Psi-series of quadratic vector fields on the plane

Publicacions Matem`atiques, Vol 41 (1997), 101–125. PSI-SERIES OF QUADRATIC VECTOR FIELDS ON THE PLANE Amadeu Delshams and Arnau Mir Abstract Psi-series (i.e., logarithmic series) for the solutions of quadratic vector fields on the plane are considered. Its existence and convergence is studied, and an algorithm for the location of logarithmic singularities is developed. Moreover, the relationship between psi-series and non-integrability is stressed and in particular it is proved that quadratic systems with psi-series that are not Laurent series do not have an algebraic first integral. Besides, a criterion about non-existence of an analytic first integral is given. 1. Introduction The study of the representation of the solutions of a system of differential equations and the relationship with the integrability of this system has been considered by many people. This relationship was initiated by Painlev´e, who gave a test of integrability based on the search of systems such that all its solutions could be represented as Laurent series of the time parameter [Inc56], [Hil76]. This Painlev´e test was later on transformed in an algorithm, the ARS algorithm [ARS78], that has been used succesfully to detect integrable systems (see [RGB89] for a general review). According to this Painlev´e principle, those systems such that all their solutions can be represented in terms of Laurent series, are in fact integrable systems. There are several results confirming this conjecture under some conditions as well as some counterexamples [Gav92], [RGB89], [BPB87]. To be able to deal with more general singularities for the solutions, like logarithmic ones, we consider a generalization of the Laurent series near a singularity tpof a solution x(t), i.e., the so called psi-series: x(t)=  n≥−k pnlog 1 ττn, 102 A. Delshams, A. Mir where pnare vectors polynomials and τ=t−tp. This kind of logarithmic expansions for solutions of systems of ordinary differential equations were called psi-series by Hille (see, for instance, [Hil76]), although they had already been studied in the beginning of the century in systems of the form t dx/dt =f(t, x) by Horn [Hor96]. An account of Horn results may be found in [Gav92]. Besides, they have also been considered in several kind of systems by the authors [MD93], and applied to measure the splitting of separatrices of rapidly forced systems [DMS97]. In this paper, we develop the study of the representation of solutions of quadratic systems on the plane. These systems are important for several applications in chemistry, population evolution, plasma physics, ... (see, for instance [CF92], [HCF93], [HCF+96] for several references and a special study of the so-called Lotka-Volterra systems), and, at the same time, contain already the main features of polynomial systems with more dimensions. Besides, the study of the algebraic integrability of planar polynomial vector fields has been revisited the last years (see [MRT95] for a connection to the existence of isochronous centers). In section 2, we give conditions for the existence of psi-series for quadratic vector fields on the plane. Thus, Theorem 1 provides a constructive algorithm under a general hypothesis H1 and a more restrictive hypothesis H2, both on the quadratic part, which ensure the existence of a formal two-parametric family of psi-series of the form x(t)=a/τ +···. Its convergence, as well as an estimate for the region of convergence is proved in Theorem 7, whose proof is deferred to section 6. It is worthmentioning that a different point of view to study this kind of representations consists on reducing the quadratic system to a system of the form studied by Horn, and can be found in [Yos83], [Yos83], [Gav88]. The knowledge of the psi-series expansion provides an algorithm to compute numerically the singularities of solutions. Such an algorithm is presented in section 3 and it is illustrated with an example. The relationship between the behaviour of different psi-series of a solution and integrability of the