Flow (mathematics)
Formal definition

A flow on a set X is a group action of the additive group of real numbers on X. More explicitly, a flow is a mapping

such that, for all x ∈ X and all real numbers s and t,

It is customary to write φ (x) instead of φ (x, t), so that the equations above can be expressed as φ 0 = Id {\displaystyle \varphi ^{0}={\text{Id}}} (the identity function) and φ s ∘ φ t = φ s + t {\displaystyle \varphi ^{s}\circ \varphi ^{t}=\varphi ^{s+t}} (group law). Then, for all t ∈ R, {\displaystyle t\in \mathbb {R},} the mapping φ t: X → X {\displaystyle \varphi ^{t}:X\to X} is a bijection with inverse φ − t: X → X. {\displaystyle \varphi ^{-t}:X\to X.} This follows from the above definition, and the real parameter t may be taken as a generalized functional power, as in function iteration.

Flows are usually required to be compatible with structures furnished on the set X. In particular, if X is equipped with a topology, then φ is usually required to be continuous. If X is equipped with a differentiable structure, then φ is usually required to be differentiable. In these cases the flow forms a one-parameter group of homeomorphisms and diffeomorphisms, respectively.

In certain situations one might also consider local flow s, which are defined only in some subset

called the flow domain of φ. This is often the case with the flows of vector fields.
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