Namespaces
Variants
Actions

Difference between revisions of "Focus"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
 
Line 1: Line 1:
 +
<!--
 +
f0407001.png
 +
$#A+1 = 29 n = 0
 +
$#C+1 = 29 : ~/encyclopedia/old_files/data/F040/F.0400700 Focus
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
 +
 +
{{TEX|auto}}
 +
{{TEX|done}}
 +
 
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
  
<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/f040700/f0407001.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
$$ \tag{* }
 +
\dot{x}  = f ( x),\ \
 +
x = ( x _ {1} , x _ {2} ),\ \
 +
f:  G \subset  \mathbf R  ^ {2} \rightarrow
 +
\mathbf R  ^ {2} ,
 +
$$
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040700/f0407002.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040700/f0407003.png" /> is the domain of uniqueness, in a neighbourhood of a singular point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040700/f0407004.png" /> (cf. [[Equilibrium position|Equilibrium position]]). This type is characterized as follows. There is a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040700/f0407005.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040700/f0407006.png" /> such that for all trajectories of the system starting in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040700/f0407007.png" />, the negative semi-trajectories are escaping (in the course of time they leave any compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040700/f0407008.png" />) and the positive semi-trajectories, without leaving <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040700/f0407009.png" />, tend to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040700/f04070010.png" />, winding round it like a [[Logarithmic spiral|logarithmic spiral]], or conversely. The point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040700/f04070011.png" /> itself is also called a focus. The nature of the approach of the trajectories of the system to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040700/f04070012.png" /> can be described more precisely if one introduces polar coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040700/f04070013.png" /> on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040700/f04070014.png" />-plane with pole at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040700/f04070015.png" />. Then for any semi-trajectory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040700/f04070016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040700/f04070017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040700/f04070018.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040700/f04070019.png" />), that tends to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040700/f04070020.png" />, the polar angle of the variable point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040700/f04070021.png" /> (a left focus) or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040700/f04070022.png" /> (a right focus) as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040700/f04070023.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040700/f04070024.png" />). Its Poincaré index is 1. The figure depicts a right unstable focus at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040700/f04070025.png" />.
+
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" />
Line 11: Line 50:
 
Figure: f040700a
 
Figure: f040700a
  
For a system (*) of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040700/f04070026.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040700/f04070027.png" />) a singular point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040700/f04070028.png" /> is a focus in case the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040700/f04070029.png" /> 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 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.
 
 
  
 
====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
This article was adapted from an original article by A.F. Andreev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article