system is also illustrated. In particular, when one considers different singularities of a solution, and as a consequence, different psi-series, the numerical experiments performed in section 3 show that the location of singularities and the behaviour of the free parameter pbehave in an irregular way, in contrast with what happens for integrable systems. So it seems that the dynamical information is hidden between the different singularities of a solution (see also [RGB89] for a related discussion), and it remains to give a rigorous criterion along these lines. The Painlev´e principle is checked in section 4, where all integrable Psi-series of quadratic vector fields 103 quadratic systems with a linear center at the origin are displayed. From the study developed there, one gathers that •the Painlev´e test cannot detect quadratic integrable systems if the first integral is a transcendent function, •all quadratic system with a complete solution in terms of psi-series (not Laurent series) do not have an algebraic first integral, and as a matter of fact this last result is proved in Theorem 8 of section 4. According to this result, the existence of logarithmic singularities should be considered not as an obstacle to integrability, but as an obstacle to algebraic non-integrability. In other words, a quadratic vector field on the plane with “genuine” psi-series can still have a transcendent first integral. Section 5 is devoted to a result about non-existence of such a trascendent first integral. An aside consequence of Theorem 1 is that there exist at least a solution ϕ(t) expressible in terms of Laurent series when hypothesis H1 holds and hypothesis H2 fails. In such general situation, the non-existence of an analytic integral near ϕ(t) is proved in Theorem 9. It is worth mentioning that although its proof uses a Ziglin lemma [Zig82], an analytical expression for ϕ(t) is not needed, as happens in the usual applications of Ziglin theory for Hamiltonian systems. 2. Existence of psi-series Let us consider the following differential system: (1)            dx1 dt = 0≤i,j≤2 a(1) ij xi 1xj 2 dx2 dt = 0≤i,j≤2 a(2) ij xi 1xj 2 which can also be written in vectorial form as (2) dx dt =X0+X1(x)+X2(x), where Xj(x) denotes a homogenous polynomial of degree j, for j= 0,1,2. Let tpa free parameter denoting a (real or complex) singularity of an arbitrary solution x(t) of system (1), and let us look for an expansion of x(t) such that its dominant term is of the form (3) x1(t)=a1τα1+··· x2(t)=a2τα2+··· 104 A. Delshams, A. Mir If we replace expansion (3) into system (1), we notice that, in order to determine the exponents αj, only the terms of order 2 have to be considered (equivalently, we could instead look for solutions of the form x1(t)=a1τα1,x2(t)=a2τα2, of the homogeneous quadratic part dx/dt =X2(x) of system (2)). Assuming that both components of X2(x) are not identically zero, we get 3 possible cases: a) α1=α2=−1. b) α2=−1 and a1is the other free parameter besides tp. In this case, it is necessary that a(1) 20 =a(1) 02 =a(2) 20 =a(2) 11 = 0 and a(1) 11 ·a(2) 02 =0. c) α1=−1 and a2is the other free parameter besides tp. In this case, it is necessary that a(2) 02 =a(2) 20 =a(1) 02 =a(1) 11 = 0 and a(2) 11 ·a(1) 20 =0. In general, expansions with non-integer exponents take place in cases b) and c): b) x1=a1τ−a(1) 11 /a(2) 02 +···,x 2=−1 a(2) 02 τ−1+··· c) x1=−1 a(1) 20 τ−1+···,x 2=a2τ−a(1) 11 /a(1) 20 +··· Unless the exponent of τbe an integer number, the singularity tpis a branching point of x(t). If the exponent of τis an integer number, the Painlev´e property is satisfied [Hil76] and thus we have already the two free parameters tpand a1(or a2) that provide the general solution of system (1), and the expansion (3) is nothing else but a Laurent series. It remains to consider case a), the only one with no restrictions. We are going to see under which conditions psi-series appear. Theorem 1. Assume that the following two hypothesis hold for system (1): H1 There exists a complex vector a=a1 a2, with both components non-zero such that X2(a)=−a. H2 