This is a preprint of: “On a conjecture on the integrability of Liénard systems”, Jaume Llibre, Adrian Murza, Clàudia Valls, Rend. Circ. Mat. Palermo (2), vol. 69, 209–216, 2020. DOI: [10.1007/s12215-018-00398-6] WE PROVE A CONJECTURE ON THE INTEGRABILITY OF LIÉNARD SYSTEMS JAUME LLIBRE, ADRIAN C. MURZA AND CLAUDIA VALLS Abstract. We consider the Liénard differential systems (1) ẋ = y + F (x), ẏ = x, in C where F (x) is an analytic function satisfying F (0) = 0 and 2 F ′ (0) ̸= 0. Then these systems have a strong saddle at the origin of coordinates. It has been conjecture that if such systems have an analytic first integral defined in a neighborhood of the origin, then the function F (x) is linear, i.e. F (x) = ax. Here we prove this conjecture, and show that when F (x) is linear and system (1) has an analytic first integral this is a polynomial. 1. Introduction and statement of the main results One of the main problems about the differential systems in C2 , and in particular for the Liénard differential systems (2) ẋ = y + F (x), ẏ = x, with the function F (x) analytic, is to know when they are integrable or not. If the function F satisfies F (0) = F ′ (0) = 0, then the eigenvalues of the linear part of system (2) at the singular point located at the origin of coordinates are ±1, and consequently the origin is a weak saddle. Recall that a saddle is weak if its eigenvalues are ±λ with 0 ̸= λ ∈ R, and a saddle is strong when its eigenvalues are λ1 , λ2 ∈ R, λ1 < 0 < λ2 and λ2 ̸= −λ1 . The vector field associated to the Liénard differential system (2) is ∂ ∂ X = (y + F (x)) +x . ∂x ∂y 2010 Mathematics Subject Classification. Primary: 34C07, 34C05, 34C40. Key words and phrases. Liénard system, first integral, strong saddle. 1 http://www.gsd.uab.cat 2 J. LLIBRE, A.C. MURZA AND C. VALLS We recall that the function H = H(x, y) is a first integral of system (2) in an open subset U or C2 if ∂H ∂H (3) X H = (y + F (x)) +x = 0 on the points of U . ∂x ∂y From Theorem 1 of Gasull and Giné [4] it follows the next result. Theorem 1. The Liénard analytic differential system (4) ẋ = y + F (x), ẏ = b x, with 0 ̸= b ∈ C, ∑ and F (x) = j≥2 aj xj , is locally integrable at the origin if and only if F (x) is an even polynomial (i.e. F (−x) = F (x)) . Note that in Theorem 1 as already noticed by the authors the origin is a weak saddle. Theorem 1 extends to C2 and for all non–zero complex number b the well known results on the existence of a local first integral in a neighborhood of the origin for a polynomial Liénard differential (4) in R2 having at the origin a center (i.e. b = −1 obtained by Poincaré [10]), or a weak saddle (i.e. b = 1, see [1, 11, 14]). In all the paper Z+ and Q+ denote the sets of non-negative integer numbers and non-negative rational numbers, respectively. Consider analytic differential systems in C2 of the form (5) u̇ = λ u + · · · , v̇ = −µ v + · · · , where λ and µ are non–zero complex numbers. In (5) the dots · · · denotes nonlinear terms. From Poincaré [10] and Furta [3] we know that a necessary condition for the existence of an analytic first integral in a neighborhood of the origin of system (2) is that λ/µ = p/q ∈ Q+ \ {0} with gcd(p, q) = 1. When λ and µ satisfies this condition we say that the origin is in [p : −q] resonance. A [p : −q] resonant differential system (5) after a scaling of time if necessary can be written as (6) u̇ = p u + · · · , v̇ = −q v + · · · , with p, q ∈ Z+ \ {0}. The next result follows from Theorem 4 of [5]. Theorem 2. The Liénard analytic differential system (2) with a strong saddle at the origin can be transformed into a system with a [p : −q] resonant saddle