This is a preprint of: “Strongly formal Weierstrass non-integrability for polynomial differential systems in C 2 ”, Jaume Giné, Jaume Llibre, Electron. J. Qual. Theo., vol. 2020(1), 1–16, 2020. DOI: [10.14232/ejqtde.2020.1.1] STRONGLY FORMAL WEIERSTRASS NON-INTEGRABILITY FOR POLYNOMIAL DIFFERENTIAL SYSTEMS IN C2 JAUME GINÉ1† AND JAUME LLIBRE2 Abstract. Recently it has been given a criterion for determin- ing the weakly formal Weierstrass non-integrability of polynomial differential systems in C2 . Here we extend this criterion for de- termining the strongly formal Weierstrass non-integrability which includes the weakly formal Weierstrass non-integrability of poly- nomial differential systems in C2 . The criterion is based on the solutions of the form y = f (x) with f (x) ∈ C[[x]] of the differen- tial system whose integrability we are studying. The results are applied to a differential system that contains the famous force-free Duffing and the Duffing-van der Pol oscillators. 1. Introduction and statement of the main result One of the main problems in the qualitative theory of differential systems is the integrability problem. For differential systems in C2 this problem consists in to determine if the system has or not an explicit first integral. When this first integral can be expressed as quadratures of elementary functions we have the so-called Liouville integrability, which is the most studied, see for instance [16, 30, 31] and references therein. The Liouville integrability is based on the cofactors of the invariant algebraic curves and the exponential factors (see definitions below). Some generalizations of the Liouville integrability theory defining the generalized cofactors have been obtained, see [7, 8, 10, 11, 19, 20, 30, 31]. Some differential systems have an explicit first integral that cannot be expressed as quadratures of elementary functions. Hence these sys- tems are not Liouville integrable. Sometimes these first integrals can be expressed in terms of special functions, as for instance functions that are solutions of second order linear differential equations (in [11, 19, 29] 2010 Mathematics Subject Classification. Primary 34C05. Secondary 37C10. Key words and phrases. Liouville integrability, polynomial differential systems. † Corresponding author. Email: gine@matematica.udl.cat. 1 2 J. GINÉ AND J. LLIBRE several examples are given). To determine when a differential system is not Liouville integrable is an open problem, see [25]. A partial answer to this question has been recently given in [23]. In this work we present a criterion to detect the strongly formal Weierstrass non–integrability which is a generalization of the crite- rion for detecting weakly formal Weierstrass non-integrability given in [23]. Finally we apply this new criterion to some differential sys- tems. Puiseux Weierstrass integrability is a generalization of formal Weierstrass integrability which includes the Liouville integrability and is based on the Puiseux Weierstrass polynomials, see again [23] and below. First we provide some preliminary definitions and results. In this paper we consider polynomial differential systems in the plane C that are given by 2 (1) ẋ = P (x, y), ẏ = Q(x, y), where the functions P and Q are polynomials in the complex variables x and y. We define by m = max{deg P, deg Q} the degree of system (1) with P (0, 0) = Q(0, 0) = 0. Along the paper we also consider the associated differential equation dy Q(x, y) (2) = , dx P (x, y) and the associated vector field X = P (x, y)∂/∂x + Q(x, y)∂/∂y. An invariant algebraic curve of system (1) is an invariant curve f = 0 with f ∈ C[x, y], such that the orbital derivative f˙ = X f = P ∂f /∂x + Q∂f /∂y vanishes on f = 0. This condition implies that there exists a polynomial K(x, y) ∈ C[x, y] of degree less than or equal to m − 1 such that ∂f ∂f (3) Xf = P +Q = Kf. ∂x ∂y This polynomial K is called the cofactor of the curve f (x, y) = 0. A function of the form ef /g with f and g polynomials is called an exponential