If {−2,λ}are the eigenvalues of DX2(a), then λis a non-negative integer. Psi-series of quadratic vector fields 105 Then, there exists a (formal) two-parametric family of solutions of system (1), formed by psi-series (4) x(t)=aτ−1+ n≥0 pnlog 1 ττn. Remark 2. Hypothesis H1 and H2 only affect X2. Due to H1,−2is an eigenvalue of DX2(a) (see equation (5)). In terms of the coefficients of system (1), the other eigenvalue λcan be written as λ= div X2(a)+2=2a1a(1) 20 +a2a(2) 02 +a(1) 11 a2+a(2) 11 a1+2. We note that the matrix DX2(a) + Id and the numbers −1,λ+ 1 are called Kowalevski’s matrix and Kowalevski’s exponents, respectively, by Yoshida [Yos83]. Proof: If we replace the expansion (3) into equation (2) with α1= α2=−1, we obtain: −aτ−2+···=X0+X1aτ−1+···+X2aτ−1+··· =X0+τ−1X1(a)+τ−2X2(a)+··· If we consider the coefficients of τ−2, we get the equation for a:−a= X2(a), which, due to hypothesis H1, has at least a solution with both components different from zero. Given ρan arbitrary number, and taking into account that X2is a quadratic vector field, it turns out that X2(ρa)=−ρ2a, for any arbitrary number ρ. Differentiating this equation and putting ρ=1,weget (5) DX2(a)a=−2a, i.e., ais an eigenvector of DX2(a) of eigenvalue −2. Now, it is very convenient to perform a change of time s=−log τ,or τ=e−s, where τ=t−tp, and a change of variables: (6) x=esa+Bw, where wis the new variable, and Bis the matrix of eigenvectors of DX2(a) associated to its eigenvalues −2,λ (λ=−2 by hypothesis H2). Denoting =d/ds =−e−sd/dt, the differential equation for wis: 106 A. Delshams, A. Mir (7) w=Λw−e−sB−1X1(Bw)−1 2e−sB−1D2(Bw,Bw) −e−sB−1X0−B−1X1(a), where Λ is the diagonal matrix Λ = B−1DX2(a)B=−20 0λ, and D2is the constant quadratic form D2=D2X2(esa)=D2X2(0). Introducing wn=B−1pn, the psi-series (4) of x(s) takes the form (8) x(s)=esa+ ∞  n=0 pn(s)e−ns =esa+ ∞  n=0 Bwn(s)e−ns, and hence, by the change (6), we look for an expansion of wof the form w(s)=n≥0wn(s)e−ns,wn(s) being polynomials in the variable s. Since w= ∞  n=0 (w n−nwn)e−ns, we get the differential equation for w0,w1and wn, for n>1: w 0=−Λw0−e−sB−1X0, w 1=−(Λ −Id) −B−1X1(Bw0)−1 2B−1D2(Bw0,Bw 0)−B−1X1(a), w n=−(Λ −nId) wn−B−1X1(Bwn−1)−1 2B−1 n−1  k=0 D2(Bwk,Bw n−k−1), which can be written in more compact form as (9) w n=Λ nwn+cn, where Λn=nId −Λ and cnis a polynomial of degree two in the variables w0,... ,w n−1. The next step is to obtain the polynomial vectors wn(s). If n<λ,Λ nis an invertible matrix and the only polynomial solution of the linear system (9) is the constant vector wn=−Λ−1 ncn. So, pn=Bwn is constant, too. If n=λ, the matrix Λnis singular and it has two eigenvalues: n+2 and 0. In this case, system (9) takes the form (10) w n,1=(n+2)wn,1+cn,1 w n,2=cn,2 and its only polynomial solution is given by wn,1=−cn,1/(n+2), wn,2= cn,2s+c. The degree of wnand pndepends on cn,2: a) If cn,2=0,wnand pnare constant. b) If cn,2=0,wnand pnare polynomials of degree 1 in the variable s. Psi-series of quadratic vector fields 107 If n>λ, in case a), pnis a constant for all nbecause Λnis a nonsingular matrix for n>λ. Under these circumstances, the expansions of x1(t) and x2(t) are Laurent series where the other free constant is c. In case b) we use the following simple lemma, whose elementary proof is omitted. Lemma 3. Let dy/ds =y=ky +qm(s)be a differential equation of first order, where kis a non-zero constant and qm(s)= m  i=0 αisiis a polynomial of degree m. Then, there exists a unique polynomial solution y=pm(s)= m  i=0 βisi, and its coefficients are given by βm=−αm/k, βi=((i+1)βi+1 −αi)/k,(i=m−1,... ,0). (In fact βi=− m−i  j=0 (i+1)···(i+j)αi+j/kj+1.) Applying this lemma to equation (9), and recalling that cnis a polynomial of degree two in the variables w0,... ,w n−1, it follows that wnis a polynomial of degree [n/λ] and therefore genuine psi-series appear. Remark 4. The coeficient cλ,2could be considered as a coefficient of non-integrability, because when it is zero, there is a biparametric family of meromorphic solutions of system (1), and according to the Painlev´e principle, it is