DARBOUX INTEGRABILITY OF A GENERALIZED FRIEDMANN-ROBERTSON-WALKER HAMILTONIAN SYSTEM

Transcription

1 This is a preprint of: Daroux integrailit of a generalized Friedmann-Roertson-Walker Hamiltonian sstem, Jaume Llire, Clàudia Valls, J. Nonlinear Math. Phs., vol. 203, , DOI: [ / ] DARBOUX INTEGRABILITY OF A GENERALIZED FRIEDMANN-ROBERTSON-WALKER HAMILTONIAN SYSTEM JAUME LLIBRE 1 AND CLÀUDIA VALLS3 Astract. We stud the Daroux first integrals of a generalized Friedmann-Roertson- Walker Hamiltonian sstem. 1. Introduction and statement of the main result Given a sstem of ordinar differential equations depending on parameters in general is ver difficult to recognize for which values of the parameters the equations have first integrals ecause there are no satisfactor methods to answer this question. In this paper we stud the first integrals of a generalized Friedmann-Roertson-Walker Hamiltonian differential sstem in R 4 1 ẋ = p x, ẏ = p, ṗ x = x ax 3 x 2, ṗ = x 2, with a, R eing parameters and the dot denotes derivative with respect to time t. The Hamiltonian of this sstem is H 0 = 1 2 p2 p 2 x x 2 + a 4 x4 + 2 x2 2. When a = 0, sstem 1 coincides with the Friedmann-Roertson-Walker sstem that modelates a universe, filled a conformall coupled a massive real scalar field see [2] for more details where the authors pesent analtical and numerical evidence of the existence of chaotic motion. For a more general point of view on the Friedmann-Roertson-Walker sstem see for instance the works [5, 7] and the references quoted therein. In this paper we stud the Daroux first integrals of a generalized Friedmann-Roertson- Walker sstem see 1. We recall that a first integral of Daroux tpe is a first integral H which is a function of Daroux tpe see elow 2 for a precise definition. The stud of the Daroux first integrals is a classical prolem in the integrailit theor of differential equations Mathematics Suject Classification. Primar 34A05, 34A34, 34C14. Ke words and phrases. Daroux first integrals, Daroux polnomials, generalized Friedman-Roertson- Walker Hamiltonian sstem. 1

2 2 J. LLIBRE AND C. VALLS The vector field associated to sstem 1 is X = p x x + p + x1 ax x 2. p x p Let U C 4 e an open set. We sa that the non-constant function H : U C is a first integral ofo the polnomial vector field X on U if Hxt, t, p x t, p t is constant for all values of t for which the solution xt, t, p x t, p t of X is defined on U. Clearl H is a first integral of X on U if and onl if H X H = p x x + p H + x1 ax2 2 H 1 + x 2 H = 0 p x p on U. In this paper we want to stud the so-called Daroux first integrals of the Friedman Roertson Walker polnomial differential sstems 1, using the Daroux theor of integrailit originated in the papers [4]. For a present state of this theor see the Chapter 8 of [6], the paper [8], and the references quoted in them. We emphasize that the stud of the existence of first integrals is a classical prolem in the theor of differential sstems, ecause the knowledge of first integrals of a differential sstem can e ver useful in order to understand and simplif the topological structure of their orits. Thus, their existence or not can also e viewed as a measure of the complexit of a differential sstem. We recall that a first integral is of Daroux tpe if it is of the form 2 f λ 1 1 f λ p p F µ 1 1 F µ q q, where f 1,..., f p are Daroux polnomials see section 2 for a definition, F 1,..., F q are exponential factors see section 2 for a definition, and λ j, µ k C for all j and k. The functions of the form 2 are called Daroux functions, and the are the ase of the Daroux theor of integrailit, which looks when these functions are first integrals or integrating factors. In this last case, the first integrals associated to integrating factors given Daroux functions are the Liouvillian first integrals, see for more details [6, 8]. The Daroux theor of integrailit is essentiall an algeraic theor of integrailit ased in the invariant algeraic hpersurfaces that a polnomial differential sstem has. In fact to ever Daroux polnomial there is associated some invariant algeraic hpersurface see again section 2, and the exponential factors appear when an invariant algeraic surface has multiplicit larger than 1, for more details see [3, 6, 8]. As far as we know is the unique theor of integrailit which is developed for studing the first integrals of polnomial differential sstems. In general the other theories of integrailit do not need that the differential sstem e polnomial. We sa that the Hamiltonian sstem 1 is completel integrale if it has two independent first integrals H 1 and H 2 in involution. That is, H 1 and H 2 are independent if their gradients are linearl independent over a set of full Leesgue measure in C 4. Moreover, H 1 and H 2 are in involution if the Poisson racket {H i, H 0 } = 0 for i {1, 2}. When = 0 sstem 1 is completel integrale. More precisel we have the following result. Theorem 1. Sstem 1 with = 0 is completel integrale with the first integrals H 1 = p 2 x x 2 + a 2 x4 and H 2 = p 2 2.

