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A method for investigating the singular points of an [[Autonomous system|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|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|Centre of a topological dynamical system]]), a [[Focus|focus]] or a [[Centro-focus|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|Centre and focus problem]]).

An account of the method was given by M. Frommer [[#References|[1]]]. It can also be adapted for investigating singular points of third-order systems.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M. Frommer,   "Die Integralkurven einer gewöhnlichen Differentialgleichung erster Ordnung in der Umgebung rationaler Unbestimmtheitsstellen"  ''Math. Ann.'' , '''99'''  (1928)  pp. 222–272</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.F. Andreev,   "Singular points of differential equations" , Minsk  (1979)  (In Russian)</TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P. Hartman,   "Ordinary differential equations" , Birkhäuser  (1982)  pp. 220–227</TD></TR></table>