Slow manifold
Encyclopedia
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 the slow manifold—center manifold
theory rigorously justifies the modelling. For example, most global and regional models of the atmosphere or oceans resolve the so-called quasi-geostrophic flow dynamics on the slow manifold of the atmosphere/oceanic dynamics,
and is thus crucial to forecasting with a climate model
.
for an evolving state vector
and with equilibrium point 
. Then the linearization of the system at the equilibrium point is
The matrix
defines four invariant subspace
s characterized by the eigenvalues
of the matrix: as described in the entry for the center manifold
three of the subspaces are the stable, unstable and center subspaces corresponding to the span of the eigenvectors with eigenvalues
that have real part negative, positive, and zero, respectively; the fourth subspace is the slow subspace given by the span of the eigenvectors, and generalized eigenvector
s, corresponding to the eigenvalue
precisely. The slow subspace is a subspace of the center subspace, or identical to it, or possibly empty.
Correspondingly, the nonlinear system has invariant manifold
s, made of trajectories of the nonlinear system, corresponding to each of these invariant subspaces. There is an invariant manifold tangent to the slow subspace and with the same dimension; this manifold is the slow manifold.
Stochastic slow manifolds also exist for noisy dynamical systems (stochastic differential equation
), as do also stochastic center, stable and unstable manifolds. Such stochastic slow manifolds are similarly useful in modeling emergent stochastic dynamics, but there are many fascinating issues to resolve such as history and future dependent integrals of the noise.
and 
has the exact slow manifold
on which the evolution is 
. Apart from exponentially decaying transients, this slow manifold and its evolution captures all solutions that are in the neighborhood of the origin. The neighborhood of attraction is, roughly, at least the half-space 
.
introduced the following dynamical system of five equations in five variables to explore the notion of a slow manifold of quasi-geostrophic flow
Linearized about the origin the eigenvalue zero has multiplicity three, and there is a complex conjugate pair of eigenvalues,
. Hence there exists a three dimensional slow manifold (surrounded by 'fast' waves in the 
and 
variables). Lorenz later argued a slow manifold did not exist! But normal form arguments suggest that there is a dynamical system that is exponentially close to the Lorenz system for which there is a good slow manifold.
to a model of one ordinary differential equation
. Consider a field
undergoing the nonlinear diffusion
with Robin boundary condition
s
Parametrising the boundary conditions by
empowers us to cover the insulating Neumann boundary condition
case
, the Dirichlet boundary condition
case
, and all cases between.
Now for a marvelous trick, much used in exploring dynamics with bifurcation theory
. Since parameter
is constant, adjoin the trivially true differential equation
Then in the extended state space of the evolving field and parameter,
, there exists an infinity of equilibria, not just one equilibrium, with 
(insulating) and 
constant, say 
. Without going into details, about each and every equilibria the linearized diffusion has two zero eigenvalues and for 
all the rest are negative (less than 
). Thus the two dimensional dynamics on the slow manifolds emerge (see emergence
) from the nonlinear diffusion no matter how complicated the initial conditions.
Here one can straightforwardly verify the slow manifold to be precisely the field
where amplitude 
evolves according to
That is, after the initial transients that by diffusion smooth internal structures, the emergent behavior is one of relatively slow decay of the amplitude (
) at a rate controlled by the type of boundary condition (constant 
).
Notice that this slow manifold model is global in
as each equilibria is necessarily in the slow subspace of each other equilibria, but is only local in parameter 
. We cannot yet be sure how large 
may be taken, but the theory assures us the results do hold for some finite parameter 
.
the two variable dynamics of this linear system forced with noise 
:
One could simply notice that the Ornstein–Uhlenbeck process
is formally the history integral
and then assert that
is simply the integral of this history integral. However, this solution then inappropriately contains fast time integrals, due to the 
in the integrand, in a supposedly long time model.
Alternatively, a stochastic coordinate transform extracts a sound model for the long term dynamics. Change variables to
where
then the new variables evolve according to the simple
In these new coordinates we readily deduce
exponentially quickly, leaving 
undergoing a random walk
to be the long term model of the stochastic dynamics on the stochastic slow manifold obtained by setting
.
