View source for Sector in the theory of ordinary differential equations

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

<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>

<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).

<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 <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]]]).

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

<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>

====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

<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>

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

(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" />):

A) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083770/s08377059.png" /> 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" />;

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" />;

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" />.

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]]].

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).

<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 <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.

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

<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>

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" />,

<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>

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" />.

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" />.

====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>