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

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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

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Simple 2D dynamical system

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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.

  • The vertical line   is invariant as when   the  -equation becomes   which ensures   remains zero. This invariant manifold,  , is a stable manifold of the origin (when  ) as all initial conditions   lead to solutions asymptotically approaching the origin.
  • The parabola   is invariant for all parameter  . One can see this invariance by considering the time derivative   and finding it is zero on   as required for an invariant manifold. For   this parabola is the unstable manifold of the origin. For   this parabola is a center manifold, more precisely a slow manifold, of the origin.
  • For   there is only an invariant stable manifold about the origin, the stable manifold including all  .

Invariant manifolds in non-autonomous dynamical systems

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A differential equation

 

represents a non-autonomous dynamical system, whose solutions are of the form   with  . In the extended phase space   of such a system, any initial surface   generates an invariant manifold

 

A fundamental question is then how one can locate, out of this large family of invariant manifolds, the ones that have the highest influence on the overall system dynamics. These most influential invariant manifolds in the extended phase space of a non-autonomous dynamical systems are known as Lagrangian Coherent Structures.[4]

See also

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References

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  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 Archived 2008-08-20 at the Wayback Machine
  3. ^ C. Chicone. Ordinary Differential Equations with Applications, volume 34 of Texts in Applied Mathematics. Springer, 2006, p.34
  4. ^ Haller, G. (2015). "Lagrangian Coherent Structures". Annual Review of Fluid Mechanics. 47 (1): 137–162. Bibcode:2015AnRFM..47..137H. doi:10.1146/annurev-fluid-010313-141322.