Constraint-defined manifolds: A legacy code approach to low-dimensional computation

C. William Gear, Ioannis G. Kevrekidis

Research output: Contribution to journalArticlepeer-review

59 Scopus citations


If the dynamics of an evolutionary differential equation system possess a low-dimensional, attracting, slow manifold, there are many advantages to using this manifold to perform computations for long term dynamics, locating features such as stationary points, limit cycles, or bifurcations. Approximating the slow manifold, however, may be computationally as challenging as the original problem. If the system is defined by a legacy simulation code or a microscopic simulator, it may be impossible to perform the manipulations needed to directly approximate the slow manifold. In this paper we demonstrate that with the knowledge only of a set of "slow" variables that can be used to parameterize the slow manifold, we can conveniently compute, using a legacy simulator, on a nearby manifold. Forward and reverse integration, as well as the location of fixed points are illustrated for a discretization of the Chafee-Infante PDE for parameter values for which an Inertial Manifold is known to exist, and can be used to validate the computational results.

Original languageEnglish (US)
Pages (from-to)17-28
Number of pages12
JournalJournal of Scientific Computing
Issue number1
StatePublished - Oct 2005

All Science Journal Classification (ASJC) codes

  • Software
  • General Engineering
  • Computational Mathematics
  • Theoretical Computer Science
  • Applied Mathematics
  • Numerical Analysis
  • Computational Theory and Mathematics


  • Differential equations
  • Inertial manifolds
  • Stiff equations


Dive into the research topics of 'Constraint-defined manifolds: A legacy code approach to low-dimensional computation'. Together they form a unique fingerprint.

Cite this