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).
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  and ). 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 , , ).
Sometimes (see, for example, ) 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
|||I. Bendixson, "Sur des courbes définiés par des équations différentielles" Acta Math. , 24 (1901) pp. 1–88|
|||A.A. Andronov, E.A. Leontovich, I.I. Gordon, A.G. Maier, "Qualitative theory of second-order dynamic systems" , Wiley (1973) (Translated from Russian)|
|||P. Hartman, "Ordinary differential equations" , Birkhäuser (1982)|
|||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|
|||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
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 .
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).
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 .
|||M. Frommer, "Die Integralkurven einer gewöhnlichen Differentialgleichung erster Ordnung in der Umgebung rationaler Unbestimtheitsstellen" Math. Ann. , 99 (1928) pp. 222–272|
|||V.V. Nemytskii, V.V. Stepanov, "Qualitative theory of differential equations" , Princeton Univ. Press (1960) (Translated from Russian)|
|||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|
|||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|
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=14776