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. 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. 

Examples

Simple 2D dynamical system

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

$dx/dt=ax-xy\quad {\text{and}}\quad dy/dt=-y+x^{2}-2y^{2}.$ The origin is an equilibrium. This system has two invariant manifolds of interest through the origin.