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An open curvilinear sector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s0837701.png" /> with vertex at an isolated singular point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s0837702.png" /> of an [[Autonomous system|autonomous system]] of second-order ordinary differential equations
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<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/s/s083/s083770/s0837703.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s0837704.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s0837705.png" /> is the domain of uniqueness, that satisfies the following four conditions: 1) each lateral boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s0837706.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s0837708.png" />-curve of the system (*) (i.e. a semi-trajectory that approaches <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s0837709.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377010.png" />, and touches a certain direction at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377011.png" />); 2) the outer boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377012.png" /> is a simple parametric arc (the homeomorphic image of a closed interval); 3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377013.png" /> does not contain singular points of (*). The fourth condition is one of the following three: 4a) all trajectories of the system (*) that start in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377014.png" /> leave this sector for both increasing and decreasing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377015.png" />; such a sector is called a hyperbolic sector, or a saddle sector (Fig. a); 4b) all trajectories of (*) that start in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377016.png" /> sufficiently near <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377017.png" /> do not leave <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377018.png" /> but approach <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377019.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377020.png" /> increases, and as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377021.png" /> decreases they leave <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377022.png" /> (or vice-versa); such a sector is called a parabolic sector or an open node sector (Fig. b); or 4c) all the trajectories of (*) that start in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377023.png" /> sufficiently near <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377024.png" /> do not leave <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377025.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377026.png" /> increases or decreases but approach <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377027.png" />, forming together with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377028.png" /> closed curves (loops), and for any two loops one encloses the other; such a sector is called an elliptic sector or a closed node sector (Fig. c).
+
An open curvilinear sector  $  S $
 +
with vertex at an isolated singular point  $  O $
 +
of an [[Autonomous system|autonomous system]] of second-order ordinary differential equations
 +
 
 +
$$ \tag{* }
 +
\dot{x}  = f ( x),\ \
 +
x \in \mathbf R  ^ {2} ,
 +
$$
 +
 
 +
$  f \in C ( G) $,  
 +
where $  G $
 +
is the domain of uniqueness, that satisfies the following four conditions: 1) each lateral boundary of $  S $
 +
is a $  TO $-
 +
curve of the system (*) (i.e. a semi-trajectory that approaches $  O $
 +
as $  | t | \rightarrow + \infty $,  
 +
and touches a certain direction at $  O $);  
 +
2) the outer boundary of $  S $
 +
is a simple parametric arc (the homeomorphic image of a closed interval); 3) $  \overline{S}\; \setminus  \{ 0 \} $
 +
does not contain singular points of (*). The fourth condition is one of the following three: 4a) all trajectories of the system (*) that start in $  S $
 +
leave this sector for both increasing and decreasing $  t $;  
 +
such a sector is called a hyperbolic sector, or a saddle sector (Fig. a); 4b) all trajectories of (*) that start in $  S $
 +
sufficiently near $  O $
 +
do not leave $  S $
 +
but approach $  O $
 +
as $  t $
 +
increases, and as $  t $
 +
decreases they leave $  S $(
 +
or vice-versa); such a sector is called a parabolic sector or an open node sector (Fig. b); or 4c) all the trajectories of (*) that start in $  S $
 +
sufficiently near $  O $
 +
do not leave $  S $
 +
as $  t $
 +
increases or decreases but approach $  O $,  
 +
forming together with $  O $
 +
closed curves (loops), and for any two loops one encloses the other; such a sector is called an elliptic sector or a closed node sector (Fig. c).
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/s083770a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/s083770a.gif" />
Line 17: Line 58:
 