at the origin. ON A CONJECTURE ON THE INTEGRABILITY OF LIÉNARD SYSTEMS 3 The study of the existence or not of a first integral in a neighborhood of a [p : −q] resonant saddle is a difficult problem, see for instance [2, 7, 8, 12, 14] and the references quoted there. Hence Theorem 2 says that the study of the existence or not of a first integral in a neighborhood of a strong saddle for the Liénard differential system (2) is also difficult. When a planar differential system has a (local) first integral we say that it is (locally) integrable. In [4] the authors left open the following problem (see the last sentence of their paper): Open problem. We do not know if there are nonlinear integrable cases in systems (2). Later on in [5] appears explicitly the following: Conjecture. The unique integrable case of the Liénard system (2) is the linear one. We remark that this conjecture is made for Liénard analytic differ- ential systems having a strong saddle at the origin. The objective of this note is to prove the previous conjecture re- stricted to polynomial first integrals and restricted to Liénard poly- nomial differential systems (2), i.e. the function F (x) is polynomial. Thus our first main result is: Theorem 3. If a Liénard analytic differential system (2) has a local analytic first integral defined in a neighborhood of the origin, then k1 − k2 (7) a = F ′ (0) = ± √ , k1 k2 where a ̸= 0 and k1 and k2 are coprime positive integers. The proof of Theorem 3 is given in section 2. Note that since we are interested in systems that are integrable, we must have a satisfying (7). When a = F ′ (0) does not satisfy (7) the analytical integrability of the Liénard analytic differential system has been studied in [9]. Theorem 4. If a Liénard analytic differential system (2) with a as in (7) has a polynomial first integral, then the degree of the polynomial F (x) must be one, i.e, F (x) = ax, and the polynomial first integral H is {√ √ √ √ √ ( k2 x − k1 y)k1 ( k1 x + k2 y)k2 if a = (k1 − k2 )/ k1 k2 , H= √ √ √ √ √ ( k2 x + k1 y)k1 ( k1 x − k2 y)k2 if a = (k2 − k1 )/ k1 k2 . 4 J. LLIBRE, A.C. MURZA AND C. VALLS Note that Theorem 4 proves the conjecture restricted to polynomial first integrals. The next result proves the conjecture. Theorem 5. If a Liénard analytic differential system (2) with a as in (7) has an analytic first integral defined in a neighborhood of the origin, then the degree of the polynomial F (x) must be one, i.e, F (x) = ax, and the polynomial first integral H is the one given in Theorem 4. Theorem 4 is proved in section 3, while Theorem 5 is proved in section 4. 2. Proof of Theorem 3 Before proving Theorem 3 we recall the following result whose proof can be found in [3, 6, 10]. Theorem 6. Assume that the eigenvalues λ1 and λ2 of the Jacobian matrix of system (2) at the singular point (0, 0) do not satisfy the con- dition k1 λ1 + k2 λ2 = 0, k1 , k2 ∈ Z+ , k1 + k2 > 0, then system (2) has no local analytic first integral defined in a neigh- borhood of the origin. We first note that the origin is the unique singular point of system (2) and that the eigenvalues of the Jacobian matrix at this point satisfy √ √ a + a 2+4 a − a2 + 4 λ2 −aλ−1 = 0, that is λ1 = > 0, λ2 = < 0. 2 2 So we have that λ1 λ2 = −1, yielding λ2 = −1/λ1 . Moreover, since by assumptions the system has a local analytic first integral in a neigh- borhood of the origin, in view of Theorem 6 we must have that k2 k1 λ21 − k2 0 = k1 λ1 + k2 λ2 = k1 λ1 − = , λ1 λ1 with k1 , k2 ∈ Z+ such that k1 + k2 > 0. So, √ √ k2 k1 λ1 = , λ2 = − . k1 k2 Note that k1 , k2 ∈ Z+ \ {0} because λ1 and λ2 are not zero. Therefore √ √ √ √ a + a2 + 4 k2 a − a2 + 4 k1 = , =− , 2 k1 2 k2 ON A CONJECTURE ON THE INTEGRABILITY OF LIÉNARD SYSTEMS 5 or equivalently √ √ a2 + a2 + 4 + 2a a2 + 4 k2 a2 + a2 + 4 − 2a a2 + 4 k1 = , = . 