Martinet - Ramis modulus for a quadratic system Текст научной статьи по специальности «Математика»

УДК 517.925

MARTINET — RAMIS MODULUS FOR A QUADRATIC SYSTEM

M. M. Turov

Chelyabinsk State University, Chelyabinsk, Russia [email protected]

The polynomial differential equations of order 2 are considered. So we used correspondence map and normalizing transformation to get coefficients of Martinet — Ramis modulus.

Keywords: Martinet — Ramis modulus, saddlenode, central manifold.

Introduction

Consider a quadratic system of ordinary differential equations

f p = P(1 — v)

[ v = v(p — v) ()

It is a limit system for well known Jouanolou system [1] in a certain sense. System (1) has a saddlenode singularity at the origin.

Martinet — Ramis modulus (C, ф) (for saddlenode singular point) [2] are constructed by transformations reducing initial system to its (orbital) formal normal form. Solutions of system (1) with given initial conditions can be found as a series (with respect to initial condition). Using these solutions it is possible to find coefficients of the normalizing transformations, and then to determinate coefficients of the modulus.

As a result we get

Theorem 1. Let (С,ф) be Martinet-Ramis modulus for system (1). Then C = 0, <f(z) = z + 2 niz2 + (2 ni — 4n2 )z3 + ...

Corollary 1. System (1) is not analytically orbital equivalent to its formal normal form.

1. Jouanolou’s systems

In the paper [1] there are considered a system of differential equations in CP2 (in homogeneous coordinates)

Г a = bn

I b = cn (2)

I c = an

As it was shown [1], these systems (for n > 2) have not algebraic solutions (and it was first example of a system with such properties).

In affine coordinates x = a/b, y = b/c system (2) turn to

dy 1 — xny

dx yn — xn+1.

Note that the rational transformation

P

1

y ■ xn ’

v

brings the last system to the quadratic form

y

n

xn+i

dv p[( 1 — v) + e(1 — p)]

dp v[(p — v) + e(1 — v)] ’

(3)

where e = 1 /n. The limit transition (e ^ 0) leads (3) to system (1). It is important to note that system (1) could be obtained from system (2) (for n = —1).

2. Construction of Martinet — Ramis modulus.

Formal normal form and its first integral

System (1) has a saddlenode singularity at the origin. Analytic classification of such singular points was given in [2]; it uses functional modulus. This functional modulus can be constructed by the next scheme.

Let V be a vector field with saddlenode singularity at the origin and V0 be its (orbital) formal normal form. Let {v = 0} be its separatrix, corresponding to the hyperbolic eigenvalue of the field linearization at the origin. Let V be holomorphic in the polydisk

О

U = {|v| < e, |p| < e} and domain U be obtained from the polydisk U by the removing of the separatrix {v = 0}.

О

As it was shown in [2], for sufficiently small e, the «cut» polydisk U can be covered by a pair of domains U±, such that there exists a normalizing transformations H± : U± ^ C2 (H± transform phase curves of V into phase curves of V0). The intersection U+ П U- consists of two connected components: U+ П U- = W+ U W_.

The transformation Ф = H+ о (H-)-1|U+ rU_ brings phase curves of formal normal form Vo into itself. Values of the first integral J (standard for this formal normal form) numerate phase curves of the formal normal form. Therefore, the maps Ф± = $|W± generate a pair of holomorphic functions ф± defined on domains w± = J(H_(W±)) such that ф± о J = J о Ф±.

As it was shown in [2] ф+( J) = J + C, and ф_(J) is a holomorphic function in (C, 0). The pair (C, ф_) (defined up to a linear transformation) is just the functional modulus of Martinet — Ramis.

3. Variations of solutions of system (1) with respect to the initial conditions

To implement the program of construction of the Martinet — Ramis modulus, here we find a solution of system (1) (as a series with respect to the initial condition). Namely, we will search a solution p = P(v,p0, v0) of the differential equation

dp p(1 — v)

dv v(p — v)

with the initial condition

p(v o) = po.

Let

p = ^ (v)pk.

k=o

(4)

(5)

Substituting (6) in (4) we get a system of differential equations

(—v + fo)f'o = L—V fo,

—vfi = ^ fi,

-Vf2 + fifi = f2,

-vf3 + fif + f2 f i = ^ f3. From (5) we find initial conditions for the functions fk

fi(vo) = l, ffc(vo) = 0, k = 1.