Figure: s083770c
 
Figure: s083770c
  
For any analytic system (*) with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377029.png" />-curves, a disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377030.png" /> of sufficiently small radius and centre at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377031.png" /> can always be divided into a finite number of sectors of a specific form: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377032.png" /> hyperbolic, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377033.png" /> parabolic and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377034.png" /> elliptic ones (see [[#References|[1]]] and [[#References|[2]]]). The [[Frommer method|Frommer method]] can be used to exhibit all these sectors, to determine the type of each, and to establish the rules of their succession in a circuit about <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377035.png" /> along the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377036.png" /> (and thereby to show the topological structure of the arrangement of the trajectories of (*) in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377037.png" />). There are a priori estimates from above for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377038.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377039.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377040.png" /> in terms of the order of smallness of the norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377041.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377042.png" /> (see [[#References|[1]]], [[#References|[4]]], [[#References|[5]]]).
+
For any analytic system (*) with $  TO $-
 +
curves, a disc $  Q $
 +
of sufficiently small radius and centre at $  O $
 +
can always be divided into a finite number of sectors of a specific form: $  h $
 +
hyperbolic, $  p $
 +
parabolic and $  e $
 +
elliptic ones (see [[#References|[1]]] and [[#References|[2]]]). The [[Frommer method|Frommer method]] can be used to exhibit all these sectors, to determine the type of each, and to establish the rules of their succession in a circuit about $  O $
 +
along the boundary of $  Q $(
 +
and thereby to show the topological structure of the arrangement of the trajectories of (*) in a neighbourhood of $  O $).  
 +
There are a priori estimates from above for $  h $,  
 +
$  p $
 +
and $  e $
 +
in terms of the order of smallness of the norm $  \| f ( x) \| $
 +
as $  x \rightarrow 0 $(
 +
see [[#References|[1]]], [[#References|[4]]], [[#References|[5]]]).
  
Sometimes (see, for example, [[#References|[3]]]) the notion of a  "sector"  is defined more freely: In hyperbolic and parabolic sectors loops are allowed that cover a set without limit points on the rear boundary of a sector, and in elliptic sectors, loops that do not contain one another. Here the first sentence of the previous paragraph remains valid also for a system (*) of general form, and the Poincaré index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377043.png" /> of the [[Singular point|singular point]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377044.png" /> of (*) is expressed by Bendixson's formula
+
Sometimes (see, for example, [[#References|[3]]]) the notion of a  "sector"  is defined more freely: In hyperbolic and parabolic sectors loops are allowed that cover a set without limit points on the rear boundary of a sector, and in elliptic sectors, loops that do not contain one another. Here the first sentence of the previous paragraph remains valid also for a system (*) of general form, and the Poincaré index $  i $
 +
of the [[Singular point|singular point]] $  O $
 +
of (*) is expressed by Bendixson's formula
  
<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/s/s083/s083770/s08377045.png" /></td> </tr></table>
+
$$
 +
= 1 +
 +
\frac{e - h }{2}
 +
.
 +
$$
  
 
====References====
 
====References====
Line 31: Line 92:
 
A Frommer sector, or Frommer normal domain, is a circular sector
 
A Frommer sector, or Frommer normal domain, is a circular sector
  
<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/s/s083/s083770/s08377046.png" /></td> </tr></table>
+
$$
 +
= \
 +
\{ {( r, \phi ) } : {0 < r \leq  \delta ,\
 +
| \phi - \phi _ {0} | \leq  \epsilon } \}
 +
$$
  
with vertex at an isolated point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377047.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377048.png" />) of the system
+
with vertex at an isolated point $  O $(
 +
$  x = x _ {0} $)  
 +
of the system
  
(see 1)) with lateral boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377049.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377050.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377051.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377052.png" />, and with the rear boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377053.png" /> satisfying the following conditions (here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377054.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377055.png" /> are polar coordinates in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377056.png" />-plane with pole at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377057.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377058.png" />):
+
(see 1)) with lateral boundary $  OA $
 +
and $  OB $,  
 +
$  \phi _ {A} = \phi _ {0} - \epsilon $,  
 +
$  \phi _ {B} = \phi _ {0} + \epsilon $,  
 +
and with the rear boundary $  AB $
 +
satisfying the following conditions (here $  r $
 +
and $  \phi $
 +
are polar coordinates in the $  x $-
 +
plane with pole at $  O $,  
 +
and $  \delta , \epsilon , \phi _ {0} \in \mathbf R $):
  
A) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377059.png" /> is an exceptional direction of the system
+
A) $  \phi = \phi _ {0} $
 +
is an exceptional direction of the system
  
at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377060.png" />, that is, there is a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377061.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377062.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377063.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377064.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377065.png" />, such that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377066.png" /> is the angle between the directions of the vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377067.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377068.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377069.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377070.png" />, and this direction is unique in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377071.png" />;
+
at $  O $,  
 +
that is, there is a sequence $  x _ {k} = x _ {0} + ( r _ {k}  \cos  \phi _ {k} , r _ {k}  \sin  \phi _ {k} ) $,  
 +
$  k = 1 \dots $
 +
$  r _ {k} \rightarrow 0 $,  
 +
$  \phi _ {k} \rightarrow \phi _ {0} $
 +
as $  k \rightarrow + \infty $,  
 +
such that if $  \alpha ( x) $
 +
is the angle between the directions of the vectors $  f ( x) $
 +
and $  x - x _ {0} $,  
 +
then $  \mathop{\rm tan}  \alpha ( x _ {k} ) \rightarrow 0 $
 +
as $  k \rightarrow + \infty $,  
 +
and this direction is unique in $  N $;
  
B) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377072.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377073.png" />;
+
B) $  \mathop{\rm tan}  \alpha ( x) \neq 0 $
 +
for any $  x \in OA \cup OB $;
  
C) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377074.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377075.png" />.
+
C) $  \alpha ( x) \neq \pi /2 $
 +
for any $  x \in N $.
  
Suppose that the angle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377076.png" /> is measured from the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377077.png" /> and has the sign of the reference direction. A sector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377078.png" /> is called a Frommer normal domain of the first type (notation: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377079.png" />) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377080.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377081.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377082.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377083.png" />; a normal domain of the second type (notation: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377084.png" />) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377085.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377086.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377087.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377088.png" />; and a normal domain of the third type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377089.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377090.png" /> has one and the same sign on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377091.png" /> and on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377092.png" />. These domains were introduced by M. Frommer [[#References|[1]]].
+
Suppose that the angle $  \alpha ( x) $
 +
is measured from the vector $  x - x _ {0} $
 +
and has the sign of the reference direction. A sector $  N $
 +
is called a Frommer normal domain of the first type (notation: $  N _ {1} $)  
 +
if $  \mathop{\rm tan}  \alpha ( x) < 0 $
 +
for $  x \in OA $
 +
and  $  \mathop{\rm tan}  \alpha ( x) > 0 $
 +
for $  x \in OB $;  
 +
a normal domain of the second type (notation: $  N _ {2} $)  
 +
if $  \mathop{\rm tan}  \alpha ( x) > 0 $
 +
on $  OA $
 +
and $  \mathop{\rm tan}  \alpha ( x) < 0 $
 +
on $  OB $;  
 +
and a normal domain of the third type $  ( N _ {3} ) $
 +
if $  \mathop{\rm tan}  \alpha ( x) $
 +
has one and the same sign on $  OA $
 +
and on $  OB $.  
 +
These domains were introduced by M. Frommer [[#References|[1]]].
  
 
The trajectories of the system
 
The trajectories of the system
  
in Frommer normal domains behave as follows. The domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377093.png" /> is covered by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377094.png" />-curves of the system (Fig. d). They form an open pencil (cf. [[Sheaf|Sheaf]] 2)), that is, a family of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377095.png" />-curves of the same type that depends continuously on a parameter which varies over an open interval. In the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377096.png" /> there is either a) a unique <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377097.png" />-curve (Fig. e), or b) infinitely many <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377098.png" />-curves (a closed pencil; cf. Fig. f). In the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377099.png" />, either a) there are infinitely many <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s083770100.png" />-curves (a semi-open pencil; Fig. g) or b) there are no <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s083770101.png" />-curves (Fig. h).
+
in Frommer normal domains behave as follows. The domain $  N _ {1} $
 +
is covered by $  O $-
 +
curves of the system (Fig. d). They form an open pencil (cf. [[Sheaf|Sheaf]] 2)), that is, a family of $  O $-
 +
curves of the same type that depends continuously on a parameter which varies over an open interval. In the domain $  N _ {2} $
 +
there is either a) a unique $  O $-
 +
curve (Fig. e), or b) infinitely many $  O $-
 +
curves (a closed pencil; cf. Fig. f). In the domain $  N _ {3} $,  
 +
either a) there are infinitely many $  O $-
 +
curves (a semi-open pencil; Fig. g) or b) there are no $  O $-
 +
curves (Fig. h).
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/s083770d.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/s083770d.gif" />
Line 71: Line 187:
 
