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 ![]() ![]() |
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