Normal forms, computer algebra and a problem of integrability of nonlinear ODEs

Normal forms, computer algebra and a problem of integrability of

nonlinear ODEs

Victor Edneral

Moscow State University

Russia

Joint work with Alexander Bruno

CADE-2007, February 20-23, Turku, Finland2

• Introduction• Normal Form of a Nonlinear System• Euler – Poisson Equations• Normal Form of the Euler – Poisson Equations in

Resonance• Structure of Integrals of the System• Necessary Conditions for Existence of Additional

Integrals• Calculation of the Normal Form of the Euler – Poisson

Equations• Conclusions


IntroductionHere, we study the connection between coefficients of normal forms and integrability of the system. For this purpose, we compute normal forms of the Euler – Poisson equations, which describe the motion of a rigid body with a fixed point. This is an autonomous sixth-order system.

A lot of books and papers are devoted to integrable systems and to methods for searching for such systems. A.D. Bruno noted that all normal forms of integrable systems are degenerated, so it is interesting to search domains with a such degeneration [Bruno, 2005].

The first attempt to calculate the normal form of the Euler – Poisson system was made in [Starzhinsky, 1977]. However, without computer algebra tools, he was unable to calculate a sufficient number of terms. We use a program for analytical computation of the normal form [V. Edneral, R. Khanin, 2003]. This is a modification of the LISP based package NORT for the MATHEMATICA system. The NORT package [Edneral, 1998] has been designed for the REDUCE system.


Normal Form of a Nonlinear System

Consider the system of order n

in a neighborhood of the stationary point X = 0 under the assumption that the vector function Φ(X) is analytical at the point X = 0 and its Taylor expansion contains no constant and linear terms.

(1.1)







Euler – Poisson Equations





Has the system additional local integrals?


Local Integrals[Lunkevich, Sibirskii, 1982].

Let us see a system

It has two stationary points

At S1 it has a center and at S2 – a focus.

It is integrable at S1 with the integral

You can see that

The system is integrable in semi plane x > -1/2 and not integrable at x < -1/2.

is an invariant line.


[Lunkevich, Sibirskii, 1982].






D2 D4

D1

D5 D3

C2 R +

R –

C3C1

(x0,y0)

Fig. 1








Normal Form of the Euler – Poisson Equations in Resonance

Let the normal form be

and vector of eigenvalues of the matrix Λ be

Let also introduce so called resonance variables. After z1 and z2 we have

(6.1)


Lemm 1 [Bruno, 2005].At the resonance in the normal form

where

are power series in

At this start from free terms but

– from linear terms.


In resonance variables we will have the system in the form

for odd values of

and

for even values of

.

G0, Hi,k and Fi,k above are linear combination of gr,m, hr.m and fr.m.

(6.2)


Structure of Integrals of the SystemAs it was shown in [Bruno, 1995] an expansion of the first integral of normal form

contains only resonance variables with the property

Thus the first integral can be written as power series

where a0, am and bm are power series in z1, z2, ρ1, ρ2.


For the resonance variables, the automorphism

if

can be rewritten as

even,odd.


If is odd then the integral A will be

Then we have am = bm and the integral is

(7.1)


Necessary Conditions for Existence of Additional Integrals

From the definition, the derivation in time of any first integralalong the system should be zero, i.e.


The identity above should be discussed at odd and even values of

separately. Corresponding coefficients in formulae below will be slight different. The lowest non vanished coefficients will be different also. It is very important for an estimation of order up to which we a need to calculate the normal form.

If we parameterize the identity above up to the common order in z1,2, ρ1,2 variables smaller than 2( ) for the odd value and up to for the even one, we will see that the identity should be right for free and linear in the common order

. (7.2)


So, if we write

o(z1,2,ρ1,2),

O(z1,2,ρ1,2).

(7.3)

o(z1,2,ρ1,2),

+ o(z1,2,ρ1,2),


then the equation for the free term has the form

Here the vector Ξ ≡ {ξi} can be calculated from the normal form. It will be a function of parameters of the system. The vector α ≡ {αi} defines a0.

If you know the first integrals of the system, you can calculate the corresponding α ≡ {αi} for each integral separately.

(A)

If Ξ ≠ 0 then equation (A) has three dimensional set of solutionsα, so only three integrals can be independent.


