In mathematics, inertial manifolds are concerned with the long term behavior of the solutions of dissipative dynamical systems.
Inertial manifolds are finite-dimensional, smooth, invariant manifolds that contain the global attractor and attract all solutions exponentially quickly.
Since an inertial manifold is finite-dimensional even if the original system is infinite-dimensional, and because most of the dynamics for the system takes place on the inertial manifold, studying the dynamics on an inertial manifold produces a considerable simplification in the study of the dynamics of the original system.
[1] In many physical applications, inertial manifolds express an interaction law between the small and large wavelength structures.
Some say that the small wavelengths are enslaved by the large (e.g. synergetics).
Computationally, numerical schemes for partial differential equations seek to capture the long term dynamics and so such numerical schemes form an approximate inertial manifold.
:[2] Hence the long term behavior of the original two dimensional dynamical system is given by the 'simpler' one dimensional dynamics on the inertial manifold
denote a solution of a dynamical system.
or may be an evolving function in an infinite-dimensional Banach space
is determined as the solution of a differential equation in
In any case, we assume the solution of the dynamical system can be written in terms of a semigroup operator, or state transition matrix,
In some situations we might consider only discrete values of time as in the dynamics of a map.
An inertial manifold[1] for a dynamical semigroup
Nonetheless, under appropriate conditions the inertial system possesses so-called asymptotic completeness:[3] that is, every solution of the differential equation has a companion solution lying in
and producing the same behavior for large time; in mathematics, for all
Researchers in the 2000s generalized such inertial manifolds to time dependent (nonautonomous) and/or stochastic dynamical systems (e.g.[4][5]) Existence results that have been proved address inertial manifolds that are expressible as a graph.
[1] The governing differential equation is rewritten more specifically in the form
for unbounded self-adjoint closed operator
Typically, elementary spectral theory gives an orthonormal basis of
denotes the orthogonal projection onto the space spanned by
We look for an inertial manifold expressed as the graph
For this graph to exist the most restrictive requirement is the spectral gap condition[1]
This spectral gap condition requires that the spectrum of
must contain large gaps to be guaranteed of existence.
Several methods are proposed to construct approximations to inertial manifolds,[1] including the so-called intrinsic low-dimensional manifolds.
[6][7] The most popular way to approximate follows from the existence of a graph.
For trajectories on the graph of an inertial manifold
Differentiating and using the coupled system form gives the differential equation for the graph: This differential equation is typically solved approximately in an asymptotic expansion in 'small'
to give an invariant manifold model,[8] or a nonlinear Galerkin method,[9] both of which use a global basis whereas the so-called holistic discretisation uses a local basis.
[10] Such approaches to approximation of inertial manifolds are very closely related to approximating center manifolds for which a web service exists to construct approximations for systems input by a user.