Flow (continuous-time dynamical system)

2020 Mathematics Subject Classification: Primary: 37-01 [MSN][ZBL]

A dynamical system determined by an action of the additive group of real numbers $ \mathbf R $( or additive semi-group of non-negative real numbers) on a phase space $ W $. In other words, to each $ t \in \mathbf R $( to each $ t \geq 0 $) corresponds a transformation $ S _ {t} : W \rightarrow W $ such that

$$ S _ {0} ( w) = w \ \textrm{ and } \ S _ {t+} s ( w) = S _ {t} ( S _ {s} ( w) ) . $$

In this case $ t $ is usually called "time" and the dependence of $ S _ {t} w $ on $ t $( for a fixed $ w $) is said to be the "motion" of the point $ S _ {t} w $; the set of all $ S _ {t} w $ for a given $ w $ is called the trajectory (or orbit) of $ w $( sometimes this term is used to describe the function $ t \rightarrow S _ {t} w $). 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 $ S _ {t} $ preserve this structure and certain conditions are imposed on the manner in which $ S _ {t} w $ depends on $ t $.

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

$$ \tag{* } \dot{w} _ {i} = f _ {i} ( w _ {1} \dots w _ {m} ) ,\ \ i = 1 \dots m , $$

or, in vector notation, $ \dot{w} = f ( w) $, $ w \in \mathbf R ^ {n} $. The immediate generalization of a flow is a flow on a differentiable manifold $ W ^ {m} $ defined ( "generated" ) by a smooth vector field $ f ( w) $ of class $ C ^ {k} $, $ k \geq 1 $( a smooth flow of class $ C ^ {k} $) given on $ W ^ {m} $. In this case the motion of a point $ S _ {t} w $, 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 $ f ( w) $ 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 $ S _ {t} w $ change as well as the expressions for the components of $ f ( w) $ 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).

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For general introductions into the theory of continuous, measurable or smooth flows, consult, respectively, [BS], [CFS] and [PM].

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