mathematics Article Invariant Algebraic Curves of Generalized Liénard Polynomial Differential Systems Jaume Giné 1, * and Jaume Llibre 2 1 Departament de Matemàtica, Universitat de Lleida, Avda. Jaume II, 69, 25001 Lleida, Catalonia, Spain 2 Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Catalonia, Spain; jllibre@mat.uab.cat * Correspondence: jaume.gine@udl.cat Abstract: In this study, we focus on invariant algebraic curves of generalized Liénard polynomial differential systems x 0 = y, y0 = − f m ( x )y − gn ( x ), where the degrees of the polynomials f and g are m and n, respectively, and we correct some results previously stated. Keywords: Liénard differential systems; invariant algebraic curve; first integrals MSC: Primary 34A05; Secondary 34C05; 37C10 1. Introduction and Statement of the Main Results In this work, we study the generalized Liénard polynomial differential systems of the following form: x 0 = y, y 0 = − f m ( x ) y − gn ( x ), (1) where the degrees of the polynomials f and g are given by the subscripts m and n, re-   spectively. These generalized Liénard systems are used to model different problems in Citation: Giné, J.; Llibre, J. Invariant numerous areas of knowledge and have been intensively studied in the last decades (see Algebraic Curves of Generalized for instance [1,2] and references therein). Liénard Polynomial Differential Consider F ( x, y) = 0 an invariant algebraic curve of the differential system (1) where F ( x, y) Systems. Mathematics 2022, 10, 209. is a polynomial, then there exists a polynomial K ( x, y) such that the following is the case. https://doi.org/10.3390/ math10020209 ∂F ∂F y+ (− f m ( x )y − gn ( x )) = KF (2) ∂x ∂y Academic Editor: Ioannis G. Stratis Received: 16 December 2021 The knowledge of the algebraic curves of system (1) allows studying modern Darboux Accepted: 5 January 2022 and Liouvillian theories of integrability (see [3] and references therein). In fact the existence of Published: 10 January 2022 invariant algebraic curves is a measure of integrability in such theories. Another problem is finding a bound on the degree of irreducible invariant algebraic curves of system (1). This Publisher’s Note: MDPI stays neutral problem goes back to Poincaré for any differential system and is known as Poincaré problem. with regard to jurisdictional claims in In 1996, Hayashi [4] stated the following result. published maps and institutional affil- iations. Theorem 1. The generalized Liénard polynomial differential system (1) with f m 6≡ 0 and m + 1 ≥ n has an invariant algebraic curve if and only if there is an invariant curve y − P( x ) = 0 satisfying gn ( x ) = −( f m ( x ) + P 0 ( x )) R Px ( x ), where P( x ) or P( x ) + F ( x ) is a polynomial with a degree of at Copyright: © 2022 by the authors. most one, such that F ( x ) = 0 f (s)ds. Licensee MDPI, Basel, Switzerland. This article is an open access article Given P and Q polynomials, an algebraic curve of the form (y + P( x ))2 − Q( x ) = 0 distributed under the terms and is called hyperelliptic curve (see for instance [5–8]). In such works, hyperelliptic curves are conditions of the Creative Commons used to determine the algebraic limit cycles of generalized Liénard systems (1). Attribution (CC BY) license (https:// Theorem 1 is also announced in [9], where the author seems to not be aware that creativecommons.org/licenses/by/ the theorem is false. Theorem 1 is not correct as the following proposition shows. More 4.0/). Mathematics 2022, 10, 209. https://doi.org/10.3390/math10020209 https://www.mdpi.com/journal/mathematics Mathematics 2022, 10, 209 2 of 5 precisely, it shows the existence of hyperelliptic invariant algebraic curves for generalized Liénard systems (1). Proposition 1. Under the assumptions of Theorem 1, the