are computed as the boundaries of an initial enclosure. The enclosures are refined to bring them closer to each other by moving vertices of the enclosure boundaries one at a time. As an application we use it to prove the existence of tangencies of invariant manifolds in the problem of singular Hopf bifurcation and to give bounds on the location of one such tangency.