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A method for investigating the singular points of an [[Autonomous system|autonomous system]] of second-order ordinary differential equations
 
A method for investigating the singular points of an [[Autonomous system|autonomous system]] of second-order ordinary differential equations
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f0418101.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
\dot{p}  = f ( p),\ \
 +
p = ( x, y),\ \
 +
f = ( X, Y): G  \rightarrow  \mathbf R  ^ {2} ,
 +
$$
 +
 
 +
where  $  f $
 +
is an analytic or a sufficiently smooth function in the domain  $  G $.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f0418102.png" /> is an analytic or a sufficiently smooth function in the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f0418103.png" />.
+
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
  
Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f0418104.png" /> is a singular point of the system (1), that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f0418105.png" />, and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f0418106.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f0418107.png" /> are analytic functions at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f0418108.png" /> with no common analytic factor that vanishes at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f0418109.png" />. The Frommer method enables one to find explicitly all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181011.png" />-curves of (1) — the semi-trajectories of the system joined to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181012.png" /> along a definite direction. Every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181013.png" />-curve of (1) not lying on the axis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181014.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181016.png" />-curve of the equation
+
$$ \tag{2 }
 +
y  ^  \prime  =
 +
\frac{Y ( x, y) }{X ( x, y) }
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181017.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$
  
(that is, can be represented near <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181018.png" /> in the form
+
(that is, can be represented near $  O $
 +
in the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181019.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$ \tag{3 }
 +
= \phi ( x),\ \
 +
\phi ( x) \rightarrow  0 \ \
 +
\textrm{ as }  x \rightarrow 0,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181020.png" /> is a solution of (2), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181021.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181024.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181025.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181026.png" />), and conversely.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181027.png" />. If it is a simple Bendixson equation, that is, if it satisfies the conditions
+
Consider equation (2) first in the domain $  x > 0 $.  
 +
If it is a simple Bendixson equation, that is, if it satisfies the conditions
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181028.png" /></td> </tr></table>
+
$$
 +
X ( x, y)  \equiv  x  ^ {h} ,\ \
 +
h \geq  1,\ \
 +
Y _ {y}  ^  \prime  ( 0, 0)  = \
 +
a  \neq  0,
 +
$$
  
then it has a unique <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181029.png" />-curve in the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181030.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181031.png" />; the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181033.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181034.png" /> is a sufficiently small positive number, is a parabolic sector for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181035.png" /> (cf. [[Sector in the theory of ordinary differential equations|Sector in the theory of ordinary differential equations]]). Otherwise, to exhibit the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181036.png" />-curves of (2) in the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181037.png" /> one applies the Frommer method. The basis for applying it is the fact that every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181038.png" />-curve (3) of equation (2), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181039.png" />, has a completely determined asymptotic behaviour at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181040.png" />, namely, it can be represented in the form
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181041.png" /></td> </tr></table>
+
$$
 +
= x ^ {v ( x) } \
 +
\mathop{\rm sign}  \phi ( x),
 +
$$
  
 
and admits a finite or infinite limit
 
and admits a finite or infinite limit
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181042.png" /></td> </tr></table>
+
$$
 +
\nu  = \lim\limits _ {x \rightarrow 0 }  v ( x)  = \
 +
\lim\limits _ {x \rightarrow 0 } \
  
which is called its order of curvature at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181044.png" />, and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181045.png" /> it also admits a finite or infinite limit
+
\frac{ \mathop{\rm ln}  | \phi ( x) | }{ \mathop{\rm ln}  x }
 +
  \in \
 +
[ 0, + \infty ],
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181046.png" /></td> </tr></table>
+
which is called its order of curvature at  $  O $,
 +
and for  $  \nu \in ( 0, + \infty ) $
 +
it also admits a finite or infinite limit
  
which is called its measure of curvature at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181048.png" />. Here the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181049.png" />-curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181050.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181051.png" />, is assigned the order of curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181052.png" />.
+
$$
 +
\gamma  = \lim\limits _ {x \rightarrow 0 } \
 +
\phi ( x) x ^ {- \nu }  \in \
 +
[- \infty , + \infty ],
 +
$$
  
