# Invariant manifold

{{ 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 ${\displaystyle dx/dt=f(x),\ x\in {\mathbb {R} }^{n},}$ with flow ${\displaystyle x(t)=\phi _{t}(x_{0})}$ being the solution of the differential equation with ${\displaystyle x(0)=x_{0}}$. A set ${\displaystyle S\subset {\mathbb {R} }^{n}}$ is called an invariant set for the differential equation if, for each ${\displaystyle x_{0}\in S}$, the solution ${\displaystyle t\mapsto \phi _{t}(x_{0})}$, defined on its maximal interval of existence, has its image in ${\displaystyle S}$. Alternatively, the orbit passing through each ${\displaystyle x_{0}\in S}$ lies in ${\displaystyle S}$. In addition, ${\displaystyle S}$ is called an invariant manifold if ${\displaystyle S}$ is a manifold. [3]

## Examples

### Simple 2D dynamical system

For any fixed parameter ${\displaystyle a}$, consider the variables ${\displaystyle x(t),y(t)}$ governed by the pair of coupled differential equations

${\displaystyle dx/dt=ax-xy\quad {\text{and}}\quad dy/dt=-y+x^{2}-2y^{2}.}$

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