3 INTEGRABILITY OF FRIEDMANN-ROBERTSON-WALKER EQUATIONS 3 When a = 0, sstem 1 is the well-known simplified Friedman-Roertson-Walker Hamiltonian sstem. Its Daroux integrailit was studiend in [10]. In that paper, the authors proved the following theorem. Theorem 2. The unique first integrals of Daroux tpe of sstem 1 with a = 0 and 0 are functions of Daroux tpe in the variale H 0 = 1 2 p2 p 2 x x x2 2. From now on we will consider the case in which ac 0. The main result of this paper is the following. Theorem 3. The unique first integrals of Daroux tpe of the generalized Friedman-Roertson- Walker Hamiltonain sstem 1 with a 0 are functions of Daroux tpe in the variale H 0. We prove Theorem 3 in section 3. Since the Daroux theor of integrailit of a polnomial differential sstem is ased on the existence of Daroux polnomials and their multiplicit, the stud of the existence or not of Daroux first integrals needs to look for the Daroux polnomials. These will e done in the next Theorems 9 and 10, whose results are the main steps for proving Theorem 3. We must mention that an comment on the existence or non existence of first integrals additional to the Hamiltonian itself appears in the reference [2]. In that paper the authors are mainl concerned with the chaotic ehavior of the Friedman Roertson Walker polnomial Hamiltonian sstem. Of course, roughl speaking the chaotic motion is against the existence of first integrals. Corollar 4. The generalized Friedman Roertson Walker Hamiltonian sstem 1 is completel integrale with first integrals of Daroux tpe if and onl if = 0. The proof is completl clear. 2. Basic results Let h = hx,, p x, p C[x,, p x, p ] \ C. As usual C[x,, p x, p ] denotes the ring of all complex polnomials in the variales x,, p x, p. We sa that h is a Daroux polnomial of sstem 1 if it satisfies X h = Kh, the polnomial K = Kx,, p x, p C[x,, p x, p ] is called the cofactor of h and has degree at most two. Ever Daroux polnomial h defines an invariant algeraic hpersurface h = 0, i.e. if a trajector of sstem 1 has a point in h = 0, then the whole trajector is contained in h = 0, see for more details [6]. When K = 0 the Daroux polnomial h is a polnomial first integral. An exponential factor E of sstem 1 is a function of the form E = expg/h C with g, h C[x,, p X, p ] satisfing g, h = 1 and X E = LE, for some polnomial L = Lx,, p x, p of degree at most 2, called the cofactor of E. A geometrical meaning of the notion of exponential factor is given the next result.

