# 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 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

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