View source for Flow (continuous-time dynamical system)

A [[Dynamical system|dynamical system]] determined by an action of the additive group of real numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040650/f0406501.png" /> (or additive semi-group of non-negative real numbers) on a phase space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040650/f0406502.png" />. In other words, to each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040650/f0406503.png" /> (to each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040650/f0406504.png" />) corresponds a transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040650/f0406505.png" /> such that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040650/f0406506.png" /></td> </tr></table>

In this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040650/f0406507.png" /> is usually called  "time"  and the dependence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040650/f0406508.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040650/f0406509.png" /> (for a fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040650/f04065010.png" />) is said to be the  "motion"  of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040650/f04065011.png" />; the set of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040650/f04065012.png" /> for a given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040650/f04065013.png" /> is called the trajectory (or orbit) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040650/f04065014.png" /> (sometimes this term is used to describe the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040650/f04065015.png" />). 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040650/f04065016.png" /> preserve this structure and certain conditions are imposed on the manner in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040650/f04065017.png" /> depends on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040650/f04065018.png" />.

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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040650/f04065019.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

or, in vector notation, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040650/f04065020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040650/f04065021.png" />. The immediate generalization of a flow is a flow on a differentiable manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040650/f04065022.png" /> defined ( "generated" ) by a smooth vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040650/f04065023.png" /> of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040650/f04065024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040650/f04065025.png" /> (a smooth flow of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040650/f04065027.png" />) given on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040650/f04065028.png" />. In this case the motion of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040650/f04065029.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040650/f04065030.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040650/f04065031.png" /> change as well as the expressions for the components of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040650/f04065032.png" /> as functions of the local coordinates. See also [[Measurable flow|Measurable flow]]; [[Continuous flow|Continuous flow]]; [[Topological dynamical system|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|cascade]]).



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

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N.P. Bhatia,   G.P. Szegö,   "Stability theory of dynamical systems" , Springer  (1970)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  I.P. [I.P. Kornfel'd] Cornfel'd,   S.V. Fomin,   Ya.G. Sinai,   "Ergodic theory" , Springer  (1982)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J. Palis,   W. de Melo,   "Geometric theory of dynamical systems" , Springer  (1982)</TD></TR></table>