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