Frommer method

A method for investigating the singular points of an autonomous system of second-order ordinary differential equations

$$\tag{1 } \dot{p} = f ( p),\ \ p = ( x, y),\ \ f = ( X, Y): G \rightarrow \mathbf R ^ {2} ,$$

where $f$ is an analytic or a sufficiently smooth function in the domain $G$.

Suppose that $O = ( 0, 0)$ is a singular point of the system (1), that is, $f ( O) = 0$, and that $X$ and $Y$ are analytic functions at $O$ with no common analytic factor that vanishes at $O$. The Frommer method enables one to find explicitly all $TO$- curves of (1) — the semi-trajectories of the system joined to $O$ along a definite direction. Every $TO$- curve of (1) not lying on the axis $x = 0$ is an $O$- curve of the equation

$$\tag{2 } y ^ \prime = \frac{Y ( x, y) }{X ( x, y) }$$

(that is, can be represented near $O$ in the form

$$\tag{3 } y = \phi ( x),\ \ \phi ( x) \rightarrow 0 \ \ \textrm{ as } x \rightarrow 0,$$

where $\phi : I \rightarrow \mathbf R$ is a solution of (2), $I = ( 0, \delta )$ or $(- \delta , 0)$, $\delta > 0$, $\phi ( x) \equiv 0$ or $\phi ( x) \neq 0$ for every $x \in I$), and conversely.

Consider equation (2) first in the domain $x > 0$. If it is a simple Bendixson equation, that is, if it satisfies the conditions

$$X ( x, y) \equiv x ^ {h} ,\ \ h \geq 1,\ \ Y _ {y} ^ \prime ( 0, 0) = \ a \neq 0,$$

then it has a unique $O$- curve in the domain $x > 0$ for $a < 0$; the domain $x > 0$, $x ^ {2} + y ^ {2} < r ^ {2}$, where $r$ is a sufficiently small positive number, is a parabolic sector for $a > 0$( cf. Sector in the theory of ordinary differential equations). Otherwise, to exhibit the $O$- curves of (2) in the domain $x > 0$ one applies the Frommer method. The basis for applying it is the fact that every $O$- curve (3) of equation (2), $\phi ( x) \not\equiv 0$, has a completely determined asymptotic behaviour at $O$, namely, it can be represented in the form

$$y = x ^ {v ( x) } \ \mathop{\rm sign} \phi ( x),$$

and admits a finite or infinite limit

$$\nu = \lim\limits _ {x \rightarrow 0 } v ( x) = \ \lim\limits _ {x \rightarrow 0 } \ \frac{ \mathop{\rm ln} | \phi ( x) | }{ \mathop{\rm ln} x } \in \ [ 0, + \infty ],$$

which is called its order of curvature at $O$, and for $\nu \in ( 0, + \infty )$ it also admits a finite or infinite limit

$$\gamma = \lim\limits _ {x \rightarrow 0 } \ \phi ( x) x ^ {- \nu } \in \ [- \infty , + \infty ],$$

which is called its measure of curvature at $O$. Here the $O$- curve $y = 0$, $x \in ( 0, \delta )$, is assigned the order of curvature $\nu = + \infty$.

The first step in the Frommer method consists in the following. One uses algebraic means to calculate all possible orders of curvature $\nu$( there is always a finite number of them), and for each order $\nu \in ( 0, + \infty )$ all possible measures of curvature for $O$- curves of (2). On the basis of the general theorems of the method, one can elucidate the question of whether (2) has $O$- curves with given possible order and measure of curvature, except for a finite number $( \geq 0)$ of so-called characteristic pairs $( \nu , \gamma )$. For each of these $\nu = r/s$, where $r$ and $s$ are natural numbers, and $0 < | \gamma | < + \infty$. Therefore the substitution $x = x _ {1} ^ {s}$, $y = ( \gamma + y _ {1} ) x _ {1} ^ {r}$ transforms (2) into a derived equation $( 2 _ {1} )$ of the same form, turning the question of whether (2) has $O$- curves with order of curvature $\nu$ and measure of curvature $\gamma$ into the question of whether $( 2 _ {1} )$ has $O$- curves in the domain $x _ {1} > 0$.

If (2) has no characteristic pairs or if each of its derived equations turns out to be a simple Bendixson equation, then all $O$- curves of (2) in the domain $x > 0$ 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 $TO$- curves of (1) in the domain $x > 0$, along with their asymptotic behaviour at $O$. Changing $x$ to $- x$ in (1) enables one to do the same for the domain $x < 0$, and a direct verification enables one to establish whether the semi-axes of the axis $x = 0$ are $TO$- curves. The behaviour of all trajectories of (1) in a neighbourhood of $O$ can be determined on the basis of this information as follows.

If the system (1) has no $TO$- curves, then $O$ is a centre (cf. Centre of a topological dynamical system), a focus or a centro-focus for it. If the set $H$ of all $TO$- curves of (1) is non-empty, then the information about its asymptotic behaviour at $O$ obtained by the Frommer method enables one to split $H$ into a finite number of non-intersecting bundles of $TO$- curves: $H _ {1} \dots H _ {k}$, $k \geq 2$, 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 $TO$- curve. The representatives $l _ {1} \dots l _ {k}$ of these bundles have different asymptotic behaviour at $O$, which enables one to establish a cyclic sequential order for the bundles as one goes round $O$ along a circle $C$ of small radius $r$, and they divide the disc bounded by $C$ into $k$ sectors $S _ {1} \dots S _ {k}$.

Suppose that the sector $S _ {i}$, $i \in \{ 1 \dots k \}$, has as its lateral edges the $TO$- curves $l _ {i}$ and $l _ {i + 1 }$, where $l _ {k + 1 }$ is the same as $l _ {1}$. Then $S _ {i}$ is: a) elliptic, b) hyperbolic or c) parabolic, according to whether the bundles $H _ {i}$ and $H _ {i + 1 }$ 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 $O$, and thereby completely to determine the topological type of the distribution of its trajectories in a neighbourhood of $O$, or to show that the problem of distinguishing between centre, focus and centro-focus arises at $O$( 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