Computing Slow Manifolds of Saddle Type

TL;DR

This paper presents a boundary value algorithm for computing saddle-type slow manifolds in dynamical systems, addressing limitations of initial value solvers and demonstrating its effectiveness through bifurcation and PDE examples.

Contribution

Introduces the SMST boundary value algorithm for accurately computing saddle-type slow manifolds in systems with multiple time scales.

Findings

The algorithm successfully computes trajectories on saddle-type slow manifolds.

It effectively handles bifurcation problems involving these manifolds.

Demonstrates applicability to PDE traveling wave profiles.

Abstract

Slow manifolds are important geometric structures in the state spaces of dynamical systems with multiple time scales. This paper introduces an algorithm for computing trajectories on slow manifolds that are normally hyperbolic with both stable and unstable fast manifolds. We present two examples of bifurcation problems where these manifolds play a key role and a third example in which saddle-type slow manifolds are part of a traveling wave profile of a partial differential equation. Initial value solvers are incapable of computing trajectories on saddle-type slow manifolds, so the slow manifold of saddle type (SMST) algorithm presented here is formulated as a boundary value method. We take an empirical approach here to assessing the accuracy and effectiveness of the algorithm.