This is a preprint of: “A characterization of the generalized Liénard polynomial differential systems having invariant algebraic curves”, Jaume Giné, Jaume Llibre, Chaos Solitons Fractals, vol. 158, 112075:1–4, 2022. DOI: [10.1016/j.chaos.2022.112075] A CHARACTERIZATION OF THE GENERALIZED LIÉNARD POLYNOMIAL DIFFERENTIAL SYSTEMS HAVING INVARIANT ALGEBRAIC CURVES JAUME GINÉ1 AND JAUME LLIBRE2 Abstract. The generalized Liénard polynomial differential systems are the differential systems of the form x′ = y, y ′ = −f (x)y − g(x), where f and g are polynomials. We characterize all the generalized Liénard polynomial differential systems having an invariant algebraic curve. We show that the first four higher co- efficients of the polynomial in the variable y, defining the invariant algebraic curve, determine completely the generalized Liénard polynomial differential system. This fact does not hold for arbitrary polynomial differential systems. 1. Introduction and statement of the main results In this work we study the generalized Liénard polynomial differential systems of the form (1) x′ = y, y ′ = −f (x)y − g(x), where the degrees of the polynomials f and g are m and n respectively. Let F (x, y) be a polynomial such that ∂F ∂F (2) y+ (−f (x)y − g(x)) = KF, ∂x ∂y for some polynomial K = K(x, y). Then F (x, y) = 0 is an invariant algebraic curve of the differential system (1), i.e. if an orbit of system (1) has a point on the curve F (x, y) = 0, the whole orbit is contained in this curve. The polynomial K is called as/or to be the cofactor of the invariant algebraic curve F (x, y) = 0. The knowledge of the algebraic curves of system (1) allows to study the Darboux and Liouvillian theories of integrability, see [4, 9, 22, 24] and references therein. In fact the existence of invariant algebraic curves is a measure of the integrability in such theories. Another problem is finding a bound for the degree of the irreducible invariant algebraic curves of system (1). This problem goes back to Poincaré for any polynomial differential system and it is known as the Poincaré problem for the invariant algebraic curves. The invariant algebraic curves of generalized Liénard systems (1) have been studied by several authors in function of degrees of f and g, see for instance [1, 3, 10, 11, 12, 13, 18, 25] and references therein. The determi- nation of invariant algebraic curves is also important when we study the algebraic 2010 Mathematics Subject Classification. Primary 34A05. Secondary 34C05, 37C10. Key words and phrases. Liénard polynomial differential systems, invariant algebraic curve, first integrals. 1 2 J. GINÉ AND J. LLIBRE limit cycles of such systems, see [14, 20, 21]. Several works are also devoted to the Liouville integrability of such systems, see [2, 16, 17, 19, 23]. Finally we remark that a new method to determine the invariant algebraic curves have been developed on [5, 6, 7, 8] based on the solutions of the differential system expressed in Puiseux series. In this note we study the reciprocal problem. This problem consists in given an invariant algebraic curve characterize the generalized Liénard poynomial differential systems having such an invariant algebraic curve. In the following we use the notation akj (x) to denote (aj (x))k . Theorem 1. Assume that a generalized Liénard polynomial differential system (1) with the polynomials f and g non-identically zero has an invariant algebraic curve that we write as s X (3) F (x, y) = aj (x)y s−j = 0 with a0 (x) ̸= 0 and s ≥ 2. j=0 Then the polynomials f and g are sa′3 (x) − (s − 1)a1 (x)a′2 (x) + (s − 1)a21 (x)a′1 (x) − sa2 (x)a′1 (x) f (x) = , (s − 1)a21 (x) − 2 s a2 (x) a1 (x)a2 (x)a′1 (x) + a1 (x)a′3 (x) − 2a2 (x)a′2 (x) g(x) = , (s − 1)a21 (x) − 2sa2 (x) a0 (x) is a constant and the cofactor of F (x, y) = 0 only depends on x. Theorem 1 is proved in section 2. Note that the common denominator in the expressions of f (x) and g(x) given in the statement of Theorem 1 must divide their numerators, otherwise f (x) and g(x) would not be polynomials. We remark that if a generalized Liénard polynomial differential system (1) has an invariant algebraic curve (3) the coefficients a1 (x), a2 (x) and a3 (x) determine completely such differential system. Of course this is not true for general polynomial differential systems, for instance the polynomial differential system ∂F ∂F ẋ = a(x, y)F − c(x, y) , ẏ = b(x, y)F + c(x, y) , ∂y ∂x where a, b and c are arbitrary polynomials, has F = F (x, y) = 0 as an invariant algebraic curve. Corollary 2. Under the assumptions of Theorem 1 if the common denominator of the expressions of f (x) and g(x) are zero, then the polynomials f (x) and g(x) become a′ (x) a′ (x) a1 (x)a′1 (x) f (x) = 1 , g(x) = 2 − . 2 s 2s P3 Proposition 3. Under the assumptions of Theorem 1 if ai (x) = j=1 aij xj for j = 1, 2, 3 are arbitrary polynomials such that the maximum degree of all them is 3, then the generalized Liénard polynomial differential systems having an irreducible invariant algebraic curve of degree 3 in the variable y are the following ones: GENERALIZED LIÉNARD POLYNOMIAL SYSTEMS 3 4a11 a11 (I) f (x) = , g(x) = − (a10 + a11 x), 9 27 1 F (x, y) = ((a10 + a11 x)(3a20 + a11 x(2a10 + a11 x)) 27 +9(3a20 + a11 x(2a10 + a11 x))y + 27(a10 + a11 x)y 2 + 27y 3 ). 5a11 2a11 (II) f (x) = , g(x) = − (3a10 + a11 x), 9 27 1 F (x, y) = (3a10 + a11 x)2 (a10 + 3a11 x) 81 1 + (3a10 + a11 x)(a10 + 3a11 x)y + (a10 + a11 x)y 2 + y 3 . 9 a11 2a2 (III) f (x) = , g(x) = − 11 x, 3 9 1 F (x, y) = 3 3 (27a30 − 4a11 x + 27a11 xy 2 + 27y 3 ). 27 2a311 − k − 6a11 a22 a4 + ka11 − 9a211 a22 + 18a222 (IV) f (x) = 2 , g(x) = 11 , 3(a11 − 3a22 ) 9(a211 − 3a22 ) 1 F (x, y) = (−2a311 x3 − 2kx3 + 9a11 a22 x3 + 27a22 x2 y + 27a11 xy 2 + 27y 3 ), 27 where k = (a211 − 3a22 )3/2 . Taking into account that the degrees of the polynomials f and g are m and n respectively, from Corollary 2 it follows that in this case the deg a1 = m + 1 and deg a2 ≤ max{n, m(m + 1)} + 1. Under the assumptions of Corollary 2 we note that the generalized Liénard poly- nomial differential system (1) having an invariant algebraic curve (3) already the coefficients a1 (x) and a2 (x) determine completely such differential system. P4 Proposition 4. Under the assumptions of Corollary 2 if a1 (x) = i=1 a1i xi and P4 i a2 (x) = i=1 a2i x are arbitrary polynomials such that the maximum degree of both is 4, then the generalized Liénard polynomial differential systems having an irreducible invariant algebraic curve of