A web service constructs such slow manifolds in finite dimensions, both deterministic and stochastic.
Dynamical system
A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a...
occurs as the most common example of a center manifold
Center manifold
In mathematics, the center manifold of an equilibrium point of a dynamical system consists of orbits whose behavior around the equilibrium point is not controlled by either the attraction of the stable manifold or the repulsion of the unstable manifold. The first step when studying equilibrium...
. One of the main methods of simplifying dynamical systems, is to reduce the dimension of the system to that of the slow manifold—center manifold
Center manifold
In mathematics, the center manifold of an equilibrium point of a dynamical system consists of orbits whose behavior around the equilibrium point is not controlled by either the attraction of the stable manifold or the repulsion of the unstable manifold. The first step when studying equilibrium...
theory rigorously justifies the modelling. For example, most global and regional models of the atmosphere or oceans resolve the so-called quasi-geostrophic flow dynamics on the slow manifold of the atmosphere/oceanic dynamics,
and is thus crucial to forecasting with a climate model
Climate model
Climate models use quantitative methods to simulate the interactions of the atmosphere, oceans, land surface, and ice. They are used for a variety of purposes from study of the dynamics of the climate system to projections of future climate...
.
Definition
Consider the dynamical systemDynamical system
A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a...
for an evolving state vector
and with equilibrium point . Then the linearization of the system at the equilibrium point isThe matrix
defines four invariant subspaceInvariant subspace
In mathematics, an invariant subspace of a linear mappingfrom some vector space V to itself is a subspace W of V such that T is contained in W...
s characterized by the eigenvalues
of the matrix: as described in the entry for the center manifoldCenter manifold
In mathematics, the center manifold of an equilibrium point of a dynamical system consists of orbits whose behavior around the equilibrium point is not controlled by either the attraction of the stable manifold or the repulsion of the unstable manifold. The first step when studying equilibrium...
three of the subspaces are the stable, unstable and center subspaces corresponding to the span of the eigenvectors with eigenvalues
that have real part negative, positive, and zero, respectively; the fourth subspace is the slow subspace given by the span of the eigenvectors, and generalized eigenvectorGeneralized eigenvector
In linear algebra, for a matrix A, there may not always exist a full set of linearly independent eigenvectors that form a complete basis – a matrix may not be diagonalizable. This happens when the algebraic multiplicity of at least one eigenvalue λ is greater than its geometric multiplicity...
s, corresponding to the eigenvalue
precisely. The slow subspace is a subspace of the center subspace, or identical to it, or possibly empty.Correspondingly, the nonlinear system has invariant manifold
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. An example is the stable manifold....
s, made of trajectories of the nonlinear system, corresponding to each of these invariant subspaces. There is an invariant manifold tangent to the slow subspace and with the same dimension; this manifold is the slow manifold.
Stochastic slow manifolds also exist for noisy dynamical systems (stochastic differential equation
Stochastic differential equation
A stochastic differential equation is a differential equation in which one or more of the terms is a stochastic process, thus resulting in a solution which is itself a stochastic process....
), as do also stochastic center, stable and unstable manifolds. Such stochastic slow manifolds are similarly useful in modeling emergent stochastic dynamics, but there are many fascinating issues to resolve such as history and future dependent integrals of the noise.
Simple case with two variables
The coupled system in two variables and has the exact slow manifold
on which the evolution is . Apart from exponentially decaying transients, this slow manifold and its evolution captures all solutions that are in the neighborhood of the origin. The neighborhood of attraction is, roughly, at least the half-space .Slow dynamics among fast waves
Edward Norton LorenzEdward Norton Lorenz
Edward Norton Lorenz was an American mathematician and meteorologist, and a pioneer of chaos theory. He discovered the strange attractor notion and coined the term butterfly effect.-Biography:...