a candidate to be an integrable system. We will see in section 4 that cλ,2is in fact a coefficient of algebraic non-integrability. Remark 5. Hypothesis H1 is very general, since it is equivalent to the existence of a complex solution b=b1 b2of det(X2(b)b) = 0, with b1·b2= 0. Indeed, such bwould satisfy X2(b)=ρb for some complex number ρ, and a=−1/ρ ·bsatisfies hypothesis H1. When it fails, other leading terms different from x(t)=a/τ +··· can take place (see, for instance, systems (11) and (15)). Only for the sake of brevity, a general classification of such cases is not considered in this paper. Remark 6. Hypothesis H2 imposes severe restrictions on X2, since it is required that λ:= div X2(a)+2∈{0,1,2,...}. When it fails, one can only assert the existence of one meromorphic solution of system (1). However, we will see in section 5 that the general case λ/∈Qgives rise to analytic non-integrability near this meromorphic solution. 2.1. Convergence of the psi-series. Once the existence of the formal expansion of psi-series has been obtained, it remains the problem of the convergence of such expansions. 108 A. Delshams, A. Mir The following theorem gives us a real region, depending on a parameter K, where the psi-series are convergent, in terms of the variable s.In the sequel, given a vector xand a p-linear application A,xwill denote a vectorial norm for x, and A= max{ A(x(1),... ,x (p)) : x(j) ≤1, j=1,... ,p}the subordinate p-linear norm. Theorem 7. Consider the constant C=M(X1+(π/2) D2·B), where Bis the matrix of eigenvectors of D2=D2X2(0) and Mis the condition number of B:M=B· B−1 . Let K>0such that the following finite number of inequalities are satisfied: wn(s)≤(n+1) −1/2(2K+Ks)n/λ n=0,1,... ,max{4λ, 9C2+1}. Then, the psi-series  n≥0 wn(s)e−ns, n≥0 pn(s)e−ns are convergent for s> s0, where s0is the positive root of the equation K(2 + s)=eλs. The convergence follows from the general theory of analytic ordinary differential equations of the form t dx/dt =f(t, x),x ∈Rnwhich was developed by Horn [Hor96]. A well-adapted proof for quadratic vector fields on the plane can be found in [Hil74], [Smi75]. However, since the estimates over the region of convergence stated above are relevant for the numerical method of the next section, its proof is presented in section 6. 3. Numerical location of singularities The problem is the following: Let (x0,y 0) be initial conditions of system (1). We want to find a singularity tpand also the other free parameter cof the expansion in psi-series or Laurent series. We solve numerically the problem in two steps. Step 1 The singularities can be of three types: a) Poles. The expansion is a Laurent series. To locate the poles, we have to find the expansion series in a Taylor series in two points. The convergence regions are two discs. One of the intersection points of these two discs is a pole (see Figure 1) and there is a well known algorithm to compute it due to Chang and Corliss [CC80]. Psi-series of quadratic vector fields 109 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • • • 1234 −1 −2 −3 −4 tpt2 t1 3 2 1 −1 −2 −3 Figure 1. Singularity of pole type. b) Singularities with rational branching x(t−tp)=x(τ)= a1 τα+ n≥0 anτα+nβ,where α, β ∈Q+. c) Singularities such that the expansion is a psi-series x(t−tp)=x(τ)=  n≥−k pnlog 1 ττn,where pnare polynomials. To locate singularities of b) or c) type we consider closed paths on the complex plane time. If the value of the solution x(t) of system (1) at the beginning of the path is different from its value at the end of the path, we can deduce that there exists at least a singularity inside the path. These paths can be chosen as rectangles. Thus, if we take smaller rectangles every time, we can find a first approximation of tp(see Figure 2). There is a difference between the singularities of b) and c) type. The convergence region of the expansions of the singularities of b) type is a Riemann surface with a finite number of sheets whereas the expansions of the singularities of c) type is a Riemann surface with an infinite number of sheets. So, to