generalized Liénard polynomial differen- tial system (1) has the following hyperelliptic invariant algebraic curves: (a) F ( x, y) = −(b + ax )λ + (y − b − ax )2 = 0 for f 0 ( x ) = −3a/2 and g1 ( x ) = a(b + ax − λ)/2 with a 6= 0; (b) F ( x, y) = − Ax2 + (y − ax )2 for f 0 ( x ) = −2a and g1 ( x ) = ( a2 − A) x with aA 6= 0; (c) F ( x, y) = −bc/(2a) − cx − acx2 /(2b) + (b + ax − y)2 = 0 for f 0 ( x ) = −2a and g1 ( x ) = (2ab − c)(b + ax )/(2b) with ab 6= 0. Proposition 1 is proved in Section 2. In fact, the correct statement of Theorem 1 is the following. Theorem 2. The generalized Liénard polynomial differential system (1) with f m 6≡ 0 and m + 1 ≥ n has the invariant algebraic curve y − P( x ) = 0 if gn ( x ) = −( f m (Zx ) + P 0 ( x )) P( x ), being P( x ) x or P( x ) + F ( x ) a polynomial of degree at most one, where F ( x ) = f (s)ds. 0 Theorem 2 is proved in Section 2. Note that the mistake in the statement of Theorem 1 is the claim that unique invariant algebraic curves are of the following form y − P( x ) = 0. Demina in [10] also detected that Theorem 1 was not correct. She found counterexam- ples to Theorem 1 with invariant algebraic curves of degree 2 and 3 in the variable y. Singer in [11] found the characterization of systems that are Liouvillian integrable. Christopher [12] rewrote this result stating that if a polynomial differential system in R2 has an inverse integrating factor of the following form:   p D ∏ Fi i , α V = exp (3) E i =1 where D, E and Fi are polynomials in C[ x, y] and αi ∈ C, then this differential system is Liouvillian integrable. For a definition of (inverse) integrating factor, see for instance Section 8.3 of [3]. We say that exp( g/h), with g and h ∈ C[ x, y], is an exponential factor of the polynomial differential system (1) if there exists a polynomial L( x, y) of a degree with at most d where d = max{m, n − 1} such that the following is the case. ∂ exp( g/h) ∂ exp( g/h) y+ (− f m ( x )y − gn ( x )) = K exp( g/h). ∂x ∂y More information on exponential factors can be found in Section 8.5 of [3]. The existence of an inverse integrating factor (3) for a polyomial differential system p in R2 is equivalent to the existence of λi and µi ∈ C is not all zero such that ∑i=1 λi Ki + q ∑ j=1 µ j L j = div( P, Q), where Ki and Li are the cofactors of some invariant algebraic curves and exponential factors of the given polynomial differential system, respectively. See, for more details, statement (iv) of Theorem 8.7 of [3]. We remark that the two kinds of invariant algebraic curves mentioned in Theorem 2 can appear simultaneously in some generalized Liénard polynomial differential systems (1) as the following example shows, which already appeared in [13]. The generalized polynomial Liénard differential system of the following: x 0 = y, y0 = −ex3 − e2 /3x − (3x2 + 4e/3)y, (4) Mathematics 2022, 10, 209 3 of 5 has invariant algebraic curves f 1 = y + ex/3 = 0 and f 2 = y + x3 + ex/3 = 0. Moreover, system (4) is Liouvillian integrable because it has an inverse integrating factor V = f 1 f 21/3 . Let U be an open subset of R2 . A C1 function H : U → R is a first integral of system (1) if it is constant on the orbits of the system contained in U, or equivalently if the following is the case. dH ∂H ∂H = y+ (− f m ( x )y − gn ( x )) = 0 on U. (5) dt ∂x ∂y Consider W as an open subset of R2 × R. A C1 function I : W → R is a Darboux invariant of system (1) if it is constant on the orbits of the system contained in W, or equivalently if the following is the case. dI ∂I ∂I + y + (− f m ( x )y − gn ( x )) = 0 on W. (6) dt ∂x ∂y p q Moreover, given λi and µi ∈ C that is not all zero such that ∑i=1 λi Ki + ∑ j=1 µ j L j = −s for some s ∈ C \ {0}, then the (multivalued) function of the following: !!