introduced the following dynamical system of five equations in five variables to explore the notion of a slow manifold of quasi-geostrophic flow
Linearized about the origin the eigenvalue zero has multiplicity three, and there is a complex conjugate pair of eigenvalues,
. Hence there exists a three dimensional slow manifold (surrounded by 'fast' waves in the and variables). Lorenz later argued a slow manifold did not exist! But normal form arguments suggest that there is a dynamical system that is exponentially close to the Lorenz system for which there is a good slow manifold.Eliminate an infinity of variables
In modeling we aim to simplify enormously. This example uses a slow manifold to simplify the 'infinite dimensional' dynamics of a partial differential equationPartial differential equation
In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...
to a model of one ordinary differential equation
Ordinary differential equation
In mathematics, an ordinary differential equation is a relation that contains functions of only one independent variable, and one or more of their derivatives with respect to that variable....
. Consider a field
undergoing the nonlinear diffusionwith Robin boundary condition
Robin boundary condition
In mathematics, the Robin boundary condition is a type of boundary condition, named after Victor Gustave Robin . When imposed on an ordinary or a partial differential equation, it is a specification of a linear combination of the values of a function and the values of its derivative on the...
s
Parametrising the boundary conditions by
empowers us to cover the insulating Neumann boundary conditionNeumann boundary condition
In mathematics, the Neumann boundary condition is a type of boundary condition, named after Carl Neumann.When imposed on an ordinary or a partial differential equation, it specifies the values that the derivative of a solution is to take on the boundary of the domain.* For an ordinary...
case
, the Dirichlet boundary conditionDirichlet boundary condition
In mathematics, the Dirichlet boundary condition is a type of boundary condition, named after Johann Peter Gustav Lejeune Dirichlet who studied under Cauchy and succeeded Gauss at University of Göttingen. When imposed on an ordinary or a partial differential equation, it specifies the values a...
case
, and all cases between.Now for a marvelous trick, much used in exploring dynamics with bifurcation theory
Bifurcation theory
Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family, such as the integral curves of a family of vector fields, and the solutions of a family of differential equations...
. Since parameter
is constant, adjoin the trivially true differential equationThen in the extended state space of the evolving field and parameter,
, there exists an infinity of equilibria, not just one equilibrium, with (insulating) and constant, say . Without going into details, about each and every equilibria the linearized diffusion has two zero eigenvalues and for all the rest are negative (less than ). Thus the two dimensional dynamics on the slow manifolds emerge (see emergenceEmergence
In philosophy, systems theory, science, and art, emergence is the way complex systems and patterns arise out of a multiplicity of relatively simple interactions. Emergence is central to the theories of integrative levels and of complex systems....
) from the nonlinear diffusion no matter how complicated the initial conditions.
Here one can straightforwardly verify the slow manifold to be precisely the field
where amplitude evolves according toThat is, after the initial transients that by diffusion smooth internal structures, the emergent behavior is one of relatively slow decay of the amplitude (
) at a rate controlled by the type of boundary condition (constant ).Notice that this slow manifold model is global in
as each equilibria is necessarily in the slow subspace of each other equilibria, but is only local in parameter . We cannot yet be sure how large may be taken, but the theory assures us the results do hold for some finite parameter .Perhaps the simplest nontrivial stochastic slow manifold
Stochastic modeling is much more complicated—this example illustrates just one such complication. Consider for small parameter the two variable dynamics of this linear system forced with noise :One could simply notice that the Ornstein–Uhlenbeck process
is formally the history integral
and then assert that
is simply the integral of this history integral. However, this solution then inappropriately contains fast time integrals, due to the in the integrand, in a supposedly long time model.Alternatively, a stochastic coordinate transform extracts a sound model for the long term dynamics. Change variables to
wherethen the new variables evolve according to the simple
In these new coordinates we readily deduce
exponentially quickly, leaving undergoing a random walkRandom walk
A random walk, sometimes denoted RW, is a mathematical formalisation of a trajectory that consists of taking successive random steps. For example, the path traced by a molecule as it travels in a liquid or a gas, the search path of a foraging animal, the price of a fluctuating stock and the...
to be the long term model of the stochastic dynamics on the stochastic slow manifold obtained by setting
.A web service constructs such slow manifolds in finite dimensions, both deterministic and stochastic.