4 k1 4 k2 Hence, k1 k2 √ = − a a2 + 4, k2 k1 that is √ k 2 − k12 a a2 + 4 = 2 . k1 k2 Hence k1 − k2 a=±√ , k1 , k2 ∈ Z+ \ {0}. k1 k2 Moreover, k1 and k2 are different, otherwise a = 0 which is not possible because the origin would be a weak saddle. Finally, we observe that k1 and k2 are coprime. Otherwise setting k1 = g.c.d{k1 , k2 }k̂1 and k2 = g.c.d{k1 , k2 }k̂2 we get g.c.d{k1 , k2 }(k̂1 − k̂2 ) k̂1 − k̂2 a = ±√ = ±√ . 2 (g.c.d{k1 , k2 }) k̂1 k̂2 k̂1 k̂2 This completes the proof of Theorem 3. 3. Proof of Theorem 4 Without loss of generality we may write the polynomial first integral H = H(x, y) as H = g0 (x)y n + g1 (x)y n−1 + . . . + gn−1 (x)y + gn (x), where the gi (x) for i = 0, . . . , d are polynomials, and g0 (x) is not the zero polynomial. Substituting H into (3) we get ( ) X h = (y + F (x)) g0′ y n + g1′ y n−1 + . . . + gn−1 ′ y + gn′ + ( ) x ng0 y n−1 + (n − 1)g1 y n−2 + . . . + 2gn−2 y + gn−1 = 0, where the prime denotes derivative with respect to the variable x. Now we rewrite this equality as g0′ y n+1 + (g0′ F + g1′ )y n + (g1′ F + g2′ + ng0 x)y n−1 + . . . ′ ′ +(gn−1 + gn−2 F + 3gn−3 x)y 2 ′ +(gn′ + gn−1 F + 2gn−2 x)y + (gn′ F + gn−1 x) = 0. 6 J. LLIBRE, A.C. MURZA AND C. VALLS Since all coefficients of the previous polynomial in the variable y must be zero, we get the following system of differential equations g0′ = 0, g1′ = 0, g2′ −F g2′ − g3′ g0 = − , g1 = , nx (n − 1)x −F g3′ − g4′ −F g4′ − g5′ (8) g2 = , g3 = , (n − 2)x (n − 3)x .. .. . . ′ −F gn−1 − gn′ −F gn′ gn−2 = , gn−1 = . 2x x From the first two equations of (8) we get that g0 and g1 are con- stants, and additionally by assumptions we have that g0 ̸= 0. From the third equation of (8) we obtain that g2 (x) is a polynomial of degree 2. From the fourth equation, since g1 , g2 and g3 are polynomials we get that F must be a polynomial. Assume that the degree of the polynomial F is d ≥ 1, then from the fourth equation of (8) it follows that the degree of the polynomial g3 is d + 2. Now from the fifth equation of (8) we get that the degree of the polynomial g4 is 2d + 2, and from the sixth we obtain that the degree of the polynomial g5 is 3d + 2. Thus recursively we have that the degree of the polynomial gk for k = 2, . . . , n is (k − 2)d + 2. From the last equation of (8) we obtain that the degree 1 + (n − 3)d + 2 of the polynomial xgn−1 must be equal to the degree d + (n − 2)d + 1 of the polynomial F gn′ , but this equality is only possible if d = 1. It is easy to check that the Liénard analytic differential system (2) of degree 1, i.e. ẋ = y + ax, ẏ = x, with a as in (7) has the polynomial first integral H as in the statement of the theorem. This completes the proof of Theorem 4. 4. Proof of Theorem 5 Consider system (2) with one of the conditions given by Theorem √ 3, namely the coefficient a of x in F (x) is equal to (k1 − k2 )/ k1 k2 (the ON A CONJECTURE ON THE INTEGRABILITY OF LIÉNARD SYSTEMS 7 √ case in which a = −(k1 − k2 )/ k1 k2 follows in the same way). Then ∑ ∞ k1 − k2 F (x) = √ x+ aj x j . k1 k2 j=2 If aj = 0 for j ≥ 2 it follows from Theorem 4 that system (2) has a polynomial first integral. Therefore we assume, first that aj ̸= 0 for some j ≥ 2, and second that system (2) has an analytic first inte- gral H defined in a neighborhood of the origin, and we will reach a contradiction. Under the assumptions on F we have ∑ ∞ k1 − k2 ẋ = y + √ x+ aj x j , (9) k1 k2 j=2 ẏ = x. Making the change of variables √ √ √ √ (10) u = k1 x + k2 y, v= k2 x − k1 