In
mathematics, a flow formalizes the idea of the motion of particles in a fluid. Flows are ubiquitous in science, including
engineering
Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...
and
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
. The notion of flow is basic to the study of
ordinary differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ...
s. Informally, a flow may be viewed as a continuous motion of points over time. More formally, a flow is a
group action
In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism ...
of the
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s on a
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
.
The idea of a
vector flow
Vector most often refers to:
*Euclidean vector, a quantity with a magnitude and a direction
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematic ...
, that is, the flow determined by a
vector field, occurs in the areas of
differential topology,
Riemannian geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to point ...
and
Lie groups. Specific examples of vector flows include the
geodesic flow
In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
, the
Hamiltonian flow In mathematics and physics, a Hamiltonian vector field on a symplectic manifold is a vector field defined for any energy function or Hamiltonian. Named after the physicist and mathematician Sir William Rowan Hamilton, a Hamiltonian vector field ...
, the
Ricci flow
In the mathematical fields of differential geometry and geometric analysis, the Ricci flow ( , ), sometimes also referred to as Hamilton's Ricci flow, is a certain partial differential equation for a Riemannian metric. It is often said to be ana ...
, the
mean curvature flow, and
Anosov flow
In mathematics, more particularly in the fields of dynamical systems and geometric topology, an Anosov map on a manifold ''M'' is a certain type of mapping, from ''M'' to itself, with rather clearly marked local directions of "expansion" and "contr ...
s. Flows may also be defined for systems of
random variables and
stochastic processes, and occur in the study of
ergodic dynamical system
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in ...
s. The most celebrated of these is perhaps the
Bernoulli flow In mathematics, the Ornstein isomorphism theorem is a deep result in ergodic theory. It states that if two Bernoulli schemes have the same Kolmogorov entropy, then they are isomorphic. The result, given by Donald Ornstein in 1970, is important ...
.
Formal definition
A flow on a set is a
group action
In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism ...
of the
additive group
An additive group is a group of which the group operation is to be thought of as ''addition'' in some sense. It is usually abelian, and typically written using the symbol + for its binary operation.
This terminology is widely used with structures ...
of
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s on . More explicitly, a flow is a
mapping
:
such that, for all and all real numbers and ,
:
It is customary to write instead of , so that the equations above can be expressed as
(the
identity function
Graph of the identity function on the real numbers
In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, un ...
) and
(group law). Then, for all the mapping is a bijection with inverse This follows from the above definition, and the real parameter may be taken as a generalized
functional power
In mathematics, function composition is an operation that takes two functions and , and produces a function such that . In this operation, the function is applied to the result of applying the function to . That is, the functions and ...
, as in
function iteration
In mathematics, an iterated function is a function (that is, a function from some set to itself) which is obtained by composing another function with itself a certain number of times. The process of repeatedly applying the same function is ...
.
Flows are usually required to be compatible with
structures furnished on the set . In particular, if is equipped with a
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
, then is usually required to be
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
. If is equipped with a
differentiable structure In mathematics, an ''n''-dimensional differential structure (or differentiable structure) on a set ''M'' makes ''M'' into an ''n''-dimensional differential manifold, which is a topological manifold with some additional structure that allows for dif ...
, then is usually required to be
differentiable
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
. In these cases the flow forms a
one-parameter group of homeomorphisms and diffeomorphisms, respectively.
In certain situations one might also consider s, which are defined only in some subset
:
called the of . This is often the case with the
flows of vector fields.
Alternative notations
It is very common in many fields, including
engineering
Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...
,
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
and the study of
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s, to use a notation that makes the flow implicit. Thus, is written for and one might say that the variable depends on the time and the initial condition . Examples are given below.
In the case of a
flow of a vector field on a
smooth manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
, the flow is often denoted in such a way that its generator is made explicit. For example,
:
Orbits
Given in , the set
is called the
orbit
In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as ...
of under . Informally, it may be regarded as the trajectory of a particle that was initially positioned at . If the flow is generated by a
vector field, then its orbits are the images of its
integral curve
In mathematics, an integral curve is a parametric curve that represents a specific solution to an ordinary differential equation or system of equations.
Name
Integral curves are known by various other names, depending on the nature and interpret ...
s.
Examples
Algebraic equation
Let be a time-dependent trajectory which is a bijective function, i.e, non-periodic function. Then a flow can be defined by
:
Autonomous systems of ordinary differential equations
Let be a (time-independent) vector field
and the solution of the initial value problem
:
Then
is the flow of the vector field . It is a well-defined local flow provided that the vector field
is
Lipschitz-continuous. Then is also Lipschitz-continuous wherever defined. In general it may be hard to show that the flow is globally defined, but one simple criterion is that the vector field is
compactly supported
In mathematics, the support of a real-valued function f is the subset of the function domain containing the elements which are not mapped to zero. If the domain of f is a topological space, then the support of f is instead defined as the smalles ...
.
Time-dependent ordinary differential equations
In the case of time-dependent vector fields , one denotes
where is the solution of
:
Then is the time-dependent flow of . It is not a "flow" by the definition above, but it can easily be seen as one by rearranging its arguments. Namely, the mapping
:
indeed satisfies the group law for the last variable:
:
One can see time-dependent flows of vector fields as special cases of time-independent ones by the following trick. Define
:
Then is the solution of the "time-independent" initial value problem
:
if and only if is the solution of the original time-dependent initial value problem. Furthermore, then the mapping is exactly the flow of the "time-independent" vector field .
