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 $dx/dt = f(x),\ x \in \mathbb R^n,$ with flow $x(t)=\phi_t(x_0)$ being the solution of the differential equation with $x(0)=x_0$. A set $S \subset \mathbb R^n$ is called an invariant set for the differential equation if, for each $x_0 \in S$, the solution $t \mapsto \phi_t(x_0)$, defined on its maximal interval of existence, has its image in $S$. Alternatively, the orbit passing through each $x_0 \in S$ lies in $S$. In addition, $S$ is called an invariant manifold if $S$ is a manifold. [3]

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.

• The vertical line $x=0$ is invariant as when $x=0$ the $x$-equation becomes $dx/dt=0$ which ensures $x$ remains zero. This invariant manifold, $x=0$, is a stable manifold of the origin (when $a\geq0$) as all initial conditions $x(0)=0,\ y(0)>-1/2$ lead to solutions asymptotically approaching the origin.
• The parabola $y=x^2/(1+2a)$ is invariant for all parameter $a$. One can see this invariance by considering the time derivative $d/dt[y-x^2/(1+2a)]$ and finding it is zero on $y=x^2/(1+2a)$ as required for an invariant manifold. For $a>0$ this parabola is the unstable manifold of the origin. For $a=0$ this parabola is a center manifold, more precisely a slow manifold, of the origin.
• For $a<0$ there is only an invariant stable manifold about the origin, the stable manifold including all $(x,y),\ y>-1/2$.