Inertial manifold

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.
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). Inertial manifolds may also appear as slow manifolds common in meteorology, or as the center manifold in any bifurcation. Computationally, numerical schemes for partial differential equations seek to capture the long term dynamics and so such numerical schemes form an approximate inertial manifold.
Introductory Example.
Consider the dynamical system in just two variables formula_1 and formula_2 and with parameter formula_3:
Hence the long term behavior of the original two dimensional dynamical system is given by the 'simpler' one dimensional dynamics on the inertial manifold formula_5, namely formula_13.
Definition.
Let formula_14 denote a solution of a dynamical system. The solution formula_14 may be an evolving vector in formula_16 or may be an evolving function in an infinite-dimensional Banach space formula_17.
In many cases of interest the evolution of formula_14 is determined as the solution of a differential equation in formula_17, say formula_20 with initial value formula_21.
In any case, we assume the solution of the dynamical system can be written in terms of a semigroup operator, or state transition matrix, formula_22 such that formula_23 for all times formula_24 and all initial values formula_25.
In some situations we might consider only discrete values of time as in the dynamics of a map.
An inertial manifold for a dynamical semigroup formula_26 is a smooth manifold formula_5 such that
The restriction of the differential equation formula_35 to the inertial manifold formula_5 is therefore a well defined finite-dimensional system called the inertial system.
Subtly, there is a difference between a manifold being attractive, and solutions on the manifold being attractive.
Nonetheless, under appropriate conditions the inertial system possesses so-called asymptotic completeness: that is, every solution of the differential equation has a companion solution lying in formula_5 and producing the same behavior for large time; in mathematics, for all formula_25 there exists formula_39 and possibly a time shift formula_40 such that formula_41 as formula_42.
Researchers in the 2000s generalized such inertial manifolds to time dependent (nonautonomous) and/or stochastic dynamical systems (e.g.)
Existence.
Existence results that have been proved address inertial manifolds that are expressible as a graph.
The governing differential equation is rewritten more specifically in the form formula_43 for unbounded self-adjoint closed operator formula_44 with domain formula_45, and nonlinear operator formula_46.
Typically, elementary spectral theory gives an orthonormal basis of formula_17 consisting of eigenvectors formula_48: formula_49, formula_50, for ordered eigenvalues formula_51.
For some given number formula_52 of modes, formula_53 denotes the projection of formula_17 onto the space spanned by formula_55, and formula_56 denotes the orthogonal projection onto the space spanned by formula_57.
We look for an inertial manifold expressed as the graph formula_58.
For this graph to exist the most restrictive requirement is the spectral gap condition formula_59 where the constant formula_60 depends upon the system.
This spectral gap condition requires that the spectrum of formula_44 must contain large gaps to be guaranteed of existence.
Approximate inertial manifolds.
Several methods are proposed to construct approximations to
inertial manifolds, including the
so-called "intrinsic low-dimensional manifolds".
The most popular way to approximate follows from the
existence of a graph.
Define the formula_52 "slow variables" formula_63, and the 'infinite'
"fast variables" formula_64.
Then project the differential equation
formula_43 onto both formula_66
and formula_67 to obtain the coupled system
formula_68 and
formula_69.
For trajectories on the graph of an inertial
manifold formula_70, the fast
variable formula_71.
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' formula_73 to
give an invariant manifold model,
or a nonlinear Galerkin method,
both of which use a global basis whereas the so-called
"holistic discretisation" uses a local basis.
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.