Flows of vector fields on manifolds
The flows of time-independent and time-dependent vector fields are defined on smooth manifolds exactly as they are defined on the Euclidean space and their local behavior is the same. However, the global topological structure of a smooth manifold is strongly manifest in what kind of global vector fields it can support, and flows of vector fields on smooth manifolds are indeed an important tool in differential topology. The bulk of studies in dynamical systems are conducted on smooth manifolds, which are thought of as "parameter spaces" in applications.
Formally: Let
be a
differentiable manifold. Let
denote the
tangent space
In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ...
of a point
Let
be the complete tangent manifold; that is,
Let
be a time-dependent
vector field on
; that is, is a smooth map such that for each
and
, one has
that is, the map
maps each point to an element of its own tangent space. For a suitable interval
containing 0, the flow of is a function
that satisfies
Solutions of heat equation
Let be a subdomain (bounded or not) of (with an integer). Denote by its boundary (assumed smooth).
Consider the following
heat equation on , for ,
:
with the following initial boundary condition in .
The equation on corresponds to the Homogeneous Dirichlet boundary condition. The mathematical setting for this problem can be the semigroup approach. To use this tool, we introduce the unbounded operator defined on
by its domain
:
(see the classical
Sobolev spaces
In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense ...
with
and
:
is the closure of the infinitely differentiable functions with compact support in for the
norm).
For any
, we have
:
With this operator, the heat equation becomes
and . Thus, the flow corresponding to this equation is (see notations above)
:
where is the (analytic) semigroup generated by .
Solutions of wave equation
Again, let be a subdomain (bounded or not) of (with an integer). We denote by its boundary (assumed smooth).
Consider the following
wave equation
The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and seism ...
on
(for ),
:
with the following initial condition in and
Using the same semigroup approach as in the case of the Heat Equation above. We write the wave equation as a first order in time partial differential equation by introducing the following unbounded operator,
:
with domain
on
(the operator is defined in the previous example).
We introduce the column vectors
:
(where
and
) and
:
With these notions, the Wave Equation becomes
and .
Thus, the flow corresponding to this equation is
:
where
is the (unitary) semigroup generated by
Bernoulli flow
Ergodic dynamical system
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in ...
s, that is, systems exhibiting randomness, exhibit flows as well. The most celebrated of these is perhaps the
Bernoulli flow In mathematics, the Ornstein isomorphism theorem is a deep result in ergodic theory. It states that if two Bernoulli schemes have the same Kolmogorov entropy, then they are isomorphic. The result, given by Donald Ornstein in 1970, is important ...
. The
Ornstein isomorphism theorem In mathematics, the Ornstein isomorphism theorem is a deep result in ergodic theory. It states that if two Bernoulli schemes have the same Kolmogorov entropy, then they are isomorphic. The result, given by Donald Ornstein in 1970, is important b ...
states that, for any given
entropy
Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynam ...
, there exists a flow , called the Bernoulli flow, such that the flow at time , ''i.e.'' , is a
Bernoulli shift
In mathematics, the Bernoulli scheme or Bernoulli shift is a generalization of the Bernoulli process to more than two possible outcomes. Bernoulli schemes appear naturally in symbolic dynamics, and are thus important in the study of dynamical syst ...
.
Furthermore, this flow is unique, up to a constant rescaling of time. That is, if , is another flow with the same entropy, then , for some constant . The notion of uniqueness and isomorphism here is that of the
isomorphism of dynamical systems
In mathematics, a measure-preserving dynamical system is an object of study in the abstract formulation of dynamical systems, and ergodic theory in particular. Measure-preserving systems obey the Poincaré recurrence theorem, and are a special ...
. Many dynamical systems, including
Sinai's billiards
A dynamical billiard is a dynamical system in which a particle alternates between free motion (typically as a straight line) and specular reflections from a boundary. When the particle hits the boundary it reflects from it Elastic collision, with ...
and
Anosov flow
In mathematics, more particularly in the fields of dynamical systems and geometric topology, an Anosov map on a manifold ''M'' is a certain type of mapping, from ''M'' to itself, with rather clearly marked local directions of "expansion" and "contr ...
s are isomorphic to Bernoulli shifts.
See also
*
Abel equation
The Abel equation, named after Niels Henrik Abel, is a type of functional equation of the form
:f(h(x)) = h(x + 1)
or
:\alpha(f(x)) = \alpha(x)+1.
The forms are equivalent when is invertible. or control the iteration of .
Equivalence
The seco ...
*
Iterated function
In mathematics, an iterated function is a function (that is, a function from some set to itself) which is obtained by composing another function with itself a certain number of times. The process of repeatedly applying the same function is ...
*
Schröder's equation
Schröder's equation, named after Ernst Schröder, is a functional equation with one independent variable: given the function , find the function such that
Schröder's equation is an eigenvalue equation for the composition operator that sen ...
*
Infinite compositions of analytic functions
In mathematics, infinite compositions of analytic functions (ICAF) offer alternative formulations of analytic continued fractions, series, products and other infinite expansions, and the theory evolving from such compositions may shed light on the ...
References
*
*
*
*
{{DEFAULTSORT:Flow (Mathematics)
Dynamical systems
Group actions (mathematics)