µ j p q gj ∏ ∏ α I= Fi i exp exp(st) (7) i =1 j =1 hj is a Darboux invariant of the differential system (see for more details statement (vi) of Theorem 8.7 of [3]). Under the assumptions of Theorem 2, there are generalized Liénard polynomial differential systems (1) that are Liouvillian integrable, as it is shown in the next result. Proposition 2. Under the assumptions of Theorem 2, if the generalized Liénard polynomial differ- ential system (1) has an invariant algebraic curve y − P( x ) = 0, then the following statements hold: (a) If P( x ) = − F ( x ) + ax + b, then system (1) has the Darboux invariant (y − P( x ))e at ; (b) If P( x ) = b, then system (1) is Liouvillian integrable with the first integral H = ey+ F( x) (y − b)b if b 6= 0, and the first integral H = y + F ( x ) if b = 0. Proposition 2 is proved in Section 2. We note that Proposition 2 shows that Theorem 2 of [13] and Theorem 4 of [14] are not correct because their proofs are based on the incorrect Theorem 1. Proposition 3. Consider the generalized Liénard polynomial differential system (1). Let P( x ) be a polynomial, then y − P( x ) = 0 is an invariant algebraic curve of system (1) if and only if gn ( x ) = −( f m ( x ) + P 0 ( x )) P( x ). Proposition 3 is proved in Section 2. In fact, the statement of Proposition 3 already appears in [8] without proof. Note that, in Proposition 3, there are no restrictions on the degrees of the polynomials f m , gn and P( x ). The Liouvillian integrability of the generalized Liénard polynomial differential system has been studied by several authors. The main result of [15] is that under restriction 2 ≤ n ≤ m, system (1) has a Liouvillian first integral if and only if gn ( x ) = a f m ( x ), where a ∈ C (see also [16] for a shorter proof). Later on, the Liouvillian integrability of differential systems (1) having hyperelliptic curves of the form (y + Q( x ) P( x ))2 − Q( x )2 = 0 was studied (see [17]). In summary, the Liouvillian integrability in the case where n > m is still open. In fact, the characterization of the invariant algebraic curves of system (1) for this case is not complete. Recently, cases m = 1 and n = 2 have been solved (see [18]). Case n = m + 1 is the still the objective of several recent works. Thus, for instance in [10,19], some particular cases for m = 2 and n = 3 have been solved. Mathematics 2022, 10, 209 4 of 5 2. Proofs Proof of Proposition 1. Assume that system (1) has an hyperelliptic invariant curve F = (y + P( x ))2 − Q( x ) = 0. Then, from (2), denoting by K = K ( x, y) the cofactor of F = 0, we obtain the following. 2gn ( x ) P( x ) + K (− P( x )2 + Q( x )) + y(−2gn ( x ) + 2P( x )(K + f m ( x ) + P 0 ( x )) − Q 0 ( x )) − y2 (K + 2 f m ( x ) + 2P 0 ( x )) = 0. From this equality, we observe that K = K ( x ) = −2( f m ( x ) + P 0 ( x )): P( x ) Q 0 ( x ) 1 P2 ( x ) Q 0 ( x ) f m (x) = −P 0 (x) − , and gn ( x ) = − Q 0 ( x ) + , 2Q( x ) 2 2Q( x ) where f m( x ) and gn ( x ) must be polynomials. If we assume that deg P = p and deg Q = q, we obtain deg f m = p − 1 and deg gn = max{q − 1, p2 − 1}. Since m + 1 ≥ n, we obtain p ≥ max{q − 1, p2 − 1}, which implies p = 1. Consequently, 1 ≥ q − 1, which implies q = 1, 2. If q = 1, then P( x ) = ax + b with a 6= 0 and Q( x ) must be proportional to P( x ); that is, Q( x ) = λP( x ). Thus, f m = −3a/2 and gn = a( ax + b − λ)/2, and F = (b + ax − y)2 − (b + ax )λ. Thus, statement (a) follows. If q = 2, then we have P( x ) = ax + b and Q( x ) = Ax2 + Bx + C with aA 6= 0, and since f m must be a polynomial, we obtain f m = −2a; moreover, in order for gm to be a polynomial, we obtain either b = B = C = 0 or A = aB/(2b) and C = (bB)/(2a) with ab 6= 0. If b = B = C = 0, then gn = ( a2 − A) x and F = − Ax2 + (y − ax )2 . Therefore, statement (b) is proven. If A = aB/(2b) and C = (bB)/(2a), then