y, with inverse change √ √ √ √ k1 u + k2 v k2 u − k1 v x= , y= k1 + k2 k1 + k2 √ and the rescaling of the time t = k1 k2 T , we have that system (9) becomes √ ∑ ∞ ( √ k u + √ k v )j ′ 1 2 u = k1 u + k1 k2 aj , j=2 k1 + k2 (11) ∞ ( √ k u + √ k v )j √ ∑ 1 2 ′ v = −k2 v + k2 k1 aj , j=2 k1 + k2 where the prime denotes derivative in the new variable T . If k1 > k2 (and so k1 > 1), we change from the variables (u, v) to the variables (u, z) where z = uk2 v k1 and so v = z 1/k1 u−k2 /k1 . If k2 > k1 (and so k2 > 1), we change from the variables (u, v) to (z, v) where z = uk2 v k1 and so u = z 1/k2 v −k1 /k2 . From now on we assume that k1 > k2 because the other case is done in a similar manner. Hence we take (12) z = uk2 v k1 that is v = z 1/k1 u−k2 /k1 . 8 J. LLIBRE, A.C. MURZA AND C. VALLS Then from (11) we have √ ∑ ∞ ( √k u + √k z 1/k1 u−k2 /k1 )j ′ 1 2 u = k1 u + k1 k2 aj , j=2 k1 + k 2 √ √ ′ (k2 −k1 )/k1 (k1 −1)/k1 (13) z = k1 k2 u z ( k1 u + k2 z 1/k1 u−k2 /k1 ) ∑∞ ( √k u + √k z 1/k1 u−k2 /k1 )j 1 2 · aj . j=2 k 1 + k2 We write H(x, y) as a formal first integral of system (9). Then Ĥ(u, v) = H(x, y) is a formal first integral of system (11) and H̃(u, z) = Ĥ(u, v) is ∑ a formal first integral of system (13). Writing Ĥ(u, v) = j≥0 Hj (u)v j with Hj a formal series in u, we can write H̃(u, z) as ∑ H̃ = H̃(u, z) = H̃j (u)z j/k1 , j≥0 where H̃j (u) = Hj (u)u−jk2 /k1 . Since H̃ is a first integral we can assume that it has no constant term. Note that H̃ satisfies ∂ H̃ ∂ H̃ (14) u′ + z′ = 0, ∂u ∂z with (u′ , z ′ ) as in (13). We will show by induction that (15) H̃j (u) = 0 for j ≥ 0. Note that to conclude the proof of the theorem it is enough to show that (15) holds, because in this case we reach to a contradiction. First note that equation (14) restricted to z = 0 becomes ( √ ∑ ∞ ( √ k u )j ) 1 k1 u + k1 k2 aj H̃0′ (u) = 0, j=2 k1 + k2 where the prime denotes derivative with respect to the variable u. Thus H̃0 is a constant. Since H̃0 has no constant terms we get H̃0 = 0. This proves (15) for j = 0. We assume that (15) is satisfied for j = 0, . . . , n − 1 with n ≥ 1 and we shall prove it for j = n. By the induction hypothesis we have that ∑ H̃ = H̃j+n (u)z (j+n)/k1 = z n/k1 g(u, z), j≥0 ON A CONJECTURE ON THE INTEGRABILITY OF LIÉNARD SYSTEMS 9 with g(u, 0) = H̃n (u). Now after simplifying equation (14) by z n/k1 , and after restricting to z = 0, equation (14) becomes ( √ ∑ ∞ ( √ k u )j ) 1 k1 u + k1 k2 aj H̃n′ (u) = 0. j=2 k1 + k2 Therefore H̃n (u) = 0. This proves (15) for j = n. In short, the theorem is proved. Acknowledgements The first author is partially supported by a FEDER-MINECO grant MTM2016-77278-P, a MINECO grant MTM2013-40998-P, and an AGAUR grant 2014SGR-568. 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Liu, Integrability and generalized center problem of resonant singular point, Appl. Math. Lett. 40 (2015), 13–16. [13] D.M. Zhu, Saddle values and integrability conditions of quadratic differential systems A Chinese summary appears in Chinese Ann. Math. Ser. A 8 (1987), no. 5, 645. Chinese Ann. Math. Ser. B 8 (1987), no. 4, 466–478. [14] H. Zoladek, The problem of center for resonant singular points of polynomial vector fields, J. Differential Equations 137 (1997), no. 1, 94–118. Jaume Llibre, Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain E-mail address: jllibre@mat.uab.cat Adrian C. Murza, Institute of Mathematics “Simion Stoilow” of the Romanian Academy, Calea Griviţei 21, 010702 Bucharest, Romania E-mail address: adrian murza@hotmail.com Claudia Valls, Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais 1049–001, Lisboa, Portugal E-mail address: cvalls@math.ist.utl.pt