degree 4 in the variable y are the following ones: a11 a211 (I) f (x) = , g(x) = − x, 2 8 a411 4 a311 3 F (x, y) = − x + x y + a11 xy 3 + y 4 ; 128 16 a11 a11 (II) f (x) = , g(x) = − (a10 + a11 x), 2 8 4 a a11 1 F (x, y) = − 10 − x(4a310 + 6a210 a11 x + 4a10 a211 x2 + a311 x3 ) + (a10 + 128 128 16 a11 x)3 y + (a10 + a11 x)y 3 + y 4 ; 1 (III) f (x) = (a11 + 2a12 x + 3a13 x2 + 4a14 x3 ), 2 1 g(x) = − f (x)(a10 + a11 x + a12 x2 + a13 x3 + a14 x4 ), 4 4 J. GINÉ AND J. LLIBRE 1 1 F (x, y) = − {a10 + x(a11 + x(a12 + x(a13 + a14 x)))}4 + {a10 + x(a11 + 128 16 x(a12 + x(a13 + a14 x)))}3 y + {a10 + x(a11 + x(a12 + x(a13 + a14 x)))}y 3 + y 4 . Corollary 2 and Propositions 3 and 4 are proved in section 2. We remark that the next polynomial Liénard differential system shows that there are invariant algebraic curves of arbitrary degree in the variable y. In [15] the authors proved that the linear polynomial Liénard differential system p−1 ẏ = x − √ y, ẋ = y, p √ √ with p > 1 has the first integral H = ( p y − x)( p y + x)p . Therefore such differential system has invariant algebraic curves of degree p + 1 for all p. 2. Proofs Proof of Theorem 1. If F (x, y) = 0 is an invariant algebraic Pr curve of system (1) it must satisfy (2) with a cofactor of the form K = j=0 Kj (x)y j . From equation (2) it follows easily that r ≤ 1. If r = 1 then the coefficient of y s+1 is a′0 (x) − a0 (x)K1 (x) = 0. Since a0 (x) must be a polynomial it follows that K1 (x) = 0 and a0 (x) is a constant. Then without loss of generality we can take a0 (x) = 1, because we can divide the invariant algebraic curve by the non-zero constant a0 (x). In summary the cofactor of F (x, y) = 0 only depends on the variable x. From the coefficient of y s in equation (2) we obtain that the cofactor K0 (x) = a′1 (x)− sf (x). And from the coefficient of y s−1 in equation (2) we get that 1 ′ (a (x) − (s − 1)a1 (x)f (x) − K0 (x)a1 (x)) . g(x) = s 2 Substituting K0 (x) is the above expression we have 1 ′ (4) g(x) = (a (x) + a1 (x)f (x) − a1 (x)a′1 (x)) . s 2 The coefficient of y s−2 in equation (2) is s−1 a′3 (x) − (s − 2)a2 (x)f (x) − a1 (x)(a′2 (x) + a1 (x)f (x) − a1 (x)a′1 (x)) s = a′1 (x)a2 (x) − s a2 (x)f (x). Consequently we obtain the function f (x) stated in Theorem 1, and substituting f (x) in (4) we get the expression of g(x) stated in the theorem. □ Example 1. Consider the polynomial Liénard differential system 1 2 2 (5) ẋ = y, ẏ = x − a (x − 4)x3 − a(x2 − 1)y, 16 having the invariant algebraic curve   2 1 3 a2 x4 a2 x6 F (x, y) = y + a x − 2x y − 4 + (a2 + 1)x2 − + 2 2 16 = y 2 + a1 (x)y + a2 (x). GENERALIZED LIÉNARD POLYNOMIAL SYSTEMS 5 Substituting a1 (x) and a2 (x) in the expressions of the polynomials f and g given in the statement of Theorem 1 we obtain 1 f (x) = a(x2 − 1), g(x) = −x + a2 (x2 − 4)x3 . 16 Proof of Corollary 2. Since the denominator of the polynomial f (x) divides its nu- merator, then f (x) must vanish when a2 (x) = (s − 1)a21 (x)/(2s). This implies that s2 a21 (x)a′1 (x) − 3sa21 (x)a′1 (x) + 2a21 (x)a′1 (x) − 2s2 a′3 (x) = 0. From this equation we get (s − 2)(s − 1)a31 (x) a3 (x) = + c1 , 6s2 where c1 is an arbitrary constant. Substituting a3 (x) in f (x) we obtain N (x) (6) f (x) = D(x) where N (x) = −2s2 a2 (x)a′1 (x) + 3s2 a21 (x)a′1 (x) − 5sa21 (x)a′1 (x) + 2a21 (x)a′1 (x) − 2s2 a1 (x)a′2 (x) + 2sa1 (x)a′2 (x) and D(x) = 2s[(s − 1)a21 (x) − 2sa2 (x)]. Dividing the numerator N (x) by the denominator D(x) the quotient is a′1 (x)/2, and the remainder R(x) is equal to 2(s−1)a1 (x)(s a1 (x)a′1 (x)−a1 (x)a′1 (x)−s a′2 (x)), which is zero when the denominator D(x) is zero. Consequently the expression (6) reduces to a′1 (x) (7) f (x) = . 