Figure: s083770h
 
Figure: s083770h
  
In a normal domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s083770102.png" /> of any type the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s083770103.png" />-curves tend to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s083770104.png" /> along the direction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s083770105.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s083770106.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s083770107.png" />), and with decreasing (increasing) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s083770108.png" /> they leave the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s083770109.png" />; all other trajectories leave <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s083770110.png" /> for both increasing and decreasing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s083770111.png" />. The problems of distinguishing between the cases a) and b) for domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s083770112.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s083770113.png" /> are called, respectively, the first and second distinction problems of Frommer.
+
In a normal domain $  N $
 +
of any type the $  O $-
 +
curves tend to $  O $
 +
along the direction $  \phi = \phi _ {0} $
 +
as $  t \rightarrow + \infty $(
 +
or $  t \rightarrow - \infty $),  
 +
and with decreasing (increasing) $  t $
 +
they leave the domain $  N $;  
 +
all other trajectories leave $  N $
 +
for both increasing and decreasing $  t $.  
 +
The problems of distinguishing between the cases a) and b) for domains $  N _ {2} $
 +
and $  N _ {3} $
 +
are called, respectively, the first and second distinction problems of Frommer.
  
 
If a system
 
If a system
  
has at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s083770114.png" /> a finite number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s083770115.png" /> of exceptional directions, each of which can be included in a normal domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s083770116.png" />, and if for all domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s083770117.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s083770118.png" /> Frommer's distinction problems are solvable, then the topological structure of the arrangement of the trajectories of the system in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s083770119.png" /> is completely explained, because the sectors with vertex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s083770120.png" /> that are positioned between normal domains are, sufficiently close to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s083770121.png" />, entirely intersected by the trajectories of the system (as in Fig. h). Such a situation holds, for example, when
+
has at $  O $
 +
a finite number $  (> 0) $
 +
of exceptional directions, each of which can be included in a normal domain $  N $,  
 +
and if for all domains $  N _ {2} $
 +
and $  N _ {3} $
 +
Frommer's distinction problems are solvable, then the topological structure of the arrangement of the trajectories of the system in a neighbourhood of $  O $
 +
is completely explained, because the sectors with vertex $  O $
 +
that are positioned between normal domains are, sufficiently close to $  O $,  
 +
entirely intersected by the trajectories of the system (as in Fig. h). Such a situation holds, for example, when
  
<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/s/s083/s083770/s083770122.png" /></td> </tr></table>
+
$$
 +
f ( x)  = P ( x) + p ( x),\ \
 +
P = ( P _ {1} , P _ {2} ),
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s083770123.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s083770124.png" /> are forms of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s083770125.png" /> in the components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s083770126.png" /> of the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s083770127.png" />,
+
where $  P _ {1} $
 +
and $  P _ {2} $
 +
are forms of degree $  n \geq  1 $
 +
in the components $  x _ {1} , x _ {2} $
 +
of the vector $  x $,
  
<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/s/s083/s083770/s083770128.png" /></td> </tr></table>
+
$$
 +
p ( x)  = o ( \| x \|  ^ {n} ) \ \
 +
\textrm{ as }  \| x \| \rightarrow 0,
 +
$$
  
and when the following conditions are fulfilled: The form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s083770129.png" /> has real linear factors, the forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s083770130.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s083770131.png" /> do not have common real linear factors, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s083770132.png" />. Here situation a) holds in each of the domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s083770133.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s083770134.png" />.
+
and when the following conditions are fulfilled: The form $  x _ {1} P _ {2} ( x) - x _ {2} P _ {1} ( x) $
 +
has real linear factors, the forms $  P _ {1} $
 +
and $  P _ {2} $
 +
do not have common real linear factors, and $  p \in C ^ {n + 1 } $.  
 +
Here situation a) holds in each of the domains $  N _ {2} $,  
 +
$  N _ {3} $.
  
