Difference between revisions of "Frommer method"
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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 } | ||
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$$ | $$ | ||
− | 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 | ||
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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 , | ||
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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 |
Frommer method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Frommer_method&oldid=52473