The first step in the Frommer method consists in the following. One uses algebraic means to calculate all possible orders of curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181053.png" /> (there is always a finite number of them), and for each order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181054.png" /> all possible measures of curvature for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181055.png" />-curves of (2). On the basis of the general theorems of the method, one can elucidate the question of whether (2) has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181056.png" />-curves with given possible order and measure of curvature, except for a finite number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181057.png" /> of so-called characteristic pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181059.png" />. For each of these <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181060.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181061.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181062.png" /> are natural numbers, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181063.png" />. Therefore the substitution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181064.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181065.png" /> transforms (2) into a derived equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181066.png" /> of the same form, turning the question of whether (2) has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181067.png" />-curves with order of curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181068.png" /> and measure of curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181069.png" /> into the question of whether <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181070.png" /> has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181071.png" />-curves in the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181072.png" />.
+
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 $.
  
If (2) has no characteristic pairs or if each of its derived equations turns out to be a simple Bendixson equation, then all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181073.png" />-curves of (2) in the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181074.png" /> 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.
+
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 $.
  
Thus, by means of a finite number of steps of the Frommer method one can exhibit all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181075.png" />-curves of (1) in the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181076.png" />, along with their asymptotic behaviour at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181077.png" />. Changing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181078.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181079.png" /> in (1) enables one to do the same for the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181080.png" />, and a direct verification enables one to establish whether the semi-axes of the axis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181081.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181082.png" />-curves. The behaviour of all trajectories of (1) in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181083.png" /> can be determined on the basis of this information as follows.
+
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.
  
If the system (1) has no <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181084.png" />-curves, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181085.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181086.png" /> of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181087.png" />-curves of (1) is non-empty, then the information about its asymptotic behaviour at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181088.png" /> obtained by the Frommer method enables one to split <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181089.png" /> into a finite number of non-intersecting bundles of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181090.png" />-curves: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181091.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181092.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181093.png" />-curve. The representatives <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181094.png" /> of these bundles have different asymptotic behaviour at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181095.png" />, which enables one to establish a cyclic sequential order for the bundles as one goes round <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181096.png" /> along a circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181097.png" /> of small radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181098.png" />, and they divide the disc bounded by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f04181099.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f041810100.png" /> sectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f041810101.png" />.
+
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.
  
Suppose that the sector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f041810102.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f041810103.png" />, has as its lateral edges the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f041810104.png" />-curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f041810105.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f041810106.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f041810107.png" /> is the same as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f041810108.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f041810109.png" /> is: a) elliptic, b) hyperbolic or c) parabolic, according to whether the bundles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f041810110.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f041810111.png" /> are respectively a) both open, b) both "closed"  or c) of different types.
+
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} $.
  
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f041810112.png" />, and thereby completely to determine the topological type of the distribution of its trajectories in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f041810113.png" />, or to show that the problem of distinguishing between centre, focus and centro-focus arises at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041810/f041810114.png" /> (cf. [[Centre and focus problem|Centre and focus problem]]).
+
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.
 
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.
Line 49: Line 190:
 
====References====
 
====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>
 
<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====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P. Hartman,  "Ordinary differential equations" , Birkhäuser  (1982)  pp. 220–227</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P. Hartman,  "Ordinary differential equations" , Birkhäuser  (1982)  pp. 220–227</TD></TR></table>

Revision as of 19:40, 5 June 2020


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)

Comments

References

[a1] P. Hartman, "Ordinary differential equations" , Birkhäuser (1982) pp. 220–227
How to Cite This Entry:
Frommer method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Frommer_method&oldid=14740
This article was adapted from an original article by A.F. Andreev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article