4 4 J. LLIBRE AND C. VALLS Proposition 5. If E = expg/h is an exponential factor for the polnomial differential sstem 1 and h is not a constant polnomial, then h = 0 is an invariant algeraic hpersurface, and eventuall e g can e exponential factors, coming from the multiplicit of the infinite invariant hperplane. The proof of Proposition 5 can e found in [3, 9]. We explain a little the last part of the statement of Proposition 5. If we extend to the projective space PR 4 the polnomial differential sstem 1 defined in the affine space R 4, then the hperplane at infinit alwas is invariant the flow of the extended differential sstem. Moreover, if this invariant hperplane has multiplicit higher than 1, then it creates exponential factors of the form e g, see for more details [9]. Theorem 6. Suppose that the polnomial vector field X of degree m defined in C 4 admits p invariant algeraic hpersurfaces f i = 0 with cofactors K i, for i = 1,..., p and q exponential factors E j = expg j /h j with cofactors L j, for j = 1,..., q. Then there exists λ i, µ j C not all zero such that p q λ i K i + µ j L j = 0 i=1 if and onl if the function of Daroux tpe is a first integral of X. Theorem 6 is proved in [6]. j=1 f λ 1 1 f p λp E µ 1 1 Eµq q From Theorem 6 it follows easil the next well known result. Corollar 7. The existence of a rational first integral for a polnomial differential sstem implies either the existence of a polnomial first integral or the existence of two Daroux polnomials with the same non zero cofactor. The following result is well known. Lemma 8. Assume that expg 1 /h 1,..., expg r /h r are exponential factors of some polnomial differential sstem 3 x = P x,, p x, p, = Qx,, p x, p, p x = Rx,, p x, p, p = Ux,, p x, p with P, Q, R, U C[x,, p x, p ] with cofactors L j for j = 1,..., r. Then expg = expg 1 /h g r /h r is also an exponential factor of sstem 3 with cofactor L = r j=1 L j. 3. Proof of Theorem 3 According with section 2 for proving Theorem 3 we need to characterize the Daroux polnomials of sstem 1. The following result characterizes the polnomial first integrals of sstem 1 with 0, i.e. it characterizes the Daroux polnomial with zero cofactor. Theorem 9. The unique polnomial first integrals of sstem 1 with a 0 are polnomials in the variale H 0.

5 INTEGRABILITY OF FRIEDMANN-ROBERTSON-WALKER EQUATIONS 5 Proof. Doing the change of variales 4 = p x p, z 2 = p x + p, sstem 1 ecomes 5 ẋ = z 2, ẏ = 1 2 z 2, ż 1 = x + + x ax 2 + x, ż 2 = x xax 2 + x +, and H 0 writes as H 0 = 1 2 x 2 z 2 + x a 2 2 x4. Now we restrict to H 0 = 0. We get that 6 z 2 = 2x x 2 + ax 4 2. Then sstem 5 on H 0 = 0 ecomes, after the rescaling dτ = dt: x = 1 2 z x x 2 + a 2 x4, 7 = 1 2 z x x 2 + a 2 x4, z 1 = x + + x ax 2 + x, where the prime denotes the derivative with respect to the variale τ. If f = fx,, p x, p is a polnomial first integral of sstem 1 then, if we denote g = gx,,, z 2 the polnomial first integral f written in the new variales x,,, z 2, we have that g is also a polnomial first integral of the differential sstem 5. Furthermore, if we denote h = hx,, the polnomial first integral g restricted to the invariant hpersurface 6, then h is a rational first integral of the differential sstem 7 in the sense that it is the quotient of a polnomial in the variales x,, and a power of. Furthermore h satisfies z x x 2 + a h 2 x4 x z x x 2 + a h 2 x4 + x + + x ax 2 + x h = 0. We write 9 h = j= n h j x,, where each h j is the quotient of a polnomial in the variale x,, with degree j and the denominator a power in the variale. Here we define the degree of h j as j. Hence, computing the terms of degree n + 4 we have x 2 4 ax hn h n x + x ax 2 + x h n = 0.

