Flow (continuous-time dynamical system)

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A dynamical system determined by an action of the additive group of real numbers (or additive semi-group of non-negative real numbers) on a phase space . In other words, to each (to each ) corresponds a transformation such that

In this case is usually called "time" and the dependence of on (for a fixed ) is said to be the "motion" of the point ; the set of all for a given is called the trajectory (or orbit) of (sometimes this term is used to describe the function ). Just as for traditional dynamical systems the phase space of a flow usually is provided with a certain structure with which the flow is compatible: the transformations preserve this structure and certain conditions are imposed on the manner in which depends on .

In applications one usually encounters flows described by autonomous systems (cf. Autonomous system) of ordinary differential equations


or, in vector notation, , . The immediate generalization of a flow is a flow on a differentiable manifold defined ( "generated" ) by a smooth vector field of class , (a smooth flow of class ) given on . In this case the motion of a point , as long as it stays within one chart (local coordinate system), is described by a system of the form (*), in the right-hand side of which one finds the components of the vector in the corresponding coordinates. When passing to another chart the description of the motion changes, since in this case both the coordinates of the point change as well as the expressions for the components of as functions of the local coordinates. See also Measurable flow; Continuous flow; Topological dynamical system.

Flows form the most important class of dynamical systems and were, moreover, the first to be studied. The term "dynamical system" is often used in a narrow sense, meaning precisely a flow (or a flow and a cascade).


For general introductions into the theory of continuous, measurable or smooth flows, consult, respectively, [a1], [a2] and [a3].


[a1] N.P. Bhatia, G.P. Szegö, "Stability theory of dynamical systems" , Springer (1970)
[a2] I.P. [I.P. Kornfel'd] Cornfel'd, S.V. Fomin, Ya.G. Sinai, "Ergodic theory" , Springer (1982) (Translated from Russian)
[a3] J. Palis, W. de Melo, "Geometric theory of dynamical systems" , Springer (1982)
How to Cite This Entry:
Flow (continuous-time dynamical system). Encyclopedia of Mathematics. URL:
This article was adapted from an original article by D.V. Anosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article