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 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, stable manifold, unsta ...
s that contain the global
attractor and attract all solutions
exponentially quickly. Since an inertial manifold is
finite-dimensional
In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e., the number of vectors) of a basis of ''V'' over its base field. p. 44, §2.36 It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to disti ...
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.
[R. Temam. Inertial manifolds. ''Mathematical Intelligencer'', 12:68–74, 1990]
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 manifold In mathematics, the slow manifold of an equilibrium point of a dynamical system occurs as the most common example of a center manifold. One of the main methods of simplifying dynamical systems, is to reduce the dimension of the system to that of ...
s common in meteorology, or as the
center manifold In the mathematics of evolving systems, the concept of a center manifold was originally developed to determine stability of degenerate equilibria. Subsequently, the concept of center manifolds was realised to be fundamental to mathematical modellin ...
in any
bifurcation. Computationally, numerical schemes for
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function.
The function is often thought of as an "unknown" to be sol ...
s 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
and
and with parameter
:
:
* It possesses the one dimensional inertial manifold
of
(a parabola).
* This manifold is invariant under the dynamics because on the manifold
:
which is the same as
:
* The manifold
attracts all trajectories in some finite domain around the origin because near the origin
(although the strict definition below requires attraction from all initial conditions).
Hence the long term behavior of the original two dimensional dynamical system is given by the 'simpler' one dimensional dynamics on the inertial manifold
, namely
.
Definition
Let
denote a solution of a dynamical system. The solution
may be an evolving vector in
or may be an evolving function in an infinite-dimensional
Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
.
In many cases of interest the evolution of
is determined as the solution of a differential equation in
, say
with initial value
.
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 control theory, the state-transition matrix is a matrix whose product with the state vector x at an initial time t_0 gives x at a later time t. The state-transition matrix can be used to obtain the general solution of linear dynamical systems ...
,
such that
for all times
and all initial values
.
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
is a smooth
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
such that
#
is of finite dimension,
#
for all times
,
#
attracts all solutions exponentially quickly, that is, for every initial value
there exist constants
such that
.
The restriction of the differential equation
to the inertial manifold
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
and producing the same behavior for large time; in mathematics, for all
there exists
and possibly a time shift
such that
as
.
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
for unbounded self-adjoint closed operator
with domain
, and nonlinear operator
.
Typically, elementary spectral theory gives an orthonormal basis of
consisting of eigenvectors
:
,
, for ordered eigenvalues
.
For some given number
of modes,
denotes the projection of
onto the space spanned by
, and
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
where the constant
depends upon the system.
This spectral gap condition requires that the spectrum of
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
''slow variables''
, and the 'infinite'
''fast variables''
.
Then project the differential equation
onto both
and
to obtain the coupled system
and
.
For trajectories on the graph of an inertial
manifold
, the fast
variable
.
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,
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 manifold In the mathematics of evolving systems, the concept of a center manifold was originally developed to determine stability of degenerate equilibria. Subsequently, the concept of center manifolds was realised to be fundamental to mathematical modellin ...
s
for which a web service exists to construct approximations
for systems input by a
user.
[{{Cite web , url=http://www.maths.adelaide.edu.au/anthony.roberts/gencm.php , title=Construct centre manifolds of ordinary or delay differential equations (autonomous)]
See also
*
Wandering set
References
Dynamical systems