6 6 J. LLIBRE AND C. VALLS We write it as 10 L[h n ] = 0, where L = x2 4 ax x + x ax 2 + x. The characteristic equations associated to the linear partial differential operator L are 11 We consider two cases. dx d = 1, d d = xax ax 2 + x. Case 1: a = 2. In this case 11 has the general solution x + = d 1, x 2 e 2/x = d 2, where d 1 and d 2 are constants of integration. According to this, we make the change of variales 12 u = x +, v = Its inverse transformation is x2 e 2/x, w =. 13 x = u w, = 2w u 2 e 2w/u w v, = w. Under the change of variales 12 and 13 equation 10 ecomes the following ordinar differential equation for fixed u and v: 2 uu 2wu w2 h n w = 0 where h n is h n written in the variales u, v and w. Solving it we have h n = h x2 n u, v = h n x +, e 2/x. Since the numerator of h h is a polnomial of degree n we must have 14 h n = α k x + k, α k R. Now using the transformations 12 and 13 and working in a similar manner to solve h n, computing the terms of degree n + 3, we get n 1 15 h n 1 = β k x + k, β k R. Now, using the transformations 12 and 13 and working in a similar manner to solve h n 2, computing the terms of degree n + 2, we get 2 uu h n w2 w = 2w u3 e 4w/u w v 2 kα k u k 1. Integrating this equations, we otain that h n 2 = h n 2 u, v, w is 1 kα k u k 2 e 4w u w 33u 2 48uw + 16w u 2 e 4 u2 v 2 Ei + k n 2 u, v u w

7 INTEGRABILITY OF FRIEDMANN-ROBERTSON-WALKER EQUATIONS 7 where Eiz is the exponential integral function, see for instance [1]. Going ack to the variales x, and we otain that h n 2 x,, is 1 kα k x + e k 2 4 x 33x x x 44 4x + x2 x + 2 e4 e 4/x z1 2 Ei + k n 2 x +, e 2/x. Since the numerator of h n 2 x,, is a polnomial of degree n 2 we must have kα k x + k 1 = 0. So α k = 0 for k = 1,..., n. Then h n = α 0, which implies that h = α 0 /z m 1 for some nonnegative integer m. Then from 8 we get 16 mα 0 z1 m x + ax 3 + x 2 x 2 = 0. Therefore, mα 0 = 0 and consequentl h is a constant, in contradiction with the fact that h is a first integral. So, this case is not possile. Case 2: a 2. In this case the general solution of 11 is ax x x + = d 1, z x 1 e 2 2 = d 2 + 2ax arctan ax where d 1 and d 2 are constants of integration. According to this, we make the change of variales ax x 17 u = x +, v = z x 1 e 2 2, w =. + 2ax arctan ax Its inverse transformation is 18 x = u w, = au w 2 + 2w 2 e 2w 2w v 1/ 1/w u 2a/ arctan e au w, = w. Under the change of variales 17 and 18 equation 10 ecomes the following ordinar differential equation for fixed u and v: u w 2 au w 2 + 2w 2 h n 4 w = 0 where h n is h n written in the variales. Solving it we have h n = h n u, v = h n x +, ax x z x 1 e ax arctan ax Since h n has degree n in the numerator we must have 19 h n = α k x + k, α k R..