g( x ) = (2ab − B)(b + ax )/(2b) and F ( x, y) = −bB/(2a) − Bx − aBx2 /(2b) + (b + ax − y)2 . By renaming B by c, we obtain statement (c). Proof of Proposition 2. We have system (1) with gn ( x ) = −( f m ( x ) + P0 ( x )) P( x ) and the invariant algebraic curve y − P( x ) = 0 is P( x ) = − F ( x ) + ax + b. Then, by using Equation (2), we obtain the result where the cofactor of the invariant algebraic curve y − P( x ) = 0 is K = − a. Consequently, system (1) has the Darboux invariant (7), which in our case becomes I = (y − P( x ))e at . Hence, statement (a) is proved. Assume now that P( x ) = b. Therefore, g( x ) = −b f ( x ), and the differential system becomes ẋ = y and ẋ = −(y + b) f ( x ), which has the Darboux first integral H = ey+ F( x) (y − b)b if b 6= 0, and the Darboux first integral H = y + F ( x ) if b = 0, as it is easy to verify using (6). Proof of Proposition 3. First, we suppose that gn ( x ) = −( f m ( x ) + P 0 ( x )) P( x ), and we shall prove that y − P( x ) = 0 is an invariant algebraic curve. From Equation (7), we have the following. − P 0 ( x )y − y f m ( x ) − gn ( x ) = K (y − P( x )). By substituting gn ( x ), we obtain the following. − P 0 ( x )(y − P( x )) − f m ( x )(y − P( x )) = K (y − P( x )). Dividing the previous equality by y − P( x ), we obtain K = − P 0 ( x ) − f m ( x ), which is a cofactor of degree p − 1 + m of system (1). Note that the degree of the polynomial Liénard differential system is the degree of, i.e., the maximum of m + p and 2p − 1. Now, we assume that y − P( x ) is an invariant algebraic curve of system (1) with cofactor K. Then, from (7), we obtain − P 0 ( x )y − y f m ( x ) − gn ( x ) = K (y − P( x )). Mathematics 2022, 10, 209 5 of 5 From this equality, we obtain K = K ( x ); then, we have (K ( x ) + f m ( x ) + P0 ( x ))y = K ( x ) P( x ) − gn ( x ). Therefore, K ( x ) = −( f m ( x ) + P0 ( x )) and gn ( x ) = K ( x ) P( x ). Hence, gn ( x ) = −( f m ( x ) + P 0 ( x )) P( x ), and the proposition is proved. Proof of Theorem 2. By Proposition 3, we only need to prove that P( x ) or P( x ) + F ( x ) are polynomials with a degree of at most one. Since m + 1 ≥ n and n are the maxima of m + p and 2p − 1 where p is the degree of the polynomial P( x ), we have m + 1 ≥ m + p; consequently, p ≤ 1, and the theorem is proved. Author Contributions: Conceptualization, J.G. and J.L.; Formal analysis, J.G. and J.L.; Investigation, J.G. and J.L.; Methodology, J.G. and J.L.; Writing—original draft, J.G. and J.L.; Writing—review & editing, J.G. and J.L. All authors have read and agreed to the published version of the manuscript. Funding: The author is partially supported by the Agencia Estatal de Investigación grant PID2020- 113758GB-I00 and an AGAUR (Generalitat de Catalunya), 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. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Not applicable. Acknowledgments: The authors are grateful to the referees for their valuable comments and sugges- tions to improve this paper. Conflicts of Interest: The authors declare no conflict of interest. References 1. Fercec, B.; Giné, J. Formal Weierstrass integrability for a Liénard differential system. J. Math. Anal. Appl. 2021, 499, 14. [CrossRef] 2. Giné, J. Liénard equation and its generalizations. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 2017, 27, 1750081. [CrossRef] 3. Dumortier, F.; Llibre, J.; Artés, J.C. Qualitative Theory of Planar Differential Systems; UniversiText; Springer: New York, NY, USA, 2006. 4. Hayashi, M. On polynomial Liénard systems which have invariant algebraic curves. Funkc. Ekvacioj 1996, 39, 403–408. 5. Liu, C.; Chen, G.; Yang, J. On the hyperelliptic limit cycles of Liéenard systems. Nonlinearity 2012, 25, 1601–1611. 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