 
Analogues of Frommer normal domains have been introduced for systems of the form
 
Analogues of Frommer normal domains have been introduced for systems of the form
  
of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s083770135.png" />.
+
of order $  \geq  3 $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M. Frommer,  "Die Integralkurven einer gewöhnlichen Differentialgleichung erster Ordnung in der Umgebung rationaler Unbestimtheitsstellen"  ''Math. Ann.'' , '''99'''  (1928)  pp. 222–272</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.V. Nemytskii,  V.V. Stepanov,  "Qualitative theory of differential equations" , Princeton Univ. Press  (1960)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.F. Andreev,  "A uniqueness theorem for a normal region of Frommer's second type"  ''Soviet Math. Dokl.'' , '''3''' :  1  (1962)  pp. 132–135  ''Dokl. Akad. Nauk SSSR'' , '''142''' :  4  (1962)  pp. 754–757</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A.F. Andreev,  "Strengthening of the uniqueness theorem for an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s083770136.png" />-curve in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s083770137.png" />"  ''Soviet Math. Dokl.'' , '''3''' :  5  (1962)  pp. 1215–1216  ''Dokl. Akad. Nauk SSSR'' , '''146''' :  1  (1962)  pp. 9–10</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M. Frommer,  "Die Integralkurven einer gewöhnlichen Differentialgleichung erster Ordnung in der Umgebung rationaler Unbestimtheitsstellen"  ''Math. Ann.'' , '''99'''  (1928)  pp. 222–272</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.V. Nemytskii,  V.V. Stepanov,  "Qualitative theory of differential equations" , Princeton Univ. Press  (1960)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.F. Andreev,  "A uniqueness theorem for a normal region of Frommer's second type"  ''Soviet Math. Dokl.'' , '''3''' :  1  (1962)  pp. 132–135  ''Dokl. Akad. Nauk SSSR'' , '''142''' :  4  (1962)  pp. 754–757</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A.F. Andreev,  "Strengthening of the uniqueness theorem for an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s083770136.png" />-curve in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s083770137.png" />"  ''Soviet Math. Dokl.'' , '''3''' :  5  (1962)  pp. 1215–1216  ''Dokl. Akad. Nauk SSSR'' , '''146''' :  1  (1962)  pp. 9–10</TD></TR></table>

Latest revision as of 08:12, 6 June 2020


An open curvilinear sector $ S $ with vertex at an isolated singular point $ O $ of an autonomous system of second-order ordinary differential equations

$$ \tag{* } \dot{x} = f ( x),\ \ x \in \mathbf R ^ {2} , $$

$ f \in C ( G) $, where $ G $ is the domain of uniqueness, that satisfies the following four conditions: 1) each lateral boundary of $ S $ is a $ TO $- curve of the system (*) (i.e. a semi-trajectory that approaches $ O $ as $ | t | \rightarrow + \infty $, and touches a certain direction at $ O $); 2) the outer boundary of $ S $ is a simple parametric arc (the homeomorphic image of a closed interval); 3) $ \overline{S}\; \setminus \{ 0 \} $ does not contain singular points of (*). The fourth condition is one of the following three: 4a) all trajectories of the system (*) that start in $ S $ leave this sector for both increasing and decreasing $ t $; such a sector is called a hyperbolic sector, or a saddle sector (Fig. a); 4b) all trajectories of (*) that start in $ S $ sufficiently near $ O $ do not leave $ S $ but approach $ O $ as $ t $ increases, and as $ t $ decreases they leave $ S $( or vice-versa); such a sector is called a parabolic sector or an open node sector (Fig. b); or 4c) all the trajectories of (*) that start in $ S $ sufficiently near $ O $ do not leave $ S $ as $ t $ increases or decreases but approach $ O $, forming together with $ O $ closed curves (loops), and for any two loops one encloses the other; such a sector is called an elliptic sector or a closed node sector (Fig. c).

