Chaos, Solitons and Fractals 158 (2022) 112075 Contents lists available at ScienceDirect Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos A characterization of the generalized Liénard polynomial differential systems having invariant algebraic curves Jaume Giné a,⁎, Jaume Llibre b a Departament de Matemàtica, Universitat de Lleida, Avda. Jaume II, 69, 25001 Lleida, Catalonia, Spain b Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain a r t i c l e i n f o a b s t r a c t Article history: The generalized Liénard polynomial differential systems are the differential systems of the form x′ = y, y′ = Received 13 December 2021 − f(x)y − g(x), where f and g are polynomials. Received in revised form 10 March 2022 We characterize all the generalized Liénard polynomial differential systems having an invariant algebraic Accepted 31 March 2022 curve. We show that the first four higher coefficients of the polynomial in the variable y, defining the invari- Available online 25 April 2022 ant algebraic curve, determine completely the generalized Liénard polynomial differential system. This fact Keywords: does not hold for arbitrary polynomial differential systems. Liénard polynomial differential systems 2010 mathematics subject classification: Primary 34A05. Secondary 34C05, 37C10. Invariant algebraic curve © 2022 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http:// First integrals creativecommons.org/licenses/by-nc-nd/4.0/). 1. Introduction and statement of the main results systems (1) have been studied by several authors in function of degrees of f and g, see for instance [5–12] and references therein. The determina- In this work we study the generalized Liénard polynomial differen- tion of invariant algebraic curves is also important when we study the tial systems of the form algebraic limit cycles of such systems, see [13–15]. Several works are also devoted to the Liouville integrability of such systems, see [16–20]. x0 ¼ y; y0 ¼ −f ðxÞy−g ðxÞ; ð1Þ Finally we remark that a new method to determine the invariant alge- braic curves have been developed on [21–24] based on the solutions where the degrees of the polynomials f and g are m and n respectively. of the differential system expressed in Puiseux series. In this note we Let F(x, y) be a polynomial such that study the reciprocal problem. This problem consists in given an invari- ∂F ∂F ant algebraic curve characterize the generalized Liénard poynomial dif- y þ ð−f ðxÞy−g ðxÞÞ ¼ KF; ð2Þ ferential systems having such an invariant algebraic curve. ∂x ∂y In the following we use the notation akj (x) to denote (aj(x))k. for some polynomial K = K(x, y). Then F(x, y) = 0 is an invariant alge- Theorem 1. Assume that a generalized Liénard polynomial differential braic curve of the differential system (1), i.e. if an orbit of system system (1) with the polynomials f and g non-identically zero has an invari- (1) has a point on the curve F(x, y) = 0, the whole orbit is contained ant algebraic curve that we write as in this curve. The polynomial K is called as/or to be the cofactor of the in- variant algebraic curve F(x, y) = 0. X s The knowledge of the algebraic curves of system (1) allows to study F ðx; yÞ ¼ a j ðxÞys− j ¼ 0 with a0 ðxÞ ≠ 0 and s ≥ 2: ð3Þ j¼0 the Darboux and Liouvillian theories of integrability, see [1–4] and refer- ences therein. In fact the existence of invariant algebraic curves is a Then the polynomials f and g are measure of the integrability in such theories. Another problem is finding a bound for the degree of the irreducible invariant algebraic curves of sa30 ðxÞ−ðs−1Þa1 ðxÞa20 ðxÞ þ ðs−1Þa21 ðxÞa10 ðxÞ−sa2 ðxÞa10 ðxÞ system (1). This problem goes back to Poincaré for any polynomial dif- f ðxÞ ¼ ; ðs−1Þa21 ðxÞ−2 sa2 ðxÞ ferential system and it is known as the Poincaré problem for the invariant a1 ðxÞa2 ðxÞa10 ðxÞ þ a1 ðxÞa30 ðxÞ−2a2 ðxÞa20 ðxÞ algebraic curves. The invariant algebraic curves of generalized Liénard g ð xÞ ¼ ; ðs−1Þa21 ðxÞ−2sa2 ðxÞ a0(x) is a constant and the cofactor of F(x, y) = 0 only depends on x. ⁎ Corresponding author. E-mail addresses: jaume.gine@udl.cat (J. Giné), jllibre@mat.uab.cat (J. Llibre). Theorem 1 is proved in Section 2. https://doi.org/10.1016/j.chaos.2022.112075 0960-0779/© 2022 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). J. Giné and J. Llibre Chaos, Solitons and Fractals 158 (2022) 112075 Note that the common denominator in the expressions of f(x) and systems having an irreducible invariant algebraic curve of degree 4 in the g(x) given in the statement of Theorem 1 must divide their numera- variable y are the following ones: tors, otherwise f(x) and g(x) would not be polynomials. a11 a2 We remark that if a generalized Liénard polynomial differential sys- f ðxÞ ¼ ;g ðxÞ ¼ − 11 x; tem (1) has an invariant algebraic curve (3) the coefficients a1(x), a2(x) 2 8 I a411 4 a311 3 and a3(x) determine completely such differential system. Of course this F ðx; yÞ ¼ − x þ x y þ a11 xy3 þ y4 128 16 is not true for general polynomial differential systems, for instance the polynomial differential system a11 a11 II f ðxÞ ¼ ; gðxÞ ¼ − ða10 þ a11 xÞ; 2 8 ∂F ∂F a410 a11 x ¼ aðx; yÞF−cðx; yÞ ; y ¼ bðx; yÞF þ cðx; yÞ ; Fðx; yÞ ¼ − − xð4a310 þ 6a210 a11 x þ 4a10 a211 x2 þ a311 x3 Þ ∂y ∂x 128 128 3 1 þ ða10 þ a11 xÞ y þ ða10 þ a11 xÞy3 þ y4 where a, b and c are arbitrary polynomials, has F = F(x, y) = 0 as an 16 invariant algebraic curve. 1 III f ðxÞ ¼ ða11 þ 2a12 x þ 3a13 x2 þ 4a14 x3 Þ 2 Corollary 2. Under the assumptions of Theorem 1 if the common denom- 1 inator of the expressions of f(x) and g(x) are zero, then the polynomials f(x) gðxÞ ¼ − f ðxÞða10 þ a11 x þ a12 x2 þ a13 x3 þ a14 x4 Þ; 4 f g and g(x) become 1 4 1 Fðx; yÞ ¼ − a10 þ xða11 þ xða12 þ xða13 þ a14 xÞÞÞ þ a10 128 16 a10 ðxÞ a20 ðxÞ a1 ðxÞa10 ðxÞ f ðxÞ ¼ ; g ðxÞ ¼ − : þxða11 þ xða12 þ xða13 þ a14 xÞÞÞ3 y 2 s 2s Proposition 3. Under the assumptions of Theorem 1 if ai(x) = ∑3j=1aijxj f g þ a10 þ xða11 þ xða12 þ xða13 þ a14 xÞÞÞ y3 þ y4 : 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 Corollary 2 and Propositions 3 and 4 are proved in Section 2. having an irreducible invariant algebraic curve of degree 3 in the variable We remark that the next polynomial Liénard differential system y are the following ones: shows that there are invariant algebraic curves of arbitrary degree in the variable y. In [25] the authors proved that the linear polynomial Liénard differential system 4a11 a11 I f ðxÞ ¼ ; g ðxÞ ¼ − ða10 þ a11 