In mathematics, a flow generalizes n-fold iteration of functions so that the iteration count n becomes a continuous parameter. It is used to formalize in mathematical terms the general idea of "a variable that depends on time" that occurs very frequently in engineering, physics and the study of ordinary differential equations. Informally, if x(t) is some coordinate of some system that behaves continuously as a function of t, then x(t) is a flow. More formally, a flow is the group action of a one-parameter group on a set.
The idea of a vector flow, that is, the flow determined by a vector field, occurs in the areas of differential topology, Riemannian geometry and Lie groups. Specific examples of vector flows include the geodesic flow, the Hamiltonian flow, the Ricci flow, the mean curvature flow, and the Anosov flow.
Formal definition
A flow on a set X is a group action of 
on X. More explicitly, a flow is a function 
with 
and that is consistent with the structure of a one-parameter group:
for all s,t in 
and 
The set 
is called the orbit of x by 
Flows are usually required to be continuous or even differentiable, when the space X has some additional structure (e.g. when X is a topological space or when 
)
A local flow is a flow which may not be defined for all times 
. (See Vector field#Flow curves.)
It is very common in many fields, including engineering, physics and the study of differential equations, to use a notation that makes the flow implicit. Thus,
- x(t)
is written for φ(x,t), and one might say that the "variable x depends on the time t". In fact, notationally, one has strict equivalence: 
. Similarly
- x0 = x(0)
is written for x = φ(x,0), and so on.
Examples
The most common examples of flows arise from describing the solutions of the autonomous ordinary differential equation
as a function of the initial condition x, when the equation has existence and uniqueness of solutions. That is, if the equation above has a unique solution 
for each 
, then 
defines a flow.
References
- D.V. Anosov (2001), "Continuous flow", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104
- D.V. Anosov (2001), "Measureable flow", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104
- D.V. Anosov (2001), "Special flow", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104
- This article incorporates material from Flow on PlanetMath, which is licensed under the GFDL.
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