Computing Slow Manifolds of Saddle Type

Details Computing Slow Manifolds of Saddle Type Computing Slow Manifolds of Saddle Type

2012-02-02

arxiv.org/abs/1202.0498v1

SIAM Journal on Applied Dynamical Systems, Vol. 4, No. 3, pp. 854-879, 2009

Mathematics

Dynamical Systems

preprint version - for final version see journal reference

Scientific paper

10.1137/080741999

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.

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