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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
+
A [[Dynamical system|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
  
<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>
+
$$
 +
S _ {0} ( w)  = w \  \textrm{ and } \  S _ {t+} s ( w)  = S _ {t} ( S _ {s} ( w) ) .
 +
$$
  
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 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|Autonomous system]]) of ordinary differential equations
 
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>
+
$$ \tag{* }
 +
\dot{w} _ {i}  = f _ {i} ( w _ {1} \dots w _ {m} ) ,\ \
 +
i = 1 \dots m ,
 +
$$
  
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]].
+
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|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]]).
 
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====
 
====Comments====
For general introductions into the theory of continuous, measurable or smooth flows, consult, respectively, [[#References|[a1]]], [[#References|[a2]]] and [[#References|[a3]]].
+
For general introductions into the theory of continuous, measurable or smooth flows, consult, respectively, {{Cite|BS}}, {{Cite|CFS}} and {{Cite|PM}}.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> N.P. Bhatia, G.P. Szegö, "Stability theory of dynamical systems" , Springer (1970) {{MR|0289890}} {{ZBL|0213.10904}} </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) {{MR|832433}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> J. Palis, W. de Melo, "Geometric theory of dynamical systems" , Springer (1982) {{MR|0669541}} {{ZBL|0491.58001}} </TD></TR></table>
+
{|
 +
|valign="top"|{{Ref|BS}}|| N.P. Bhatia, G.P. Szegö, "Stability theory of dynamical systems" , Springer (1970) {{MR|0289890}} {{ZBL|0213.10904}}
 +
|-
 +
|valign="top"|{{Ref|CFS}}|| I.P. Cornfel'd, S.V. Fomin, Ya.G. Sinai, "Ergodic theory" , Springer (1982) (Translated from Russian) {{MR|832433}} {{ZBL|}}
 +
|-
 +
|valign="top"|{{Ref|PM}}|| J. Palis, W. de Melo, "Geometric theory of dynamical systems" , Springer (1982) {{MR|0669541}} {{ZBL|0491.58001}}
 +
|}

Latest revision as of 19:39, 5 June 2020


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).

Comments

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

References

[BS] N.P. Bhatia, G.P. Szegö, "Stability theory of dynamical systems" , Springer (1970) MR0289890 Zbl 0213.10904
[CFS] I.P. Cornfel'd, S.V. Fomin, Ya.G. Sinai, "Ergodic theory" , Springer (1982) (Translated from Russian) MR832433
[PM] J. Palis, W. de Melo, "Geometric theory of dynamical systems" , Springer (1982) MR0669541 Zbl 0491.58001
How to Cite This Entry:
Flow (continuous-time dynamical system). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Flow_(continuous-time_dynamical_system)&oldid=23609
This article was adapted from an original article by D.V. Anosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article