xÞ; 9 27 p−1 1  x ¼ y; y ¼ x− pffiffiffi y; F ðx; yÞ ¼ ða10 þ a11 xÞð3a20 þ a11 xð2a10 þ a11 xÞÞ p 27 þ9ð3a20 þ a11 xð2a10 þ a11 xÞÞy pffiffiffi pffiffiffi with p N 1 has the first integral H ¼ ð p y−xÞð p y þ xÞp . Therefore such 2 3 þ27ða10 þ a11 xÞy þ 27y Þ: differential system has invariant algebraic curves of degree p + 1 for all p. 5a11 2a11 II f ðxÞ ¼ ; gðxÞ ¼ − ð3a10 þ a11 xÞ; 2. Proofs 9 27 2 1 Fðx; yÞ ¼ ð3a10 þ a11 xÞ ða10 þ 3a11 xÞ Proof of Theorem 1. If F(x, y) = 0 is an invariant algebraic curve of sys- 81 1 tem (1) it must satisfy (2) with a cofactor of the form K = ∑rj=0Kj(x)yj. þ ð3a10 þ a11 xÞða10 þ 3a11 xÞy þ ða10 þ a11 xÞy2 þ y3 : From Eq. (2) it follows easily that r ≤ 1. If r = 1 then the coefficient of 9 ys+1 is a0′(x) − a0(x)K1(x) = 0. Since a0(x) must be a polynomial it a11 2a2 follows that K1(x) = 0 and a0(x) is a constant. Then without loss of III f ðxÞ ¼ ; gðxÞ ¼ − 11 x; 3 9 generality we can take a0(x) = 1, because we can divide the invariant 1 algebraic curve by the non-zero constant a0(x). In summary the Fðx; yÞ ¼ ð27a30 −4a11 x þ 27a11 xy2 þ 27y3 Þ: 3 3 27 cofactor of F(x, y) = 0 only depends on the variable x. 2a311 −k−6a11 a22 a411 þ ka11 −9a211 a22 þ 18a222 From the coefficient of ys in Eq. (2) we obtain that the cofactor IV f ðxÞ ¼ Þ; gðxÞ ¼ Þ; 3ða211 −3a22 9ða211 −3a22 K 0 (x) = a 1′ (x) − sf(x). And from the coefficient of y s−1 in Eq. (2) 1 we get that F ðx; yÞ ¼ ð−2a311 x3 −2kx3 þ 9a11 a22 x3 þ 27a22 x2 y þ 27a11 xy2 þ 27y3 Þ; where 27 k ¼ ða211 −3a22 Þ 3=2 : 1 g ðxÞ ¼ ða 0 ðxÞ−ðs−1Þa1 ðxÞf ðxÞ−K 0 ðxÞa1 ðxÞÞ: s 2 Substituting K0(x) is the above expression we have 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 1 a1 = m + 1 and deg a2 ≤ max {n, m(m + 1)} + 1. g ðxÞ ¼ ða 0 ðxÞ þ a1 ðxÞ f ðxÞ−a1 ðxÞa10 ðxÞÞ: ð4Þ s 2 Under the assumptions of Corollary 2 we note that the generalized Liénard polynomial differential system (1) having an invariant algebraic The coefficient of ys−2 in Eq. (2) is curve (3) already the coefficients a1(x) and a2(x) determine completely such differential system. s−1 Proposition 4. Under the assumptions of Corollary 2 if a1(x) = ∑4i=1a1ixi a30 ðxÞ−ðs−2Þa2 ðxÞf ðxÞ − a1 ðxÞða20 ðxÞ þ a1 ðxÞf ðxÞ−a1 ðxÞa1 0 ðxÞÞ s and a2(x) = ∑4i=1a2ixi are arbitrary polynomials such that the maximum degree of both is 4, then the generalized Liénard polynomial differential ¼ a10 ðxÞa2 ðxÞ−sa2 ðxÞ f ðxÞ: 2 J. Giné and J. Llibre Chaos, Solitons and Fractals 158 (2022) 112075 Consequently we obtain the function f(x) stated in Theorem 1, and i = 1, 2, 3. The equation for the highest power in x which corresponds substituting f(x) in (4) we get the expression of g(x) stated in the to x11 is 3a213(4a13a33 − a223) = 0. theorem. □ First we consider a33 = a223/(4a13) with a13 ≠ 0. Then vanishing the coefficients associated to the next powers of x we obtain Example 1. Consider the polynomial Liénard differential system a32=(2a13a22a23 − a12a223)/(4a213), a31 = (a213a222 + 2a213a21a23 − 1     2a12a13a22a23 + a212a223 − a11a13a223)/(4a313), and x ¼ y; y ¼ x− a2 x2 −4 x3 −a x2 −1 y; ð5Þ 16 1 a30 ¼ ð4a313 a21 a22 −2a12 a213 a222 þ 4a313 a20 a23 −4a12 a213 a21 a23 þ 4a212 a13 a22 a23 having the invariant algebraic curve 8a413   −4a11 a213 a22 a23 −2a312 a223 þ 