A single possibility to have an additional integral in this case is that the tree known integrals are dependent each from other. Mathematically it can be written as the vector equality

0

lkji

detV

)3(

4

)3(

3

)3(

2

)3(

1

)2(

4

)2(

3

)2(

2

)2(

1

)1(

4

)1(

3

)1(

2

)1(

1

aaaaaaaaaaaa

Because for checking this condition we need calculate only the lowest orders of the normal form it is possible to calculate it in analytical form in variables of the system.

a)1(

1


If Ξ ≡ 0 then from (5.2), (5.3) we will have the condition

(B)

The dimension of solutions (α, β) of the system above is

rank(M),– 5

where M is a matrix (4 x 5) which consists from vectors

η and ζ are coefficients which can be calculated fromcoefficients of the normal form as functions of parameters.


Let us say that formal integral (7.1) is local independent on known integrals if its linear approximation in z, ρ, ω is linear independent from first approximations of known integrals.

Main theorem [Bruno, 2005]. For existence of an additional formal integral at the family of the stationary point Fδ, it is necessary a satisfaction of one of two sets of conditions:

1. Ξ ≠ 0 and V = 0,2. Ξ = 0 and rank(M) < 2.

Note, that if the original system has five first integrals, thenright hand side of normalized equation is linear, and rank(M) = 0.


Calculation of the Normal Form of the Euler – Poisson Equations

Near stationary points of families Sσ we computed normal forms of the System up to terms of some order m

For that, we used the program [Edneral, Khanin, 2003]. All calculations were lead in rational arithmetic and float point numbers in this paper are approximations of exact results.


Case of Resonance (1:2)

We will use the uniformization

then domain corresponds the interval

and at δ = 1 we have


Components of external products will be

The system

will have only two solutions

So at δ = 1 in the interval above


At δ = -1

Solutions


So only solution h4 lies in the mechanical semi-interval

But h4 is a special point with all zero eigenvalues and we can conclude that

With respect of the Main theorem of existence of an additional integral

1. Ξ ≠ 0 and V = 0,2. Ξ = 0 and rank(M) < 2,

we should look for points where Ξ = 0.


Due to automorphism (5.1) and to Property 1, the normal form has corresponding automorphism and the sum

k ≡ q3 + q4 + q5 + q6

is even for all its terms. We considered sums

For the normal form, it occurs that, for m = q1 + q2 + k = 4 we will have


and all lower terms cancel. Here, the quantities with a hat ĝk, k = 1, …, 6, denote the normal forms calculated up to order four.

It can be demonstrated that the vector Ξ has a components

,11

2

6

4

3^

zzzg

,22

2

6

4

3

2

^

zzzg

,3

a ,4

bSo we can calculate Ξ now. Coefficients a and b depend on δ2 and c. For δ2 = 1, both coefficients a and b are pure imaginary. For δ2 = –1, they are pure imaginary if c (0, c2) and are real if c (c2, 2].


c3 c2



c4 c5 c6c2


Finally we opened that ξ3 = ξ4 = 0 at

for

for

Thus we satisfy case 2 of the main theorem about the necessary condition of local integrability, if the rank of matrix M is smaller then 2.

We found also that at all points above ξ1 = ξ2 = 0.


We calculated the matrix M and its rank at the all points where Ξ = 0. Particularly at c = ½ the matrix M is

for

for


Case of Resonance (1:3)


c7=1/9


We calculated rank(M) at the points above and opened, thatit is equal to 2 in all cases except point c = ½, where this rankis equal 1 and point c = 1, where it has a zero value.


Conclusions

• The normalized system of Euler – Poisson has no additional formal integrals at the mechanical values of the parameter c in resonances (1:2) and (1:3), i.e., it is nonintegrable, except known cases c = ½ and c = 1.

• We have a new workable approach for searching additional formal integrals.

• We have a new method for a proof of nonintegrability in some domains.


References

Bruno, A.D.: Theory of normal forms of the Euler – Poisson equations. Preprint of the Keldysh Institute of Applied Mathematics of RAS No 100. Moscow (2005).

Lunkevich, V.A,, Sibirskii, K.S., Integrals of General Differential System at theCase of Center. Differential Equation, v. 18, No 5 (1982) 786–792, in Russian.


Absence of formal integral

Nonintegrability of shortened system

Nonintegrability of original system

Absence of local integral

Normal forms, computer algebra and a problem of integrability of nonlinear ODEs

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