8 8 J. LLIBRE AND C. VALLS Now using the transformations 17 and 18 and working in a similar manner to solve h n, computing the terms of degree n + 3, we get n 1 20 h n 1 = β k x + k, β k R. Now, using the transformations 17 and 18 and working in a similar manner to solve h n 2, computing the terms of degree n + 2, we get 21 u w 2 h n 2 4 w = au w2 + 2w 2 e 2w 2w v 1/ 2/w u 2 2a/ arctan e au w kα k u k 1. When u = 0, if we denote h n 2 = h n 2 v, w the restriction of h n 2 on u = 0 we have h n 2 = 4a + 2α 1 e 2 2a/ arctan 2 /a e 2w v 1/ 2/w w + 2e 4 Ei loge2w v 1/ w 2w loge 2w v 1/ + c n 2 v, where again Eiz is the exponential integral function. Going ack to the variales x, and if we denote ĥn 2 the function h n 2 in the variales x,, we get ĥ n 2 = 4a + 2α 1 e 2 2a/ arctan 2 /a ax /x 2 2 2a arctan e x ax + 2e 4 Ei ax a 2 log x arctan ax 2 1 ax 2 x log ax x + c n 2 2a arctan z x 1 e ax arctan ax. 2 ax Taking into account that ĥn 2 must e a polnomial, we must have c n 2 = 0 and α 1 = 0 ecause a Therefore, h n 2 = x + g n 2 for some polnomial g n 2. Going to the variales u, v, w we have that h n 2 = uḡ n 2 where, after simplifing u, ḡ n 2 satisfies u w 2 ḡ n 2 4 w = au w2 + 2w 2 e 2w 2w v 1/ 2/w u 2 2a/ arctan e au w kα k u k 2. Now proceeding in the same wa as we did for h n 2 we otain that α 2 = 0 compare the aove equation with 21. Proceeding inductivel we otain that α k = 0 for k = 1,..., n. Then h n = α 0, which implies that h = α 0 /z1 m for some nonnegative integer m. Then from 8 we get 16, i.e., mα 0 = 0 and consequentl h is a constant, in contradiction with the fact that h is a first integral. This concludes the proof of the theorem. k=2

9 INTEGRABILITY OF FRIEDMANN-ROBERTSON-WALKER EQUATIONS 9 Now we characterize the existence of Daroux polnomials. Theorem 10. Sstem 1 with a 0 has no Daroux polnomial with non-zero cofactor. Proof. Let f e a Daroux polnomial with non-zero cofactor K. Then K has cofactor of degree at most 2. We write it as K = a 0 + a 1 x + a 2 + a 3 p x + a 4 p + a 5 x 2 + a 6 x + a 7 xp x + a 8 xp + a a 10 p x + a 11 p + a 12 p 2 x + a 13 p x p + a 14 p 2. Note that sstem 1 is invariant under the transformation σ : C 4 C 4 with σx = x, σ =, σp x = p x, σp = p. If f is a Daroux polnomial of sstem 1 then g = f σf is a Daroux polnomial of sstem 1 invariant σ and with cofactor K 1 = K + σk = 2a 0 + a 5 x 2 + a 6 x + a 7 xp x + a 8 xp + a a 10 p x + a 11 p + a 12 p 2 x + a 13 p x p + a 14 p 2. Let f e a Daroux polnomial of sstem 1 with non-zero cofactor K. If f is invariant σ then K = K 1 and if f is not invariant σ then we consider g = f σf a new Daroux polnomial of sstem 1 invariant σ and with cofactor K 1. We have g 22 p x x p g + x1 ax2 2 g 1 + x 2 g = K 1 g. p x p We write g as a polnomial in the variale p x as k g = g i x,, p p j x, i=0 where each g i is a polnomial in the variales x,, p. Without loss of generalit we can assume that g k x,, p 0. Then computing the coefficient of p k+2 x in 22 we get 2a 12 g k = 0 which ields a 12 = 0. Now computing the coefficient of p k+1 x in 22 we otain g k x = 2a 7x + a 10 + a 13 p g k. Solving it we get g k = K k, p e a 7x 2 +2a 10 x+2a 13 p x. Since g k must e a polnomial we get a 7 = a 10 = a 13 = 0. Now we write g as a polnomial in the variale p as m g = ḡ i x,, p x p j, i=0 where each ḡ i is a polnomial in the variales x,, p x. Without loss of generalit we can assume that ḡ m x,, p x 0. Then computing the coefficient of p m+2 in 22 we get 2a 14 ḡ m = 0 which ields a 14 = 0. Now computing the coefficient of p m+1 in 22 we otain ḡ m = 2a 8x + a 11 ḡ m.