4a11 a12 a13 a223 −2a10 a213 a223 þ a13 a323 Þ: 1 3   a2 x4 a2 x6 F ðx; yÞ ¼ y2 þ a x −2x y−4 þ a2 þ 1 x2 − þ 2 2 16 ¼ y2 þ a1 ðxÞy þ a2 ðxÞ: 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 2a212a223 = 0. The Substituting a1(x) and a2(x) in the expressions of the polynomials f case a23 = 0 with a12 ≠ 0 implies a23 = a33 = 0 and a32 = a222/(4a12), and g given in the statement of Theorem 1 we obtain 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 1 2 2  f ðxÞ ¼ a x2 −1 ; g ðxÞ ¼ −x þ a x −4 x3 : the polynomials f, g and F of the statement of the proposition. □ 16 Proof of Corollary 2. Since the denominator of the polynomial f(x) di- Proof of Proposition 4. This proof follows in a similar way to the proof vides its numerator, then f(x) must vanish when a2(x) = (s − 1)a21(x)/ of Proposition 3, but firstly imposing that f and g satisfy (7) and (8). We (2s). This implies that omit the details that the reader can compute easily. □ s2 a21 ðxÞa1 0 ðxÞ−3sa21 ðxÞa1 0 ðxÞ þ 2a21 ðxÞa1 0 ðxÞ−2s2 a3 0 ðxÞ ¼ 0: CRediT authorship contribution statement From this equation we get Jaume Giné: Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Writing – original draft, Writing – review ðs−2Þðs−1Þa31 ðxÞ & editing, Visualization. Jaume Llibre: Conceptualization, Methodology, a3 ðxÞ ¼ þ c1 ; 6s2 Software, Validation, Formal analysis, Investigation, Writing – original draft, Writing – review & editing, Visualization. where c1 is an arbitrary constant. Substituting a3(x) in f(x) we obtain NðxÞ Declaration of competing interest f ðxÞ ¼ ð6Þ DðxÞ The authors declare that they have no known competing financial where NðxÞ ¼ −2s2 a2 ðxÞa1 0 ðxÞ þ 3s2 a21 ðxÞa1 0 ðxÞ−5sa21 ðxÞa1 0 ðxÞ þ 2a21 ðxÞ interests or personal relationships that could have appeared to influ- a1 0 ðxÞ−2s2 a1 ðxÞa2 0 ðxÞ þ 2sa1 ðxÞa2 0 ðxÞ and D(x) = 2s[(s − 1)a21 ence the work reported in this paper. (x) − 2sa2(x)]. Dividing the numerator N(x) by the denominator D(x) the quotient Acknowledgements is a1′(x)/2, and the remainder R(x) is equal to 2ðs−1Þa1 ðxÞðsa1 ðxÞa1 0 ðxÞ −a1 ðxÞa1 0 ðxÞ−sa2 0 ðxÞÞ, which is zero when the denominator D(x) is The author is partially supported by the Agencia Estatal de zero. Consequently the expression (6) reduces to Investigación grant PID2020-113758GB-I00 and an AGAUR (Generalitat de Catalunya) grant number 2017SGR 1276. The second a1 0 ðxÞ author is partially supported by the Agencia Estatal de Investigación f ðxÞ ¼ : ð7Þ 2 grant PID2019-104658GB-I00, and the H2020 European Research Coun- cil grant MSCA-RISE-2017-777911. In this case g(x) takes the form   References a1 ðxÞa1 0 ðxÞ ðs−2Þðs−1Þa21 ðxÞ þ 2s2 a2 ðxÞ −4s2 a2 ðxÞa2 0 ðxÞ gðxÞ ¼   : ð8Þ 2s2 ðs−1Þa21 ðxÞ−2sa2 ðxÞ [1] Christopher CJ. Liouvillian first integrals of second order polynomial differential equations. Electron J Differential Equations. 1999.(49) 7 pp. In the above expression of g(x) since the denominator divides the [2] Dumortier F, Llibre J, Artés JC. Qualitative theory of planar differential systems. New York: UniversiText, Springer–Verlag; 2006. numerator, working in the same way as we did for the polynomial f(x) [3] Singer MF. Liouvillian first integrals of differential equations. 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