View source for Sector in the theory of ordinary differential equations

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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" />

Figure: s083770a

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/s083770b.gif" />

Figure: s083770b

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/s083770c.gif" />

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 [[#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  $  i $
of the [[Singular point|singular point]]  $  O $
of (*) is expressed by Bendixson's formula

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

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I. Bendixson,   "Sur des courbes définiés par des équations différentielles"  ''Acta Math.'' , '''24'''  (1901)  pp. 1–88</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.A. Andronov,   E.A. Leontovich,   I.I. Gordon,   A.G. Maier,   "Qualitative theory of second-order dynamic systems" , Wiley  (1973)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  P. Hartman,   "Ordinary differential equations" , Birkhäuser  (1982)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  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</TD></TR></table>

====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 [[#References|[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|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" />

Figure: s083770d

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/s083770e.gif" />

Figure: s083770e

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/s083770f.gif" />

Figure: s083770f

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/s083770g.gif" />

Figure: s083770g

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/s083770h.gif" />

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====
<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>