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)

References

[a1]	P. Hartman, "Ordinary differential equations" , Birkhäuser (1982) pp. 220–227