# Invariant manifold

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.

- 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

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]

## References

- Hirsh M.W., Pugh C.C., Shub M., Invariant Manifolds, Lect. Notes. Math., 583, Springer, Berlin — Heidelberg, 1977
- 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
- C. Chicone. Ordinary Differential Equations with Applications, volume 34 of Texts in Applied Mathematics. Springer, 2006, p.34
- Haller, G. (2015). "Lagrangian Coherent Structures".
*Annual Review of Fluid Mechanics*.**47**: 137. Bibcode:2015AnRFM..47..137H. doi:10.1146/annurev-fluid-010313-141322.