Frommer method
A method for investigating the singular points of an autonomous system of second-order ordinary differential equations
![]() | (1) |
where is an analytic or a sufficiently smooth function in the domain
.
Suppose that is a singular point of the system (1), that is,
, and that
and
are analytic functions at
with no common analytic factor that vanishes at
. The Frommer method enables one to find explicitly all
-curves of (1) — the semi-trajectories of the system joined to
along a definite direction. Every
-curve of (1) not lying on the axis
is an
-curve of the equation
![]() | (2) |
(that is, can be represented near in the form
![]() | (3) |
where is a solution of (2),
or
,
,
or
for every
), and conversely.
Consider equation (2) first in the domain . If it is a simple Bendixson equation, that is, if it satisfies the conditions
![]() |
then it has a unique -curve in the domain
for
; the domain
,
, where
is a sufficiently small positive number, is a parabolic sector for
(cf. Sector in the theory of ordinary differential equations). Otherwise, to exhibit the
-curves of (2) in the domain
one applies the Frommer method. The basis for applying it is the fact that every
-curve (3) of equation (2),
, has a completely determined asymptotic behaviour at
, namely, it can be represented in the form
![]() |
and admits a finite or infinite limit
![]() |
which is called its order of curvature at , and for
it also admits a finite or infinite limit
![]() |
which is called its measure of curvature at . Here the
-curve
,
, is assigned the order of curvature
.
The first step in the Frommer method consists in the following. One uses algebraic means to calculate all possible orders of curvature (there is always a finite number of them), and for each order
all possible measures of curvature for
-curves of (2). On the basis of the general theorems of the method, one can elucidate the question of whether (2) has
-curves with given possible order and measure of curvature, except for a finite number
of so-called characteristic pairs
. For each of these
, where
and
are natural numbers, and
. Therefore the substitution
,
transforms (2) into a derived equation
of the same form, turning the question of whether (2) has
-curves with order of curvature
and measure of curvature
into the question of whether
has
-curves in the domain
.
If (2) has no characteristic pairs or if each of its derived equations turns out to be a simple Bendixson equation, then all -curves of (2) in the domain
have been exhibited in the first step of the process. Otherwise one performs the second step — one studies, according to the plan of the first step, the derived equations that are not simple Bendixson equations. In doing this one arrives at derived equations of a second series, etc. At each stage the process, generally speaking, branches, but for a fixed equation (2) the number of branches of the process is finite and every branch terminates in a reduced equation which is either a simple Bendixson equation or has no characteristic pairs.
Thus, by means of a finite number of steps of the Frommer method one can exhibit all -curves of (1) in the domain
, along with their asymptotic behaviour at
. Changing
to
in (1) enables one to do the same for the domain
, and a direct verification enables one to establish whether the semi-axes of the axis
are
-curves. The behaviour of all trajectories of (1) in a neighbourhood of
can be determined on the basis of this information as follows.
If the system (1) has no -curves, then
is a centre (cf. Centre of a topological dynamical system), a focus or a centro-focus for it. If the set
of all
-curves of (1) is non-empty, then the information about its asymptotic behaviour at
obtained by the Frommer method enables one to split
into a finite number of non-intersecting bundles of
-curves:
,
, each of which is either open: it consists of semi-trajectories of one type (positive or negative) that fill a domain, or "closed" : it consists of a single
-curve. The representatives
of these bundles have different asymptotic behaviour at
, which enables one to establish a cyclic sequential order for the bundles as one goes round
along a circle
of small radius
, and they divide the disc bounded by
into
sectors
.
Suppose that the sector ,
, has as its lateral edges the
-curves
and
, where
is the same as
. Then
is: a) elliptic, b) hyperbolic or c) parabolic, according to whether the bundles
and
are respectively a) both open, b) both "closed" or c) of different types.
Thus, the Frommer method enables one, in a finite number of steps, either to find, for the system (1), a cyclic sequence of hyperbolic, parabolic and elliptic sectors joined to the point , and thereby completely to determine the topological type of the distribution of its trajectories in a neighbourhood of
, or to show that the problem of distinguishing between centre, focus and centro-focus arises at
(cf. Centre and focus problem).
An account of the method was given by M. Frommer [1]. It can also be adapted for investigating singular points of third-order systems.
References
[1] | M. Frommer, "Die Integralkurven einer gewöhnlichen Differentialgleichung erster Ordnung in der Umgebung rationaler Unbestimmtheitsstellen" Math. Ann. , 99 (1928) pp. 222–272 |
[2] | A.F. Andreev, "Singular points of differential equations" , Minsk (1979) (In Russian) |
Comments
References
[a1] | P. Hartman, "Ordinary differential equations" , Birkhäuser (1982) pp. 220–227 |
Frommer method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Frommer_method&oldid=46995