Computing Slow Manifolds of Saddle Type

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.

preprint version - for final version see journal reference

Cited by in corpus (7)

• Homoclinic Orbits of the FitzHugh-Nagumo Equation: Bifurcations in the Full System
• Canards and curvature: nonsmooth approximation by pinching
• From First Lyapunov Coefficients to Maximal Canards
• Shilnikov Homoclinic Bifurcation of Mixed-Mode Oscillations
• Rigorous Enclosures of a Slow Manifold
• Discretized Fast-Slow Systems with Canards in Two Dimensions
• Geometric analysis of fast-slow PDEs with fold singularities via Galerkin discretisation