10 10 J. LLIBRE AND C. VALLS Solving it we get ḡ m = K n x, p x e 2a 8x+a Since ḡ m must e a polnomial we get a 8 = a 11 = 0. Hence, K 1 = 2a 0 +a 5 x 2 +a 6 x+a 9 2. Then g p x x + p g + x1 ax2 2 g 1 + x 2 g = 2a 0 + a 5 x 2 + a 6 x + a 9 2 g. p x p Now we write g = l j=0 g jx,, p x, p is written as a sum of homogeneous polnomials of degree j. Then computing the terms of degree l + 2 we get x ax g l p x + x g l p = 2a 5 x 2 + a 6 x + a 9 2 g l. Solving it we get g l = K x,, ap x 2 p x x + p 2 ax e 2pxa 5 x 2 +a6 x+a 9 2 x 2. Since it must e a polnomial we get a 5 = a 6 = a 9 = 0. Then, K 1 = 2a 0. Now proceeding as in the proof of Theorem 9 introducing the change of variales 4 and the restriction H 0 = 0 see 7 we get that gx,, p x, p = hx,, with z x x 2 + a h 2 x4 x + + x + + x ax 2 + x h = 2a 0 h. 1 2 z x x 2 + a h 2 x4 We write h as in 9. Now we consider two cases. Case 1: a = 2. In this case, we get h n and h n 1 as in 14 and 15. Now, using the transformations 12 and 13 and working in a similar manner to solve h n 2, computing the terms of degree n + 2 in 23, we get 2 uu h n 2 w2 w = 2w u3 e 4w/u w v 2 kα k u k 1 + 2a 0 2w ue 2w/u w v α k u k. Integrating this equation and going ack to the variales x, and we get that 1 x2 h n 2 = e 2+2/x 2x + e 2/x e 2 4a 0 α k u k e 2/x 33x x x2 e 2/x kα k u k 1 2x + + 8e 2x+/x 2a 0 α k u k Ei x + 11 kα k u k 1 2 x2 4x + x2 x + e 2/x Ei + k n 2 x +, x e 2/x.

11 INTEGRABILITY OF FRIEDMANN-ROBERTSON-WALKER EQUATIONS 11 Since the numerator of h n 2 must e a polnomial of degree n 2 we have kα k x + k 1 = 0, and a 0 α k x + k = 0. This implies that α k = 0 for k = 1,..., n and a 0 α 0 = 0. Case 2: a 2. In this case, we get h n and h n 1 as in 14 and 15. Now, using the transformations 17 and 18 and working in a similar manner to solve h n 2, computing the terms of degree n + 2 in 23, we get u w 2 h n 2 4 w = au w2 + 2w 2 e 2w 2w v 1/ 2/w u 2 2a/ arctan e au w kα k u k 1 + 2a 0 e 2w v 1/ 1/w u e 2a/ arctan 2w au w α k u k. When u = 0 denoting h n 2 the restriction of h n 2 to u = 0, we get w 2 h n 2 4 w = a + 2α 1w 2 e 2w v 1/ 2 2/we 2 2a/ arctan a 2 + 2a 0 α 0 e 2w v 1/ 1/w 2a/ arctan e a. Integrating this equation, and going ack to the variales x, and denoting ĥn 2 the function h n 2 in the variales x,, we get that ĥ n 2 = 4e 2 2a/ arctan a 2 2α 1 a + 2e Ei 4 log z1 x 2 2a a e arctan ax 4 2 log α 1 a + 2 log z x 1 a e 2a z x 1 a e z1 x e a+2 2 2α 0 a 0 2a 2a arctan 2 ax + arctan 2 ax 1/ arctan 2 ax 2 + k n 2 z x 1 a e 2a arctan 2 ax 1/ ax x z x 1 e ax arctan ax Since the numerator of ĥn 2 must e a polnomial of degree n 2 we have α 1 = 0 and a 0 α 0 = 0. Proceeding inductivel, as we did in the proof of Theorem 9 we otain that α k = 0 for k = 1,..., n and a 0 α 0 = 0. In summar, in oth cases we get that α k = 0 for k = 1,..., n and a 0 α 0 = 0. If α 0 = 0 then h = 0, in contradiction that h is a Daroux polnomial. Therefore a 0 = 0 and K 1 = 0. Thus g = f σf is a polnomial first integral. B Theorem 9 the unique polnomial first integrals are polnomials in the variale H 0, we must have that f is a polnomial in the variale H 0 which is not possile ecause f is a Daroux polnomial of sstem 1 with non zero cofactor. This concludes the proof of the theorem. Theorem 11. The unique rational first integrals of sstem 1 with 0 are rational functions in the variale H 0.