factor if there is a polynomial L of degree at most m − 1 such that ∂ef /g ∂ef /g X (ef /g ) = P +Q = L ef /g . ∂x ∂y The polynomial L is called the cofactor of the exponential factor ef /g . FORMAL WEIERSTRASS NON-INTEGRABILITY CRITERION 3 A non-locally constant function H : U ⊂ C2 → C is a first integral of system (1) in the open set U if this function is constant on each solution (x(t), y(t)) of system (1) contained in U . In fact if H ∈ C 1 (U ) is a first integral of system (1) on U if and only if X H = P ∂H/∂x+Q∂H/∂y ≡ 0 on U . A non-constant function M : U ⊂ C2 → C is an integrating factor in U if   ∂M ∂M ∂P ∂Q (4) P +Q =− + M = −div(X )M. ∂x ∂y ∂x ∂y This integrating factor M is associated to a first integral H when M P = −∂H/∂y and M Q = ∂H/∂x. Moreover V = 1/M is an in- verse integrating factor in U \ {M = 0}. A polynomial differential system (1) has a Liouville first integral H if its associated integrating factor is of the form  Y D (5) M = exp Ciαi , E i where D, E and the Ci are polynomials in C[x, y] and αi ∈ C, see [3, 17, 30, 31]. The curves Ci = 0 and E = 0 are invariant algebraic curves of the differential system (1), and the exponential exp(D/E) is a product of some exponential factors associated to the multiple invariant algebraic curves of system (1) or to the invariant straight line at infinity, see for instance [2, 4, 5, 15] or Chapter 8 of [10]. The Liouville integrability is based on the existence of algebraic co- factors for the invariant algebraic curves and for the exponential fac- tors. The first generalization of this theory is to consider non–algebraic invariant curves but still with algebraic cofactors, see [11]. In [12] a method for detecting non–algebraic invariant curves for polynomial dif- ferential systems was given. However there exist non-algebraic invari- ant curves without an algebraic cofactor, see [20]. Now we are recall the definition of Puiseux Weierstrass integrability introduced in [23]. Let C((x))) be the set of series in fractionary powers in the variable x with coefficients in C (these series are called Puiseux series), and C[y] the set of the polynomials in the variable y with coefficients in the ring C. A function of the form X̀ (6) ai (x)y i ∈ C((x))[y] i=0 4 J. GINÉ AND J. LLIBRE is a Puiseux Weierstrass polynomial in y of degree `, i.e. a polynomial in the variable y with coefficients in C((x)). Here we have privileged the variable y but of course we can privileged the variable x instead of the y. In the next result we provide the expression of the cofactor of an invariant curve y − g(x) = 0 with g(x) being a Puiseux series, for a proof see [23], see also [13]. Proposition 1. Let g(x) ∈ C((x)). An invariant curve of the form y − g(x) = 0 of a polynomial differential system (1) of degree m has a Puiseux Weierstrass polynomial cofactor of the form (7) K(x, y) = km−1 (x)y m−1 + · · · + k1 (x)y + k0 (x). A planar autonomous differential system is Puiseux Weierstrass in- tegrable if admits an integrating factor of the form (5) where D, E and the Ci ’s are Puiseux Weierstrass polynomials. This definition is a generalization of the Weierstrass integrability given in [19] and studied in [21, 22, 24, 28]. We remark that by definition that all the Liouvil- lian integrable systems are particular cases of the Puiseux Weierstrass integrable systems. Let C[[x, y]] be the set of all formal power series in the variables x and y with coefficients in C. Theorem 2. If f ∈ C[[x, y]] then it has a unique decomposition of the form Ỳ (8) f = uxs (y − gj (x)), j=1 where gj (x) are Puiseux series and s ∈ Z, s ≥ 0 and u ∈ C[[x, y]] is invertible inside the ring C[[x, y]]. For a proof of Theorem 2 see Corollary 1.5.6 of [1]. We note that a Darboux integrating factor (5) is analytic function where it is defined consequently by Theorem 2 it can be written into the form (8). The first aim of this work was to give a necessary condition for de- tecting the Puiseux Weierstrass integrability but when gj (x) ∈ C[[x]]] of a polynomial