2 In this case g(x) takes the form  a1 (x)a′1 (x) (s − 2)(s − 1)a21 (x) + 2s2 a2 (x) − 4s2 a2 (x)a′2 (x) (8) g(x) = . 2s2 ((s − 1)a21 (x) − 2sa2 (x)) In the above expression of g(x) since the denominator divides the numerator, work- ing in the same way as we did for the polynomial f (x) we obtain that a′2 (x) a1 (x)a′1 (x) g(x) = − , s 2s and this completes the proof of the corollary. □ We observe that the generalized Liénard polynomial differential system of Ex- ample 1 does not satisfy the assumption that the common denominator of the expressions of f and g stated in Theorem 1 vanishes. So we cannot apply Corollary 2 to differential system (5). Proof of Proposition 3. Under the assumptions of the proposition we arrive to an algebraic system between the parameters of ai (x) for i = 1, 2, 3. The equation for the highest power in x which corresponds to x11 is 3a213 (4a13 a33 − a223 ) = 0. First we consider a33 = a223 /(4a13 ) with a13 ̸= 0. Then vanishing the coefficients associated to the next powers of x we obtain a32 = (2a13 a22 a23 − a12 a223 )/(4a213 ), 6 J. GINÉ AND J. LLIBRE a31 = (a213 a222 + 2a213 a21 a23 − 2a12 a13 a22 a23 + a212 a223 − a11 a13 a223 )/(4a313 ), and 1 a30 = 4 (4a313 a21 a22 − 2a12 a213 a222 + 4a313 a20 a23 − 4a12 a213 a21 a23 + 4a212 a13 a22 a23 8a13 − 4a11 a213 a22 a23 − 2a312 a223 + 4a11 a12 a13 a223 − 2a10 a213 a223 + a13 a323 ). But the solutions obtained vanishing the coefficients of next powers of x provides a reducible invariant algebraic curve F = 0. Second we consider a13 = 0. The next coefficient is 2a212 a223 = 0. The case a23 = 0 with a12 ̸= 0 implies a23 = a33 = 0 and a32 = a222 /(4a12 ), and all the invariant curves F = 0 are reducible. The case a12 = 0 implies a23 = 0. Imposing that the rest of equations vanish we obtain the polynomials f , g and F of the statement of the proposition. □ Proof of Proposition 4. This proof follows in a similar way to the proof of Propo- sition 3, but firstly imposing that f and g satisfy (7) and (8). We omit the details that the reader can compute easily. □ Acknowledgements. The author is partially supported by the Agencia Estatal de Investigación grant PID2020-113758GB-I00 and an AGAUR (Generalitat de Cata- lunya) grant number 2017SGR 1276. The second author is partially supported by the Agencia Estatal de Investigación grant PID2019-104658GB-I00, and the H2020 European Research Council grant MSCA-RISE-2017-777911. References [1] J. Chavarriga, I.A. Garcı́a, J. Llibre, H. 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Soc. 350 (1998), no. 4, 1681–1701. 1 Departament de Matemàtica, Universitat de Lleida, Avda. Jaume II, 69; 25001 Lleida, Catalonia, Spain Email address: jaume.gine@udl.cat 2 Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bel- laterra, Barcelona, Catalonia, Spain Email address: jllibre@mat.uab.cat