Sector in the theory of ordinary differential equations

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

(*)

, where is the domain of uniqueness, that satisfies the following four conditions: 1) each lateral boundary of is a -curve of the system (*) (i.e. a semi-trajectory that approaches as , and touches a certain direction at ); 2) the outer boundary of is a simple parametric arc (the homeomorphic image of a closed interval); 3) does not contain singular points of (*). The fourth condition is one of the following three: 4a) all trajectories of the system (*) that start in leave this sector for both increasing and decreasing ; such a sector is called a hyperbolic sector, or a saddle sector (Fig. a); 4b) all trajectories of (*) that start in sufficiently near do not leave but approach as increases, and as decreases they leave (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 sufficiently near do not leave as increases or decreases but approach , forming together with 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 -curves, a disc of sufficiently small radius and centre at can always be divided into a finite number of sectors of a specific form: hyperbolic, parabolic and 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 along the boundary of (and thereby to show the topological structure of the arrangement of the trajectories of (*) in a neighbourhood of ). There are a priori estimates from above for , and in terms of the order of smallness of the norm as (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 of the singular point of (*) is expressed by Bendixson's formula

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

with vertex at an isolated point () of the system

(see 1)) with lateral boundary and , , , and with the rear boundary satisfying the following conditions (here and are polar coordinates in the -plane with pole at , and ):

A) is an exceptional direction of the system

at , that is, there is a sequence , , as , such that if is the angle between the directions of the vectors and , then as , and this direction is unique in ;

B) for any ;

C) for any .

Suppose that the angle is measured from the vector and has the sign of the reference direction. A sector is called a Frommer normal domain of the first type (notation: ) if for and for ; a normal domain of the second type (notation: ) if on and on ; and a normal domain of the third type if has one and the same sign on and on . These domains were introduced by M. Frommer [1].

The trajectories of the system

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

Figure: s083770d

Figure: s083770e

Figure: s083770f

Figure: s083770g

Figure: s083770h

In a normal domain of any type the -curves tend to along the direction as (or ), and with decreasing (increasing) they leave the domain ; all other trajectories leave for both increasing and decreasing . The problems of distinguishing between the cases a) and b) for domains and are called, respectively, the first and second distinction problems of Frommer.

If a system

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

where and are forms of degree in the components of the vector ,

and when the following conditions are fulfilled: The form has real linear factors, the forms and do not have common real linear factors, and . Here situation a) holds in each of the domains , .

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

of order .

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