differential system (1). However this has been im- possible using only the formal solutions of the form y = f (x) of the associated differential equation for the reasons that we will see later on. FORMAL WEIERSTRASS NON-INTEGRABILITY CRITERION 5 We say that a polynomial differential system (1) is strongly formal Weierstrass integrable if it has an integrating factor of the form Ỳ (9) M (x, y) = α(x) (y − gk (x))αk , k=1 where the functions α(x), gk (x) ∈ C[[x]] for i = 1, . . . , k. Note that the definition of strongly formal Weierstrass integrability is a generalization of the definition of weakly formal Weierstrass integrability given in [23], where that the functions α(x) is constant equal to one. In this work we give a criterion for detecting when a polynomial differential system (1) is not strongly formal Weierstrass integrable with α(x), gk (x) ∈ C[[x]]. This criterion is based on the following result which provides a necessary condition in order that a polynomial differential system (1) be strongly formal Weierstrass integrable with α(x), gk (x) ∈ C[[x]]. Our main result is the following one. Theorem 3. Assume that a polynomial differential system (1) is strongly formal Weierstrass integrable with α(x), gk (x) ∈ C[[x]], and let H(x, y) be a first integral provided by the strongly formal Weierstrass integra- bility. (a) Let h(x) ∈ C[[x]] and y = h(x) be an invariant curve of the system such that H(x, y) is defined on the curve y = h(x). Then there exists an integrating factor M (x, y) of the form (9) such that M (x, h(x)) = 0. (b) Assume that the origin of system (1) is a singular point, and the first integral H(x, y) and M (x, y) of statement (a) are well- defined at the origin. Then a linear combination of the formal Weierstrass polynomial cofactors up to order r of the solutions of the form y = f (x) satisfying Eq := ẋdy/dx − ẏ = 0 must be equal to minus the divergence of system (1) up to order r. Theorem 3 is proved in Section 2. Now we apply Theorem 3 to a differential system that contains the force-free Duffing and Duffing-Van der Pol oscillators. Hence we con- sider the differential system (10) ẋ = y, ẏ = −(ζx2 + α)y − (εx3 + σx). This system contains the famous force-free Duffing (ζ = 0, ε 6= 0) and the Duffing-Van der Pol (ζ 6= 0, ε 6= 0) oscillators that appear in 6 J. GINÉ AND J. LLIBRE several fields of mathematics, physics, biology, see [18] and references therein. The Liouville integrability of system (10) was studied in [9] where the following results were established. Theorem 4. System (10) with ζ = 0 and ε 6= 0 is Liouvillian integrable if and only if either α = 0, or σ = 2α2 /9. In the case ζ 6= 0 by a suitable rescaling of the variables for the Duffing-Van der Pol system we can take ζ = 3 without loss of generality. Theorem 5. System (10) with ζ = 3 and ε 6= 0 is Liouvillian integrable if and only if α = 4ε/3 and σ = ε2 /3. Applying Theorem 3 to system (10) we obtain the following result. Theorem 6. System (10) can be strongly formal Weierstrass integrable with α(x), gk (x) ∈ C[[x]] if, and only if, one of the following cases holds: (a) σ = 2α2 /9, (b) σ 6= 2α2 /9, σ 6= 0 and 3αε − 4ζσ = 0, (c) σ 6= 2α2 /9, σ = 6 0 and −21αε2 + 6α2 εζ + 24εζσ − 7αζ 2 σ = 0, (d) σ 6= 2α2 /9, σ = 0 and −6ε(7ε − 2αζ) = 0. We can see that all the Liouvillian integrable cases given in Theorems 4 and 5 are included in Theorem 6. In particular the case ζ = 3 with α = 4ε/3 and σ = ε2 /3 vanish the condition −21αε2 + 6α2 εζ + 24εζσ − 7αζ 2 σ = 0. Theorem 6 is proved in Section 3. The following proposition shows that if a polynomial differential sys- tem has a Puiseux Weierstrass first integral of the form (5) then it has an integrating factor of the same form. Proposition 7. If system (1) has a Puiseux Weierstrass first integral of the form (5), then it has a Puiseux Weierstrass integrating factor of the form (5). The proof is straightforward because M = (∂H/∂y)/P (x, y) which has the form (5). This proposition was generalized in [17] for non- Liouville integrable systems. 