know the type of singularity, we turn round it a big enough number of times. If the value of solution (x1(t),x 2(t)) is the same at the end and at the beginning of the path, we have a singularity of b) type; 116 A. Delshams, A. Mir For each system that appears in Table 3, we have checked if hypothesis H1 and H2 of Theorem 1 are verified. If this is the case, we put YES in the column “psi-series” of Table 4, we write λ+ 1 and we check if the psi-series are in fact Laurent series. Case λ+1 Psi-series Laurent series I 3 YES YES II 3−1/α YES if α=−1 n,n=1,2,... YES for n=1,2,... ,7 III 0 YES NO IV 2 YES NO (YES if γ=−2) V1 YES NO (YES if γ=−1/2) VI ? NO NO VII 1/2,−2 YES YES VIII 0 YES YES IX −2(∆±b√∆) c(1−c)YES if r=1,2,... YES X1 YES NO Table 4. Occurrence of psiand Laurent series. In cases I, II and α=1/n, IV and V, Laurent series appear only when the general integral H(x, y) is an algebraic function: p(x, y, H)=0, where pis a polynomial in its three variables x, y, H. This phenomenum also takes place in case IX, but it requires more cumbersome computations. In view of this fact, we can conjecture that for the algebraically integrable systems, psi-series reduce to Laurent series. This is confirmed by cases III and X where psi-series are not Laurent series, and the integrals are not algebraic. We notice that the reciprocal assertion is false, since system VIII, for integer b2+ 1, is not algebraically integrable and only Laurent series appear. It is worth mentioning that the Painlev´e test cannot detect integrable systems such that the first integral is not algebraic. For instance, case III cannot be detected by this test. System VI is a special one, in the sense that standard psi-series do not appear, but instead one can find the following expansions, where β Psi-series of quadratic vector fields 117 is the other free parameter, x(t)=±(log τ)−1/2τ−1+β(log τ)−3/2τ−1+···, y(t)=−3 2τ−1−3 4(log τ)−1τ−1+···, which can be seen as generalized psi-series. After all the above motivation, we are now in position to prove that quadratic systems on the plane with psi-series not reducible to Laurent series cannot have an algebraic first integral. This result was also proved by Gavrilov [Gav88], but the proof that we present here is immediate thanks to the framework introduced in the proof of Theorem 1. It is worth mentioning here that the proof of Theorem 1 provides us with an algorithm to detect the existence of genuine psi-series, and therefore, in view of next theorem, of an obstruction to the existence of algebraic first integrals. Theorem 8. Assume that system (1) has a non-trivial algebraic integral: p(x, H)=0, where pis a non-constant polynomial in the variable xand the constant of integration H. Then, if hypothesis H1,H2 hold, the psi-series provided by Theorem 1 are Laurent series. Proof: Consider the change of time s=−log τand the change of variables (6) x=esa+Bw, as in section 2, and let mbe the degree of the polynomial p. If we expand the first integral p(esa+Bw,H) = 0 of system (7) in Taylor series with respect to w, we get: p(esa+Bw,H)=p(esa, H)+Dp(esa, H)Bw +1 2D2p(esa, H)(Bw,Bw) +1 m!Dmp(esa, H)(Bw, m) ...,Bw)=0. Substituting the expansion w= n≥0 wne−ns, in the previous formula, and equating to zero the coefficients of e−ns, we arrive at an expression of the form: pn(wn,w n−1,... ,w 0,H)=0, 118 A. Delshams, A. Mir where pnis a polynomial in the variables wi,i=0,... ,n and H, which is a first integral of the equation (10), once fixed wn−1,... ,w 0. Next, we choose n=λ. This means that wn−1,... ,w 0are constants and as a consequence pn(wn,w n−1,... ,w 0,H) = 0 is an algebraic first integral of the equation (10) satisfied by wn. But the general solution of equation (10) is: wn,1=−cn,1/n +2+c1e(n+2)s, wn,2=cn,2s+c2, where c1and c2are the free constants of integration. Therefore, equation (10) possesses an algebraic first integral only if cn,2= 0 (this first integral is simply wn,2=c2), and this implies that the expansions of x1(t), x2(t) are Laurent series. 