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] Consider the differential equation
ϕ
being the solution of the differential equation with
is called an invariant set for the differential equation if, for each
, the solution
ϕ
, defined on its maximal interval of existence, has its image in
Alternatively, the orbit passing through each
is called an invariant manifold if
[3] For any fixed parameter
governed by the pair of coupled differential equations The origin is an equilibrium.
This system has two invariant manifolds of interest through the origin.
A differential equation represents a non-autonomous dynamical system, whose solutions are of the form
ϕ
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.