Figure: s083770a

Figure: s083770b

Figure: s083770c

For any analytic system (*) with $ TO $- curves, a disc $ Q $ of sufficiently small radius and centre at $ O $ can always be divided into a finite number of sectors of a specific form: $ h $ hyperbolic, $ p $ parabolic and $ e $ elliptic ones (see [1] and [2]). The Frommer method can be used to exhibit all these sectors, to determine the type of each, and to establish the rules of their succession in a circuit about $ O $ along the boundary of $ Q $( and thereby to show the topological structure of the arrangement of the trajectories of (*) in a neighbourhood of $ O $). There are a priori estimates from above for $ h $, $ p $ and $ e $ in terms of the order of smallness of the norm $ \| f ( x) \| $ as $ x \rightarrow 0 $( see [1], [4], [5]).

Sometimes (see, for example, [3]) the notion of a "sector" is defined more freely: In hyperbolic and parabolic sectors loops are allowed that cover a set without limit points on the rear boundary of a sector, and in elliptic sectors, loops that do not contain one another. Here the first sentence of the previous paragraph remains valid also for a system (*) of general form, and the Poincaré index $ i $ of the singular point $ O $ of (*) is expressed by Bendixson's formula

$$ i = 1 + \frac{e - h }{2} . $$

References

[1] I. Bendixson, "Sur des courbes définiés par des équations différentielles" Acta Math. , 24 (1901) pp. 1–88
[2] A.A. Andronov, E.A. Leontovich, I.I. Gordon, A.G. Maier, "Qualitative theory of second-order dynamic systems" , Wiley (1973) (Translated from Russian)
[3] P. Hartman, "Ordinary differential equations" , Birkhäuser (1982)
[4] A.N. Berlinskii, "On the structure of the neighborhood of a singular point of a two-dimensional autonomous system" Soviet Math. Dokl. , 10 : 4 (1969) pp. 882–885 Dokl. Akad. Nauk SSSR , 187 : 3 (1969) pp. 502–505
[5] M.E. Sagalovich, "Classes of local topological structures of an equilibrium state" Diff. Equations , 15 : 2 (1979) pp. 253–255 Differentsial'nye Urnveniya , 15 : 2 (1979) pp. 360–362

Comments

The lateral boundaries are sometimes called base solutions.

A Frommer sector, or Frommer normal domain, is a circular sector

$$ N = \ \{ {( r, \phi ) } : {0 < r \leq \delta ,\ | \phi - \phi _ {0} | \leq \epsilon } \} $$

with vertex at an isolated point $ O $( $ x = x _ {0} $) of the system

(see 1)) with lateral boundary $ OA $ and $ OB $, $ \phi _ {A} = \phi _ {0} - \epsilon $, $ \phi _ {B} = \phi _ {0} + \epsilon $, and with the rear boundary $ AB $ satisfying the following conditions (here $ r $ and $ \phi $ are polar coordinates in the $ x $- plane with pole at $ O $, and $ \delta , \epsilon , \phi _ {0} \in \mathbf R $):

A) $ \phi = \phi _ {0} $ is an exceptional direction of the system

at $ O $, that is, there is a sequence $ x _ {k} = x _ {0} + ( r _ {k} \cos \phi _ {k} , r _ {k} \sin \phi _ {k} ) $, $ k = 1 \dots $ $ r _ {k} \rightarrow 0 $, $ \phi _ {k} \rightarrow \phi _ {0} $ as $ k \rightarrow + \infty $, such that if $ \alpha ( x) $ is the angle between the directions of the vectors $ f ( x) $ and $ x - x _ {0} $, then $ \mathop{\rm tan} \alpha ( x _ {k} ) \rightarrow 0 $ as $ k \rightarrow + \infty $, and this direction is unique in $ N $;

B) $ \mathop{\rm tan} \alpha ( x) \neq 0 $ for any $ x \in OA \cup OB $;

C) $ \alpha ( x) \neq \pi /2 $ for any $ x \in N $.

Suppose that the angle $ \alpha ( x) $ is measured from the vector $ x - x _ {0} $ and has the sign of the reference direction. A sector $ N $ is called a Frommer normal domain of the first type (notation: $ N _ {1} $) if $ \mathop{\rm tan} \alpha ( x) < 0 $ for $ x \in OA $ and $ \mathop{\rm tan} \alpha ( x) > 0 $ for $ x \in OB $; a normal domain of the second type (notation: $ N _ {2} $) if $ \mathop{\rm tan} \alpha ( x) > 0 $ on $ OA $ and $ \mathop{\rm tan} \alpha ( x) < 0 $ on $ OB $; and a normal domain of the third type $ ( N _ {3} ) $ if $ \mathop{\rm tan} \alpha ( x) $ has one and the same sign on $ OA $ and on $ OB $. These domains were introduced by M. Frommer [1].