2. Proof of Theorem 3 Proof of statement (a) of Theorem 3. By assumptions the first integral H(x, y) is defined on the invariant curve y = h(x). So H(x, h(x)) = c ∈ FORMAL WEIERSTRASS NON-INTEGRABILITY CRITERION 7 C, and the first integral H̄(x, y) = H(x, y) − c satisfies H̄(x, h(x)) = 0. Now we consider the integrating factor M (x, y) associated to the first integral H̄. Perhaps this inverse integrating factor does not vanish at y = h(x), but we consider the function M̄ = M F (H̄) being F an arbitrary function of H̄ such that F (0) = 0. This function M̄ is also an inverse integrating factor of system (1) because X (M̄ ) = X (M F (H̄)) = X (M )F (H̄) + M X (F (H̄)) = F (H̄)X (M ) = −F (H̄)div(X ) M = −div(X ) M F (H̄) = −div(X ) M̄ . Hence we obtain that M̄ (x, h(x)) = 0 because F (0) = 0.  We can repeat this process to obtain an integrating factor that van- ish in a finite number of the solutions of the form y = h(x) such H(x, h(x)) = c ∈ C. In the proof of statement (b) of Theorem 3 we shall need the following result, for a proof see for instance Proposition 8.4 of [10]. Proposition 8. Assume that f ∈ C[x, y] and let f = f1n1 . . . frnr be its factorization into irreducible factors over C[x, y]. Then for a polyno- mial system (1), f = 0 is an invariant algebraic curve with cofactor Kf if and only if fi = 0 is an invariant algebraic curve for each i = 1, . . . , r with cofactor Kfi . Moreover Kf = n1 Kf1 + . . . + nr Kfr . Proof of statement (b) of Theorem 3. We assume that the system is strongly formal Weierstrass integrable with α(x), gk (x) ∈ C[[x]] this means by definition that the system has an integrating factor of the form (9). Hence we know that a first integral H and an integrating factor M of the form given in statement (a) P∞can bej found. We com- pute the solutions y = fi (x) where fi (x) = j=0 aj x with ai arbitrary coefficients that must satisfy the equation Eq := ẋdy/dx − ẏ = 0 up to certain order r. Note that these solutions satisfy that either M (x, fi (x)) = O(xr ), or M (x, fi (x)) = c2 + O(xr ), with c2 6= 0, this case appears when the integrating factor (8) has s = 0. The first ones correspond to the fi (x) that approximate the invariant curves y = gk (x) that appear in the integrating factor (9). For such fi (x) we compute the cofactor Ki up to certain order r though the equation (11) X (y − fi (x)) = K̄i (y − fi (x)) + O(xr ). 8 J. GINÉ AND J. LLIBRE Hence these cofactors K̄i of the solutions y − fi (x) are the approxima- tions up to order r of the cofactors Kk of the invariant curves y − gk (x) of the integrating factor (9). The second ones satisfy Ỳ (12) M (x, fi (x)) = α(x) (fi (x) − gk (x))αk = c2 + O(xr ). k=1 Hence, since c2 6= 0, M (x, fi (x)) = c2 + O(xr ), and from (9) we have that α(0) 6= 0. Then up order r we have Ỳ h c i 2 (13) (fi (x) − gk (x))αk = + O(xr ), k=1 α(x) r where here [ ]r means up to order r. Consequently y = fi (x) is an approximation up to order r of the equation Ỳ c2 (14) (y − gk (x))αk = . k=1 α(x) We apply the vector field operator to (14) and we obtain  Ỳ    αk c2 c2 α0 (x) c2 (15) X (y − gk (x)) =X =− P = −K α , k=1 α(x) α(x)2 α(x) because X (α(x)) = Kα (x, y)α(x) where Kα is a formal Weierstrass polynomial cofactor. This happens because α(x) = 0 is an invariant algebraic curve of the vector field X . Indeed, α(x) is a factor of the integrating factor M (x, y) given in (9), and M (x, y) = 0 is an invariant curve because it satisfies (4), and the factors of an invariant curve are also invariant curves. Moreover we have taken into account that X (α(x)) = α0 (x)ẋ = α0 (x)P (x, y) and then Kα = α0 (x)P (x, y)/α(x). In summary from equations (14) and (15) we have  Ỳ  Ỳ αk (16) X (y − gk (x)) = −Kα (y − gk (x))αk k=1 k=1 Now we apply the vector field operator to (13) and we obtain ! h  Ỳ αk c2 i (17) X (fi (x) − gk (x)) =X + O(xr ), k=1 α(x) r FORMAL WEIERSTRASS NON-INTEGRABILITY CRITERION 9 where X (O(xr )) = O(xr−1 ) P (x, fi (x)) = O(xr ). Taking into account equation (15) we define the new cofactor K̃α