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 $t\ge 0$) corresponds a transformation $S_t:W\to W$ such that

$$S_0(w)=w\text{ and }{S}_{t+}s(w)=S_t(S_s(w)).$$

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 $t\to S_{t}w$). 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

$$\begin{array}{}\text{(*)}& {\dot{w}}_i=f_i(w_1\dots w_m),i=1\dots m,\end{array}$$

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 $C^k$, $k\ge 1$( 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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