Solving (7) and using (8), we get the functions

(7)

(8)

fo = 0,

f3

v 1

—5-ev

2v3

_3_

v0

fi = — ev vo, vo

f

2 2e

V 1_^_

v vo

vo

1

e s i m

—as + ev — evo S

V

(9)

v e i

Vo

vdS

2

4e vo

V

vo

1

es , i —dS + 4e v S

V

vo

1

e s 2

—dS + 3e v + 2 S

V

vo

1 e s

^ dS +

S

2 1 + X"

+ e vo — 4ev+vo

Remark 1. In all formulas the integration is done along some curve connecting points v0 and v. The choice of these curves will be specified in section 5.

4. Correspondence map

Let us consider a transversal TVo = {v = v0}; as parameter on TVo we will use p-variable. We define a correspondence map Л^ V1 : TVo M TV1 by the following way. Construct a solution p = P(v,p0, v0) of (4) with the initial condition p(v0) = p0. These solution is holomorphic in a neighbourhood of v0. Let 7 be a curve on the central manifold {p = 0, v = 0}, connecting v0 and vi. Let us continue the function P(v,p0, v0) analytically along the curve 7. And let pi be the result of this continuation, pi = Л7P(vi,p0,v0). Then, by definition, Л^(Ы = pi.

Example of a correspondence map for an orbital formal normal form.

Orbital formal normal form for (1) is

p

v -

p,

__VL

i—v

(10)

Note that the function J = pv ie v is constant on solutions of (10). Thus, correspondence map f : p0 M pi for (11) is defined from the equation

J (v0,p0) = J (vi, pi).

Solving (11), we get

vi _1_L

pi = ЛY0 (p0) = p0—ev1 vo. Y v0

Note that this map p0 M pi does not depend on 7.

5. Normalizing transformation

Figure 2 illustrates this construction of normalizing transformation. A phase curve of the field V, going through the point A of the transversal Tvi = {v = v\}, intersects the transversal Tv0 = {v = vo} at a point A0. A phase curve of field V0, going through the point A0 of the transversal Tv0 = {v = v0}, intersects the transversal Tvi = {v = vi} in a point A. Then, by definition, H(A) = A.

Here, the point vo is selected on the real line {Im(v) = 0, Re(v) > 0,p = 0}. For v\, such that Re(vi) < 0, the curve 7, that used in the construction of H-(v,p) (H+(v,p)), have to go around above (below) of the deleted point v = 0 on the central manifold. Finally, for H± we obtain H± : (v,p) M (v,p±), where

p±

v 1 —eJ vo

_1_

-o P±(vo,p,v),

and P± are the functions, constructed in section 3, where the paths of integration in (9) are defined as above, with the correspondence of the sign.

6. Formulas for Ф. Calculations of ф-component

In the notation of section 6 for Ф we obtain formulas

where

and

From section 2

$(v,po) = (v,p),

v i__L

p=— eJ voP+(vo,po,v), vo

~ r> ( vo —— 1 ,vo\

po = P- (v,po—e-o J J .

ф-J ) = J O T(vo,po) = pvo 1e vo =

-1 - v 1 - -in / D / vo J, - 1

= vo e Jo— ev vo P+ [vo,P- (v,po—e Jo j ,vo),v o vo v

Substituting (9) into (12)-(14), from (15) we get

ф^) = J + 2niJ2 + (2ni — 4n2) J3 + ...

(12)

(13)

(14)

(15)

Note that the transition functions Ф constructed here do not coincide with ones from section 2, but are conjugated with them. It means that the first coefficients of ф are found correctly. The proof of the theorem is finished.

The corollary is a sequence of the nontriviality of ф-component of the modulus [2].

References

1. Jouanolou J.P. Equations de Pfaff Algebriques. Lecture Notes in Mathematics, 1979, vol. 708, pp. 157-198.

2. Martinet J., Ramis J.-P. Classification analytique des equations differentielles non lineaires resonnantes du premier ordre. Annales scientifiques de I’Ecole normale superieure, 1983, vol. 16, pp. 571-621.

Article received 18.09.2016 Corrections received 28.09.2016

Chelyabinsk Physical and Mathematical Journal. 2016. Vol. 1, iss. 3. P. 86-91.

МОДУЛИ МАРТИНЕ — РАМИСА

ДЛЯ ОДНОЙ КВАДРАТИЧНОЙ СИСТЕМЫ

М. М. Туров

Челябинский государственный университет,, Челябинск, Россия

[email protected]

В работе рассматривается полиномиальное дифференциальное уравнение степени 2.

С помощью отображения соответствия и нормализующего преобразования получены коэффициенты модулей Мартине — Рамиса.

Ключевые слова: модуль Мартине —Рамиса, седлоузел, центральное многообразие.

Поступила в редакцию 18.09.2016 После переработки 28.09.2016

Сведения об авторе

Туров Михаил Михайлович, студент математического факультета, Челябинский государственный университет, Челябинск, Россия; [email protected].