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. 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.
Definition.
Consider the differential equation formula_1
with flow formula_2 being the solution of the differential equation with formula_3. 
A set formula_4 is called an "invariant set" for the differential equation if, for each formula_5, the solution formula_6, defined on its maximal interval of existence, has its image in formula_7. Alternatively, the orbit
passing through each formula_5 lies in formula_7. In addition, formula_7 is called an "invariant manifold" if formula_7 is a manifold.
Examples.
Simple 2D dynamical system.
For any fixed parameter formula_12, consider the variables formula_13 governed by the pair of coupled differential equations
The origin is an equilibrium. This system has two invariant manifolds of interest through the origin. 
Invariant manifolds in non-autonomous dynamical systems.
A differential equation 
represents a non-autonomous dynamical system, whose solutions are of the form formula_32 with formula_33. In the extended phase space formula_34 of such a system, any initial surface formula_35 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.