The trajectories of the system

in Frommer normal domains behave as follows. The domain $ N _ {1} $ is covered by $ O $- curves of the system (Fig. d). They form an open pencil (cf. Sheaf 2)), that is, a family of $ O $- curves of the same type that depends continuously on a parameter which varies over an open interval. In the domain $ N _ {2} $ there is either a) a unique $ O $- curve (Fig. e), or b) infinitely many $ O $- curves (a closed pencil; cf. Fig. f). In the domain $ N _ {3} $, either a) there are infinitely many $ O $- curves (a semi-open pencil; Fig. g) or b) there are no $ O $- curves (Fig. h).

Figure: s083770d

Figure: s083770e

Figure: s083770f

Figure: s083770g

Figure: s083770h

In a normal domain $ N $ of any type the $ O $- curves tend to $ O $ along the direction $ \phi = \phi _ {0} $ as $ t \rightarrow + \infty $( or $ t \rightarrow - \infty $), and with decreasing (increasing) $ t $ they leave the domain $ N $; all other trajectories leave $ N $ for both increasing and decreasing $ t $. The problems of distinguishing between the cases a) and b) for domains $ N _ {2} $ and $ N _ {3} $ are called, respectively, the first and second distinction problems of Frommer.

If a system

has at $ O $ a finite number $ (> 0) $ of exceptional directions, each of which can be included in a normal domain $ N $, and if for all domains $ N _ {2} $ and $ N _ {3} $ Frommer's distinction problems are solvable, then the topological structure of the arrangement of the trajectories of the system in a neighbourhood of $ O $ is completely explained, because the sectors with vertex $ O $ that are positioned between normal domains are, sufficiently close to $ O $, entirely intersected by the trajectories of the system (as in Fig. h). Such a situation holds, for example, when

$$ f ( x) = P ( x) + p ( x),\ \ P = ( P _ {1} , P _ {2} ), $$

where $ P _ {1} $ and $ P _ {2} $ are forms of degree $ n \geq 1 $ in the components $ x _ {1} , x _ {2} $ of the vector $ x $,

$$ p ( x) = o ( \| x \| ^ {n} ) \ \ \textrm{ as } \| x \| \rightarrow 0, $$

and when the following conditions are fulfilled: The form $ x _ {1} P _ {2} ( x) - x _ {2} P _ {1} ( x) $ has real linear factors, the forms $ P _ {1} $ and $ P _ {2} $ do not have common real linear factors, and $ p \in C ^ {n + 1 } $. Here situation a) holds in each of the domains $ N _ {2} $, $ N _ {3} $.

Analogues of Frommer normal domains have been introduced for systems of the form

of order $ \geq 3 $.

References

[1] M. Frommer, "Die Integralkurven einer gewöhnlichen Differentialgleichung erster Ordnung in der Umgebung rationaler Unbestimtheitsstellen" Math. Ann. , 99 (1928) pp. 222–272
[2] V.V. Nemytskii, V.V. Stepanov, "Qualitative theory of differential equations" , Princeton Univ. Press (1960) (Translated from Russian)
[3] A.F. Andreev, "A uniqueness theorem for a normal region of Frommer's second type" Soviet Math. Dokl. , 3 : 1 (1962) pp. 132–135 Dokl. Akad. Nauk SSSR , 142 : 4 (1962) pp. 754–757
[4] A.F. Andreev, "Strengthening of the uniqueness theorem for an -curve in " Soviet Math. Dokl. , 3 : 5 (1962) pp. 1215–1216 Dokl. Akad. Nauk SSSR , 146 : 1 (1962) pp. 9–10
How to Cite This Entry:
Sector in the theory of ordinary differential equations. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Sector_in_the_theory_of_ordinary_differential_equations&oldid=48642
This article was adapted from an original article by A.F. Andreev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article