Difference between revisions of "Focus"
From Encyclopedia of Mathematics
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A type of arrangement of the trajectories of an autonomous system of first-order ordinary differential equations | A type of arrangement of the trajectories of an autonomous system of first-order ordinary differential equations | ||
| − | + | $$ \tag{* } | |
| + | \dot{x} = f ( x),\ \ | ||
| + | x = ( x _ {1} , x _ {2} ),\ \ | ||
| + | f: G \subset \mathbf R ^ {2} \rightarrow | ||
| + | \mathbf R ^ {2} , | ||
| + | $$ | ||
| − | + | $ f \in C ( G) $, | |
| + | where $ G $ | ||
| + | is the domain of uniqueness, in a neighbourhood of a singular point $ x _ {0} $( | ||
| + | cf. [[Equilibrium position|Equilibrium position]]). This type is characterized as follows. There is a neighbourhood $ U $ | ||
| + | of $ x _ {0} $ | ||
| + | such that for all trajectories of the system starting in $ U \setminus \{ x _ {0} \} $, | ||
| + | the negative semi-trajectories are escaping (in the course of time they leave any compact set $ V \subset U $) | ||
| + | and the positive semi-trajectories, without leaving $ U $, | ||
| + | tend to $ x _ {0} $, | ||
| + | winding round it like a [[Logarithmic spiral|logarithmic spiral]], or conversely. The point $ x _ {0} $ | ||
| + | itself is also called a focus. The nature of the approach of the trajectories of the system to $ x _ {0} $ | ||
| + | can be described more precisely if one introduces polar coordinates $ r, \phi $ | ||
| + | on the $ ( x _ {1} , x _ {2} ) $- | ||
| + | plane with pole at $ x _ {0} $. | ||
| + | Then for any semi-trajectory $ r = r ( t) $, | ||
| + | $ \phi = \phi ( t) $, | ||
| + | $ t \geq 0 $( | ||
| + | $ t \leq 0 $), | ||
| + | that tends to $ x _ {0} $, | ||
| + | the polar angle of the variable point $ \phi ( t) \rightarrow + \infty $( | ||
| + | a left focus) or $ - \infty $( | ||
| + | a right focus) as $ t \rightarrow \infty $. | ||
| − | A focus is either asymptotically Lyapunov stable or completely unstable (asymptotically stable as | + | A focus is either asymptotically Lyapunov stable or completely unstable (asymptotically stable as $ t \rightarrow - \infty $). |
| + | Its Poincaré index is 1. The figure depicts a right unstable focus at $ x _ {0} = ( 0, 0) $. | ||
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/f040700a.gif" /> | <img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/f040700a.gif" /> | ||
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Figure: f040700a | Figure: f040700a | ||
| − | For a system (*) of class | + | For a system (*) of class $ C ^ {1} $( |
| + | $ f \in C ^ {1} ( G) $) | ||
| + | a singular point $ x _ {0} $ | ||
| + | is a focus in case the matrix $ A = f ^ { \prime } ( x _ {0} ) $ | ||
| + | has complex conjugate eigen values with non-zero real part, but it may also be a focus in case this matrix has purely imaginary or multiple real eigen values (see also [[Centre|Centre]]; [[Centre and focus problem|Centre and focus problem]]). | ||
For references see [[Singular point|Singular point]] of a differential equation. | For references see [[Singular point|Singular point]] of a differential equation. | ||
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| − | |||
====Comments==== | ====Comments==== | ||
| − | |||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> P. Hartman, "Ordinary differential equations" , Birkhäuser (1982)</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> P. Hartman, "Ordinary differential equations" , Birkhäuser (1982)</TD></TR></table> | ||
Latest revision as of 19:39, 5 June 2020
A type of arrangement of the trajectories of an autonomous system of first-order ordinary differential equations
$$ \tag{* } \dot{x} = f ( x),\ \ x = ( x _ {1} , x _ {2} ),\ \ f: G \subset \mathbf R ^ {2} \rightarrow \mathbf R ^ {2} , $$
$ f \in C ( G) $, where $ G $ is the domain of uniqueness, in a neighbourhood of a singular point $ x _ {0} $( cf. Equilibrium position). This type is characterized as follows. There is a neighbourhood $ U $ of $ x _ {0} $ such that for all trajectories of the system starting in $ U \setminus \{ x _ {0} \} $, the negative semi-trajectories are escaping (in the course of time they leave any compact set $ V \subset U $) and the positive semi-trajectories, without leaving $ U $, tend to $ x _ {0} $, winding round it like a logarithmic spiral, or conversely. The point $ x _ {0} $ itself is also called a focus. The nature of the approach of the trajectories of the system to $ x _ {0} $ can be described more precisely if one introduces polar coordinates $ r, \phi $ on the $ ( x _ {1} , x _ {2} ) $- plane with pole at $ x _ {0} $. Then for any semi-trajectory $ r = r ( t) $, $ \phi = \phi ( t) $, $ t \geq 0 $( $ t \leq 0 $), that tends to $ x _ {0} $, the polar angle of the variable point $ \phi ( t) \rightarrow + \infty $( a left focus) or $ - \infty $( a right focus) as $ t \rightarrow \infty $.
A focus is either asymptotically Lyapunov stable or completely unstable (asymptotically stable as $ t \rightarrow - \infty $). Its Poincaré index is 1. The figure depicts a right unstable focus at $ x _ {0} = ( 0, 0) $.
Figure: f040700a
For a system (*) of class $ C ^ {1} $( $ f \in C ^ {1} ( G) $) a singular point $ x _ {0} $ is a focus in case the matrix $ A = f ^ { \prime } ( x _ {0} ) $ has complex conjugate eigen values with non-zero real part, but it may also be a focus in case this matrix has purely imaginary or multiple real eigen values (see also Centre; Centre and focus problem).
For references see Singular point of a differential equation.
Comments
References
| [a1] | P. Hartman, "Ordinary differential equations" , Birkhäuser (1982) |
How to Cite This Entry:
Focus. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Focus&oldid=12561
Focus. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Focus&oldid=12561
This article was adapted from an original article by A.F. Andreev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article