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are analytic functions at 
 
are analytic functions at    O
 
with no common analytic factor that vanishes at    O .  
 
with no common analytic factor that vanishes at    O .  
The Frommer method enables one to find explicitly all    TO -
+
The Frommer method enables one to find explicitly all    TO -curves of (1) — the semi-trajectories of the system joined to    O
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
along a definite direction. Every    TO -
+
is an    O -curve of the equation
curve of (1) not lying on the axis    x = 0
 
is an    O -
 
curve of the equation
 
  
 
$$ \tag{2 }
 
$$ \tag{2 }
Line 69: Line 66:
 
$$
 
$$
  
then it has a unique    O -
+
then it has a unique    O -curve in the domain    x > 0
curve in the domain    x > 0
 
 
for    a < 0 ;  
 
for    a < 0 ;  
 
the domain    x > 0 ,  
 
the domain    x > 0 ,  
 
  x  ^ {2} + y  ^ {2} < r  ^ {2} ,  
 
  x  ^ {2} + y  ^ {2} < r  ^ {2} ,  
 
where    r
 
where    r
is a sufficiently small positive number, is a parabolic sector for    a > 0 (
+
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
cf. [[Sector in the theory of ordinary differential equations|Sector in the theory of ordinary differential equations]]). Otherwise, to exhibit the    O -
+
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 ,  
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 ,  
 
has a completely determined asymptotic behaviour at    O ,  
 
namely, it can be represented in the form
 
namely, it can be represented in the form
Line 110: Line 103:
  
 
which is called its measure of curvature at    O .  
 
which is called its measure of curvature at    O .  
Here the    O -
+
Here the    O -curve    y = 0 ,  
curve    y = 0 ,  
 
 
  x \in ( 0, \delta ) ,  
 
  x \in ( 0, \delta ) ,  
 
is assigned the order of curvature    \nu = + \infty .
 
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 (
+
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 )
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)
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 ) .  
 
of so-called characteristic pairs    ( \nu , \gamma ) .  
 
For each of these    \nu = r/s ,  
 
For each of these    \nu = r/s ,  
Line 128: Line 117:
 
  y = ( \gamma + y _ {1} ) x _ {1}  ^ {r}
 
  y = ( \gamma + y _ {1} ) x _ {1}  ^ {r}
 
transforms (2) into a derived equation    ( 2 _ {1} )
 
transforms (2) into a derived equation    ( 2 _ {1} )
of the same form, turning the question of whether (2) has    O -
+
of the same form, turning the question of whether (2) has    O -curves with order of curvature    \nu
curves with order of curvature    \nu
 
 
and measure of curvature    \gamma
 
and measure of curvature    \gamma
 
into the question of whether    ( 2 _ {1} )
 
into the question of whether    ( 2 _ {1} )
has    O -
+
has    O -curves in the domain    x _ {1} > 0 .
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 -
+
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
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.
 
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 -
+
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 ,  
curves of (1) in the domain    x > 0 ,  
 
 
along with their asymptotic behaviour at    O .  
 
along with their asymptotic behaviour at    O .  
 
Changing    x
 
Changing    x
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in (1) enables one to do the same for the domain    x < 0 ,  
 
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
 
and a direct verification enables one to establish whether the semi-axes of the axis    x = 0
are    TO -
+
are    TO -curves. The behaviour of all trajectories of (1) in a neighbourhood of    O
curves. The behaviour of all trajectories of (1) in a neighbourhood of    O
 
 
can be determined on the basis of this information as follows.
 
can be determined on the basis of this information as follows.
  
If the system (1) has no    TO -
+
If the system (1) has no    TO -curves, then    O
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
 
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 -
+
of all    TO -curves of (1) is non-empty, then the information about its asymptotic behaviour at    O
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
 
obtained by the Frommer method enables one to split    H
into a finite number of non-intersecting bundles of    TO -
+
into a finite number of non-intersecting bundles of    TO -curves:    H _ {1} \dots H _ {k} ,  
curves:    H _ {1} \dots H _ {k} ,  
 
 
  k \geq  2 ,  
 
  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 -
+
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}
curve. The representatives    l _ {1} \dots l _ {k}
 
 
of these bundles have different asymptotic behaviour at    O ,  
 
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
 
which enables one to establish a cyclic sequential order for the bundles as one goes round    O
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Suppose that the sector    S _ {i} ,  
 
Suppose that the sector    S _ {i} ,  
 
  i \in \{ 1 \dots k \} ,  
 
  i \in \{ 1 \dots k \} ,  
has as its lateral edges the    TO -
+
has as its lateral edges the    TO -curves    l _ {i}
curves    l _ {i}
 
 
and    l _ {i + 1 }  ,  
 
and    l _ {i + 1 }  ,  
 
where    l _ {k + 1 } 
 
where    l _ {k + 1 } 
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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 ,  
 
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 ,  
 
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 (
+
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]]).
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.

Latest revision as of 01:42, 23 June 2022


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=52473
This article was adapted from an original article by A.F. Andreev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article