5. Non-analytic first integral In this section, we are going to establish a theorem about analytic non-integrability for a quadratic system on the plane satisfying hypothesis H1, assuming that div DX2(a) is an irrational number. In such situation, there exists a particular solution represented in terms of a Laurent series expansion that will allow us to apply a lemma by Ziglin to the variational equations. It is important to notice that a “closed” expression for the particular solution will not be needed. Theorem 9. Assume that hyphotesis H1 holds for the quadratic system (1), an that λ:= div DX2(a)+2is not a rational number: λ/∈Q. Then, (i) There exist a solution ϕ(t)of system (1) with a Laurent series expansion: ϕ(t)=aτ−1+ n≥0 anτn. (ii) There exists no non-constant analytic first integral of system (1) in a neighbourhood of ϕ(t). Proof: From the proof of Theorem 1, as the matrix Λn=nId −Λ is not a singular matrix for all n, it follows readily that there exists a solution ϕ(t) with a Laurent series expansion. Next, we consider the variational equation of system (1) associated to the solution ϕ(t): (13) ˙v(t)=DX(ϕ(t))v, Psi-series of quadratic vector fields 119 which takes the form ˙v(t)=τ−1(DX2(a)+···)v, due to hypothesis H1. Next, we perform the change of variables: v= Bw, as in section 2, where Bis the matrix of eigenvectors of DX2(a) associated to its eigenvalues −2,λ (λ=−2 by the hypothesis of an irrational λ). The equation for wis: ˙w=τ−1−20 0λ+···w, and consequently, we can write the expansion of the components of w as: w1=C1 τ2+···,w 2=C2τλ+···, where the component w1corresponds to the solution ˙ϕ(t) of the variational equation (13). Thus, the expansion of w1is a Laurent series. Now, assume that there exists an analytic non trivial integral of system (1): G(x)=H. By using Ziglin lemma (see [Zig82], [Yos87]), there exists a natural k>0 such that the following function is a first integral of the variational system (13): DkG(ϕ(t))(v,... k),v)=C, or written as a function of the new variables w: (14) DkG(ϕ(t))(Bw,... k),Bw)=C. The expression above is an homogeneous polynomial of degree kin the variables w1and w2. If we expand the previous expression as a function of w2and take into account the Laurent series caracter of w1, we get: γk(τ)wk 2+...+γ1(τ)w2+γ0(τ)=C, where γi(τ), i=0,... ,k are functions of τwhose expansions only contain integer exponents of τ. From w2=C2τλ+O(τµ), µ>λ,we get: γk(τ)Ck 2τλk +Oτ(k−1)λ+µ+Oτ(k−1)λ=C. The only term with τλk in the above expression is γk(τ)Ck 2τλk. So, we deduce γk(τ)Ck 2= 0. We conclude C2= 0 and we have arrived at a contradiction, since 14 was supposed to be a first integral for all the solutions of the variational equation. It follows that there cannot exist a non-constant analytic first integral in a neighbourbood of ϕ(t). 120 A. Delshams, A. Mir An example. We consider the following quadratic system: (15) ˙x=y+2αxy ˙y=−x+γx2+βy2 which corresponds, for β=1−α, to case II of Table 3. The vector ais in this case: a=±(2α−β)/(4γα2),−1/(2α) , and the matrix DX2(a) is: DX2(a)=−1±(2α−β)/γ ±γ(2α−β)/α2−β/α . The eigenvalues of DX2(a) are −2 and λ:= 1 −β/α. So, using the theorem above, if β/α /∈Q, there exists a solution ϕ(t) with a Laurent series expansion and the differential system cannot have a non-trivail analytic first integral in a neighbourhood of ϕ(t). For β=1−α, the condition for non-existence of an analytic first integral reads as α/∈Q. However, system (15) for β=1−αis nothing else but case II of Table 3 which possesses a general first integral. This does not give rise to a contradiction, since the solution ϕ(t) with a Laurent series expansion can be explicitly computed in this case: Φ:=γ+3α−1+2(α−1)(2α−1)(γx2−(3α−1)y2) −2(α−1)(γ+3α−1)x=0, and the general first integral can be written as: (1+2αx)α−1·Φα=C, which, if α∈ Q, is not analytic in a neighbourhood of ϕ(t). 