through the equation h  h  c2 i c2 i (18) X = −K̃α α(x) r α(x) r which is equation (15) taking the lower terms up to r and where K̃α is an approximation up to r of the cofactor Kα . Therefore from (13), (17) and (18) we obtain an approximation of the cofactor of α(x) up to order r computing Q  ` αk X k=1 (fi (x) − gk (x)) (19) Q` = −K̃α + O(xr ). α k=1 (fi (x) − gk (x)) k By the definition of integrating factor (9) and from the extension of the Darboux theory to Weierstrass functions, see for instance Theorem 3 of [23], we have that (20) X (M ) = −div(X )M. In short the cofactors K̄i of the solutions y − fi (x) passing through the origin are the approximations up to order r of the cofactors Ki of the solution y = gi (x). By Proposition 8 the other solutions y − fi (x) not passing through the origin with cofactor K̃i give by equation (19) an approximation up to order r of the cofactor K̃α of α(x), i.e. s X (21) µi K̃i = −K̃α . i=1 Therefore, from (12), (20) and (21) we obtain that X̀ s X (22) λi K̄i + µi K̃i = −divr (X ) + O(xr+1 ). i=1 i=1 This proves statement (b) of the theorem.  In summary, if condition (22) is not satisfied then system (1) does not admit an integrating factor of the form (9) and consequently is not strongly formal Weierstrass integrable. Hence we have a necessary condition to have P`strongly formal Ps Weierstrass r+1 integrability. Note that if we have that i=1 λi K̄i + i=1 µi K̃i = O(x ) system (1) satisfies a necessary condition to have a first integral of the form (9), see for more details statement (i) of Theorem 8.7 of [10]. 10 J. GINÉ AND J. LLIBRE 3. Proof of Theorem 6 We apply the criterion provided by statement (b) of Theorem 3 to detect if system (10) can be strongly formal Weierstrass integrable, that is, if it can has an inverse integrating factor of the form (9). We propose a solution curve of the form y = f (x) = a0 + a1 x + a2 x2 + a3 x3 + a4 x4 + a5 x5 + · · · Substituting this solution in the first ordinary differential equation Eq := ẋdy/dx − ẏ = 0 we get an infinite system of equations. First we have studied the case when a0 6= 0, and in this case it is easy to see that we find two solutions not passing through the origin but we do not find any possible integrable case. So we consider the case a0 = 0. In order to determine the first coefficients we fix up to certain order the developments of f (x) and Eq in power series of the variable x. If we compute the solutions up to order 6 we obtain the following finite system of equations a2 (3a1 + α) = 0, a21 + a1 α + σ = 0, 2a22+ 4a1 a3 + a3 α + ε + a1 ζ = 0, 7a3 a4 + 7a2 a5 + a4 ζ = 0, 5a2 a3 + 5a1 a4 + a4 α + a2 ζ = 0, 3a23 + 6a2 a4 + 6a1 a5 + a5 α + a3 ζ = 0. From the first equation we have two possibilities a2 = 0 or a1 = −α/3. First we take a2 = 0. The obtained system is compatible and we get two solutions. We denote them y1 and y2 but we do not write them here due to their long extensions. Now we study the case a1 = −α/3 with a2 6= 0. In this case the equation a21 + a1 α + σ = 0 takes the form σ − 2α2 /9 = 0. Hence we must impose σ = 2α2 /9 in order that the finite system of equations be compatible. Under this condition we find four more solutions that we denote by y3 , y4 , y5 and y6 , but again we do not write them here due to their big extensions. We recall that all these solutions pass through the origin, i.e., yi (0) = 0 for i = 1, . . . , 6. Now we compute their cofactors using equation (11), that we denote by K̄i . Finally we verify if the equation 6 X λi K̄i = −div6 X + O(x7 ), i=1 has any solution, and since it has a solution statement (a) of the the- orem follows. Now we consider the case σ 6= 2α2 /9. In this case the solutions yi for i = 3, . . . , 6 do not exist and we only have the solutions y1 and y2 . We compute their cofactors from equation (11), that we denote by K̄1 FORMAL WEIERSTRASS NON-INTEGRABILITY CRITERION 11 and K̄2 and we verify if the equation λ1 K̄1 + λ2 K̄2 = −div6 X + O(x7 ), is satisfied. This equation gives a system of three equations. The