Difference between revisions of "Flow (continuous-time dynamical system)"
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{{MSC|37-01}} | {{MSC|37-01}} | ||
| − | A [[Dynamical system|dynamical system]] determined by an action of the additive group of real numbers | + | 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 | ||
| − | + | $$ | |
| + | S _ {0} ( w) = w \ \textrm{ and } \ S _ {t+} s ( w) = S _ {t} ( S _ {s} ( w) ) . | ||
| + | $$ | ||
| − | In this case | + | 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 | ||
| − | + | $$ \tag{* } | |
| + | \dot{w} _ {i} = f _ {i} ( w _ {1} \dots w _ {m} ) ,\ \ | ||
| + | i = 1 \dots m , | ||
| + | $$ | ||
| − | or, in vector notation, | + | 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==== | ||
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 |
Flow (continuous-time dynamical system). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Flow_(continuous-time_dynamical_system)&oldid=46945