2 1 −1 −2 1 23 −1 −2 −3 Figure 5. Phase portrait of differential system (15) for parameters α=2, γ= 2 and β=1−α=−1. Psi-series of quadratic vector fields 121 The phase portrait of the integrable system (15) can be seen in Figure 5 for the following values of the parameters: α=2,γ= 2 and β= 1−α=−1. The three dots of the figure are the equilibrium points of the system. The Laurent series solution ϕ(t) corresponds to the separatrix solution passing through two equilibrium points in the line x=−1 4: −1 2α,±1 2α−1−2α β. 6. Proof of Theorem 7 The proof of this theorem is based on the following lemmas. Lemma 10. Let us consider the following linear differential system (16) w n=Anwn+cn, where Anis m×mmatrix with positive eigenvalues and cn(s)is a polynomial vector. Then, the only polynomial solution of system (16) is given by (17) wn(s)=− ∞  s e(s−u)Ancn(u)du =− m  j=0 A−j−1 nc(j) n(s), where c(j) n(s)is the derivative of order jof cn(s). Proof: If we use the method of variations of constants, we can write the solution of system (16) as wn(s)=esAnUn(s), where Un(s) is given by (18) Un(s)=Un(s0)+s s0 e−uAncn(u)du. Since we look for polynomial solutions, Un(s) has to satisfy lims→∞ Un(s) = 0 in order to avoid an exponential behaviour at infinity. Taking into account this asymptotic behaviour and using expression (18), we can write Un(s0)as (19) Un(s0)=− ∞  s0 e−Anucn(u)du, and we can conclude, using formulas (19) and (18), that the function wn(s) can be written as wn(s)=− ∞  s e−(s−u)Ancn(u)du. This is the first equality of equation (17). Integrating it by parts, we get the second equality. 122 A. Delshams, A. Mir Lemma 11. Given K>0and n>4λ, let (20) U(s)=∞ s e(s−u)(n−λ)(2K+Ku)n/λ du be a function of the variable s>0. Then, U(s)<3 n(2K+Ks)n/λ. Proof: Following [Hil74], we introduce γ:= n/λ and X:= X(s)= 2K+Ks. Integrating by parts formula (20), we obtain U(s)<Xγ n−λ1+ Kγ n−λX−1+K2γ(γ−1) (n−λ)2X−2 +···+Km+1γ(γ−1) ···(γ−m) (n−λ)m+1 , where m=[γ]. Since n−λ>3n/4, it follows that (γ−j)/(n−λ)<1/λ and K/X < 1/2 for j>0, and so we can deduce that U(s)<4 3nXγ1+ 1 2λ+1 (2λ)2+···=4 3n 2λ (2λ−1)Xγ<3 nXγ. Proof of the Theorem: We only have to prove that the bound (21) wj(s)≤(j+1) −1/2(2K+Ks)j/λ holds for n≥0. If this is the case, we can conclude that      ∞  n=0 wn(s)e−ns    ≤ ∞  n=0 (n+1) −1/2(2K+Ks)n/λe−ns, and the theorem follows applying the radical test to the series of the right part. By hypothesis, bound (21) is valid for j<max{4λ, 9C2+1}. Proceeding by induction, we assume that the bound (21) is also true for j<n, where n>max{4λ, 9C2+1}. In particular, Λn=nId −Λ has positive eigenvalues, and we can use Lemma 10, and write the solution of system (9) as wn(s)=−∞ s e(s−u)Λncn(u)du, Psi-series of quadratic vector fields 123 where cn(u)=e−sB−1X1(Bwn−1)−1 2B−1 n−1  k=0 D2(Bwk,Bw n−k−1). As a consequence, we can bound wn(s)as wn(s)≤∞ s e(s−u)(n−λ)cn(u)du, and it only remains to bound cn(u): cn(u)≤e−s B−1 ·B·X1·wn−1 +1 2 B−1 ·B·B·D2 n−1  k=0 wk·wn−k−1 ≤MX1 √n+1 2B·D2 n−1  k=0 1 k(n−k−1)(2K+Ks)(n−1)/λ ≤MX1+π 2B·D2(2K+Ks)(n−1)/λ =C(2K+Ks)(n−1)/λ. Using now Lemma 11, we get (22) wn(s)≤C∞ s e(s−u)(n−λ)(2K+Ku)n/λ du <3C n(2K+Ks)n/λ, and since n>9C2+1, we have 3C/n < (n+1) −1/2, and we get bound (21) for j=n. The study of the convergence for sc∈Ccan be reduced to the real case. Namely, let sc=s−iθbe a complex number with s>0 and θ∈[0,2π) (or another interval of length 2π). 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Yoshida, A criterion for the non-existence of an aditional integral in hamiltonian systems with a homogeneous potential, Phys. D29 (1987), 128–142. [Zig82] S. L. Ziglin, Branching of solutions and nonexistence of first integrals in Hamiltonian mechanics I, Functional Anal. Appl. 16(3) (1982), 181–189. Internet access. All the authors’ preprints quoted in the references list are available at http://www-ma1.upc.es in the preprints page, or at ftp://ftp-ma1.upc.es, in the pub/preprints directory. Amadeu Delshams: Dept. de Matem`atica Aplicada I Universitat Polit`ecnica de Catalunya Diagonal 647 08028 Barcelona SPAIN e-mail: [email protected]c.es Arnau Mir: Dept. de Matem`atiques i Inform`atica Universitat de les Illes Balears Crta. de Valldemossa, Km. 7.5 07071 Palma de Mallorca SPAIN e-mail: dmiam[email protected] Primera versi´o rebuda el 30 de Novembre de 1996, darrera versi´o rebuda el 5 de Febrer de 1997