first one is √ (23) α(2 + λ1 + λ2 ) − (λ1 − λ2 ) α2 − 4σ = 0. From this condition we can isolate λ1 if σ 6= 0 (we will consider σ = 0 below) and we have √ α(2 + λ2 ) + λ2 α2 − 4σ λ1 = √ . −α + α2 − 4σ From the second equation we obtain √ √ (−α + 2 α2 − 4σ + λ2 α2 − 4σ)(3αε − 4ζσ) = 0. Hence we have two possibilities: If 3αε − 4ζσ = 0 the third equation can vanish choosing the √ value of λ2 √ and this proves statement (b) of the theorem. If −α+2 α2 − 4σ +λ2 α2 − 4σ = 0 we isolate the value of λ2 , i.e. √ α − 2 α2 − 4σ λ2 = √ , α2 − 4σ and the third equation provides the condition −21αε2 +6α2 εζ +24εζσ− 7αζ 2 σ = 0, which shows statement (c) of the theorem. Now we study the case σ = 0. In this case condition (23) becomes α(1 + λ2 ) = 0. Taking into account that we are in the case σ 6= 2α2 /9, we must take λ2 = −1. The second condition is ε(3 + λ1 ) = 0. The case ε = 0 gives a trivial integrable case. Hence we must consider λ1 = −3. In this case the third condition gives −6ε(7ε − 2αζ) = 0 which proves statement (d) of the theorem. Hence this completes the proof of theorem. 4. Examples Example 1. Consider the differential system (24) ẋ = y + xy + x2 , ẏ = 2y(y + x). This system was studied in [14] where an algorithmic method to de- termine integrability was given. Using the method developed in [14] 12 J. GINÉ AND J. LLIBRE it was shown that system (24) has an integrating factor of the form 2 M (x, y) = ex /(2y) y −5/2 and the a Liouville first integral x2 Z x/√2y e 2y √ 2 H(x, y) = √ + 2 et dt. y 0 Now we are going to apply the criterion provided by statement (b) of Theorem 3 for detecting if system (24) can have an inverse integrating factor of the form (9). We propose a solution curve of the form y = f (x) = a0 + a1 x + a2 x2 + a3 x3 + a4 x4 + a5 x5 + · · · Substituting this solution in the first ordinary differential equation Eq := ẋdy/dx − ẏ = 0 we get an infinite system of equations. In order to determine the first coefficients we fix up to certain order the developments of f (x) and Eq in power series in variable x. If we do that up to order 6 and we solve the finite system of equations we obtain the following solutions. 1) a6 = a5 = a4 = a3√= a2 = a1 = 0, √ √ 2) a6 = −1368989−4007 150829 607500 √ , a5 = 781+3 150829 750 √ , a4 = 173− 150829 900 √ , 2 497− 150829 427− 150829 427− 150829 a3 = − 3 , a2 = √ 70 , a1 = √35 , a0 = 70 √ , 3) a6 = −1368989+4007 150829) 607500 √ , a5 = 781−3750 √ 150829) , a4 = 173+√900 150829) , 2 497+ 150829 427+ 150829 427+ 150829 a3 = − 3 , a2 = 70 , a1 = 35 , a0 = 70 . The solutions correspond to the solution curves 1) y1 = 0 + O(x7 ), √ √ √ 427− 150829 427− 150829 497− 150829 2 2) y2 = f√ 2 (x) = 70 + 35 x + x − 32 x3 √ √70 + 173− 900 150829) 4 x +√781+3750150829 x√ 5 − 1368989+4007 607500 √ 150829 6 x + O(x7 ), 427+ 150829 3) y3 = f√ 3 (x) = 70 + 427+ 35150829 x + 497+ 70150829 x2 − 32 x3 √ √ + 173+ 900 150829) 4 x + 781−3750150829 x5 − 1368989−4007 607500 150829 6 x + O(x7 ), respectively. The first one corresponds to the invariant algebraic curve y = 0 whose cofactor is 2x + 2y. However, in general, we can have an approximation of a solution of the form y = gk (x) and an approxima- tion of its cofactor. To compute the approximation of the Weierstrass polynomial cofactor of the solution curve y = 0, since the system is of degree 2, it must be of the form K̄1 = k0 (x) + k1 (x)y. Hence we have the equation ∂(y − y1 ) ∂(y − y1 ) ẋ + ẏ = (k0 (x) + k1 (x)y)(y − y1 ) + O(x7 ), ∂x ∂y From here we obtain k0 = 2x and k1 = 2. FORMAL WEIERSTRASS NON-INTEGRABILITY CRITERION 13 For determining the cofactors of the other two solutions y = fi (x) for i = 2, 3 we use equation (17) that in this case are X (y2 (x) − y1 (x)) = K̃2 (x)(y2 (x) − y1 (x)) + O(x7 ), X (y3 (x) − y1 (x)) = K̃3 (x)(y3 (x) − y1 (x)) + O(x7 ), We do not write here the expressions of K̃2 and K̃3 due to their exten- sion but the reader can compute them straightforward. Now we study if the cofactors K̄1 , K̄2 and K̄3 satisfy (22), i.e. λ1 K̄1 + µ1 K̃2 + µ2 K̃3 = −div6 X + O(x7 ), and this equation has not solution. Hence system (24) has not an integrating factor of the form (9), this implies that system (24) is not strongly formal Weierstrass integrable. If we try to see if there is a linear combination that gives zero, then the system has the unique solution λ1 = µ2 = µ3 = 0. Therefore the system has not a first integral of the form (9). The conclusion is that system (24) is not strongly formal Weierstrass integrable in the original coordinates (x, y). However we can ask if system (24) is strongly formal Weierstrass integrable after a change of variable. The answer to this question is positive as we will see below. System (24) after doing the change of variables x √ z=√ , u = y, 2y takes the form √ u̇ = 2u2 + 2uz, ż = 1. First we rename the new variables of the form u := x and z := y. So the equation associated √ 2 to this differential system is the Bernoulli equation dx/dy = 2x + 2xy, and then its integrability is straightforward. In 2 fact an integrating factor is given by M (x, y) = e−y x2 and a first integral is ey 2 √ Z y t2 H(x, y) = + 2 e dt. x 0 Anyway we are going to apply the necessary condition of strongly for- mal Weierstrass integrability to this system. Attending to the form of the integrating factor in this case the answer must be positive. Hence consider the system of the form √ (25) ẋ = 2x2 + 2xy, ẏ = 1. 14 J. GINÉ AND J. LLIBRE Now we study if system (25) is strongly formal Weierstrass integrable. We propose a solution curve of the form y = f (x) = a0 + a1 x + a2 x2 + a3 x3 + a4 x4 + a5 x5 + · · · Substituting this solution in the ordinary differential equation Eq := ẋdy/dx − ẏ = 0 we get an infinite system of equations without any so- lution. Therefore privileging the variable y the system has no solutions curves. Next we propose a solution curve of the form x = f (y) = a0 + a1 y + a2 y 2 + a3 y 3 + a4 y 4 + a5 y 5 + · · · Substituting this solution in the first ordinary differential equation Eq := ẋdx/dy − ẋ = 0 we get an infinite system of equations. We determine the first parameters fixing certain order in the developments of f (y) and Eq in power series of the variable y. If we do that up to order 4 and we solve the finite system of equations we obtain the following solutions. 5 1) x1 = O(xq ), q √ i √ 225−15 15 −15+ √ 15 i 15− 15 2 √ 2 √ 2) x2 = − 15 2 + √ y− 15 2 y − 45 (6+ 15)y 3 + q √ √ 15 2 i 15− 15 450 2 (10 + 3 15)y 4 + O(x5 ), q √ √ q √ √ √ i 225−15 15 −15+ 15 i 15− 15 2 2 3) x3 = 15 2 + √ y+ 15 2 y − 45 (6+ 15)y 3 − q √ √ 15 2 i 15− 15 450 2 (10 + 3 15)y 4 + O(x5 ), q √ √ q √ √ √ 15+ 15 15+√ 15 i 15+ 15 2 2 4) x4 = −i 30 − y + 15 2 y − 45 (6 − 15)y 3 q √ 15 √ 2 − 450 15+2 15 (10 − 3 15)y 4 + O(x5 ), i q √ √ q √ √ √ 15+ 15 15+√ 15 i 15+ 15 2 2 5) x5 = i 30 − y − 15 2 y − 45 (6 − 15)y 3 q √ 15 2 √ i 15+ 15 + 450 2 (10 − 3 15)y 4 + O(x5 ), Next we compute their Weierstrass polynomial cofactor for the solution curve y1 through the equation ∂(x − x1 ) ∂(x − x1 ) ẋ + ẏ = (k0 (y) + k1 (y)x)(x − x1 ) + O(x5 ), ∂x ∂y √ which is K̄1 = 2x + 2y, and the cofactors of the other solutions through the equations X (xi (x) − x1 (x)) = K̃i (x)(xi (x) − x1 (x)) + O(x5 ), for i = 2, 3, 4, 5. We do not write here the expressions of these cofac- tors due to their extension. Finally we try to see if there is a linear FORMAL WEIERSTRASS NON-INTEGRABILITY CRITERION 15 combination of these cofactors equals to minus the divergence, that is, λ1 K̄1 + µ2 K̃2 + µ3 K̃3 + µ4 K̃4 + µ5 K̃5 = −div4 X + O(x5 ), p and this p system has the solution p λ 1 = −2, µ 2 = p 5/6 − 5/3, µ3 = 5/6 − 5/3, µ4 = 5/6 + 5/3 and µ5 = 5/6 + 5/3. Hence system (25) satisfies the strongly formal Weierstrass integrability condition and it can have an integrating factor of the form (9) as indeed it has. Moreover we can also study if the system has a first integral of the form (9) using the equation λ1 K̄1 + µ2 K̃2 + µ3 K̃3 + µ4 K̃4 + µ5 K̃5 = O(x5 ), and this system has the only solution λ1 = µ2 = µ3 = µ4 = µ5 = 0. Consequently system (25) has not a first integral of the form (9). Example 2. In 1944 Kukles [27] studied the following system (26) ẋ = y, ẏ = −x + Q(x, y), where Q(x, y) = a1 x2 + a2 xy + a3 y 2 + a4 x3 + a5 x2 y + a6 xy 2 + a7 y 3 , giving the conditions in order that the origin of (26) be a center. How- ever some decades later in [6, 26] was proved that the conditions were uncompleted showing that the origin of the following system has also a center. Consider the system x3 x2 y y3 (27) ẋ = y, ẏ = −x + x2 − − √ − 2y 2 + √ . 