Flow (continuous-time dynamical system)

2010 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).