Invariant manifold

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{{ safesubst:#invoke:Unsubst||$N=Refimprove |date=__DATE__ |$B= {{#invoke:Message box|ambox}} }} In dynamical systems, a branch of mathematics, an invariant manifold is a topological manifold that is invariant under the action of the dynamical system.[1] Examples include the slow manifold, center manifold, stable manifold, unstable manifold, subcenter manifold and inertial manifold.

Typically, although by no means always, invariant manifolds are constructed as a 'perturbation' of an invariant subspace about an equilibrium. In dissipative systems, an invariant manifold based upon the gravest, longest lasting modes forms an effective low-dimensional, reduced, model of the dynamics. [2]

Definition

Consider the differential equation with flow being the solution of the differential equation with . A set is called an invariant set for the differential equation if, for each , the solution , defined on its maximal interval of existence, has its image in . Alternatively, the orbit passing through each lies in . In addition, is called an invariant manifold if is a manifold. [3]

Examples

Simple 2D dynamical system

For any fixed parameter , consider the variables governed by the pair of coupled differential equations

The origin is an equilibrium. This system has two invariant manifolds of interest through the origin.

See also

References

  1. Hirsh M.W., Pugh C.C., Shub M., Invariant Manifolds, Lect. Notes. Math., 583, Springer, Berlin — Heidelberg, 1977
  2. A. J. Roberts. The utility of an invariant manifold description of the evolution of a dynamical system. SIAM J. Math. Anal., 20:1447–1458, 1989. http://locus.siam.org/SIMA/volume-20/art_0520094.html
  3. C. Chicone. Ordinary Differential Equations with Applications, volume 34 of Texts in Applied Mathematics. Springer, 2006, p.34