3 2 3 2 System (27) has an inverse integrating factor of the form x √ √ V (x, y) = e−x(1− 2 ) (3 2(1 − x) + x( 2 x + y))3 , and the following first integral √ y 2 (x + 1) + 2 2xy(x − 2) + 6(3x − 2) + 2x3 − 10x2 H(x, y) = √ √ · (x(y + 2x) + 3 2(1 − x))2 Z x(1− x2 ) x e + ex(1− 2 ) dx . The analyticity of this first integral around the origin implies that the origin is a center. Now we are going to apply the criterion to detect if system (24) can have a strongly formal Weierstrass P first integral. We propose a solution i curve of the form y = f (x) = i=0 ai x and substitute this solution into the differential equation Eq := ẋdy/dx − ẏ = 0 and we get an infinite system of equations. If we develop up to order 3 and we solve the finite system of equations we obtain the solutions curves. 16 J. GINÉ AND J. LLIBRE 1) y1 = ix − ix2 + O(x3 ), 2 2) y2 = −ix + ixq + O(x3 ), √ √ √ √  √ √ 3) y3,4 = 3 2 ± 3(4 − 6) − ( 2 + 3)x + 1/12 6 2 − 6 3 ± √ √ 3/2 q √  2 3(4 − 6) ∓ 10 3(4 − 6) x + O(x3 ), √ q √ √ √  √ √ 4) y5,6 = 3 2 ± 3(4 ∓ 6) + (− 2 + 3)x + 1/12 6 2 + 6 3 ± √ √ 3/2 q √  2 3(4 + 6) ∓ 10 3(4 + 6) x + O(x3 ). Now we compute the Weierstrass polynomial cofactor of the first two solutions curves. These cofactors, as the system is of degree 3 must be of the form K = k0 (x) + k1 (x)y + k2 (x)y 2 . Applying equation (11) to the solution curves y − f (x) = 0 we obtain the Weierstrass polynomial cofactors up to order 3 in the variable x √ √ √ 1) K1 =√−1/6(6i + 4(−3i + 2)x2 ) − 1/6(12 − 2ix + 2ix2 )y + 1/(3 2)y 2 + O(x3 ), √ √ √ 2) K2 =√ 1/6(6i − 4(3i + 2)x2 ) − 1/6(12 + 2ix − 2ix2 ))y + 1/(3 2)y 2 + O(x3 ), The cofactors of the other solutions must be computed through the equation (19) that in this case are   X (yi (x) − y1 (x))(yi (x) − y2 (x))   = K̃i (x) (yi (x) − y1 (x))(yi (x) − y2 (x)) + O(x3 ), for i = 3, 4, 5, 6. We do not write here the expressions of these cofactors due to their extension. From statement (b) of Theorem 3 we study if system (27) has a strongly formal Weierstrass first integral using the equation λ1 K1 + λ2 K2 + µ3 K3 + µ4 K4 + µ5 K5 + µ6 K6 = O(x3 ), FORMAL WEIERSTRASS NON-INTEGRABILITY CRITERION 17 and this system has the solution λ1 = λ2 = 0 and  √ √ p √ p √  µ5 6 10 + 5 15 − 6 24 − 6 6 − 15 4 − 6 µ6 = √ √ p √ p √ , 6 10 + 5 15 + 6 24 − 6 6 + 15 4 − 6 q q µ5 h √ √ √ √ µ3 = 24 10 + 20 15 + 4 60 − 15 6 + 5 40 − 10 6 D1 q q q √ √ √ + 129 24 − 6 6 + 316 4 − 6 + 8 4 + 6 q √ q √ q √ i + 3 6(4 + 6) − 49 10(4 + 6) − 40 15(4 + 6) , q q µ5 h √ √ √ √ µ4 = − 6 10 − 4 15 + 60 − 15 6 + 40 − 10 6 D2 q q q q √ √ √ √ − 29 24 − 6 6 − 71 4 − 6 + 4 + 6 + 6(4 + 6) q √ q √ i + 11 10(4 + 6) + 9 15(4 + 6) , √ p √ p √ where D√1 = 372 p+ 152 6 +√80 60p − 15 6 +√98 40 − 10 6 and D2 = 84 + 34 6 + 18 60 − 15 6 + 22 40 − 10 6. Consequently system (27) can have a strong formal Weierstrass first integral as indeed it has as we have seen before. 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[29] G.R. Nicklason, An Abel type cubic system Electron. J. Differential Equa- tions 2015, No. 189, 17 pp. [30] M.J. Prelle and M.F. Singer, Elementary first integrals of differential equations, Trans. Amer. Math. Soc. 279 (1983), 613–636. [31] M.F. Singer, Liouvillian first integrals of differential equations, Trans. Amer. Math. Soc. 333 (1992), 673–688. 1 Departament de Matemàtica, Inspires Research Centre, Universi- tat de Lleida, Avda. Jaume II, 69; 25001 Lleida, Catalonia, Spain Email address: gine@matematica.udl.cat 2 Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain Email address: jllibre@mat.uab.cat