12 12 J. LLIBRE AND C. VALLS Proof. It follows directl from Corollar 7 and Theorems 9 and 10. Now we proceed as in the proof of Theorem 10. Proof of Theorem 3. It follows from Theorems 6, 9 and 10 and Proposition 5 that in order to have a first integral of Daroux tpe we must have q exponential factors E j = expg j /h j H 0 with cofactors L j such that q j=1 µ jl j = 0. Let G = q j=1 µ jg j /h j H 0, then E = expg is an exponential factor of sstem 1 with cofactor L = q j=1 µ jl j see Lemma 8 and G satisfies that X G = 0, that is, G must e a rational first integral of sstem 1. B Theorem 11 it must e a rational function in the variale H 0. Acknowledgements The first author was partiall supported the MICINN/FEDER grant MTM , AGAUR grant 2009SGR-410 and ICREA Academia. The second author has een partiall supported FCT through the project PTDC/MAT/117106/2010 and through CAMGD, Lison. References [1] M. Aramowitz and I.A. Stegun, Handook of mathematical functions with formulas, graphs, and mathematical tales, National Bureau of Standards Applied Mathematics Series 55, Government Printing Office, Washington, D.C., [2] E. Calzeta and C.E. Hasi, Chaotic Friedmann-Roertson-Walker cosmolog, Class. Quantum Grav , [3] C. Christopher, J. Llire and J.V. Pereira, Multiplicit of invariant algeraic curves in polnomial vector fields, Pacific J. Math , [4] G. Daroux, Mèmoire sur les équations différentielles algéraiques du premier ordre et du premier degré Métanges, Bull. Sci. Math. 2éme série , 60 96; ; [5] C. Dariescu and M.A. Dariescu, Vacuum electromagnetic modes in Friedman-Roertson-Walker universe, Internat. J. Modern Phs. A , [6] F. Dumortier, J. Llire and J.C. Artés, Qualitative theor of planar differential sstems, Universitext, Springer Verlag, New York, [7] K. Kleidis, A. Kuiroukidis, D.B. Papadopoulos and L. Vlahos, Gravitomagnetic instailities in anisotropicall expanding fluids, Internat. J. Modern Phs. A , [8] J. Llire and X. Zhang, On the Daroux integrailit of the polnomial differential sstems, Qualitative Theor and Dnamical Sstems , [9] J. Llire and X. Zhang, Daroux theor of integrailit for polnomial vector fields in R n taking into account the multiplicit at infinit, Bull. Sci. Math , [10] J. Llire and C. Valls, Daroux integrailit of a simplified Friedman-Roertson-Walker Hamiltonian sstem, to appear in J. Nonlinear Math. Phs. 1 Departament de Matemàtiques, Universitat Autònoma de Barcelona, Bellaterra, Barcelona, Catalonia, Spain address: jllire@mat.ua.cat 3 Departamento de Matemática, Instituto Superior Técnico, Universidade Técnica de Lisoa, Av. Rovisco Pais , Lisoa, Portugal address: cvalls@math.ist.utl.pt