On the polynomial differential systems having polynomial first integrals

Transcription

1 Bull. Sci. math. 136 (2012) On the olynomial differential systems having olynomial first integrals Belén García a,, Jaume Llibre b, Jesús S. Pérez del Río a a Deartamento de Matemáticas, Universidad de Oviedo, Avda Calvo Sotelo, s/n., Oviedo, Sain b Deartament de Matemàtiques, Universitat Autònoma de Barcelona, Bellaterra, Barcelona, Catalonia, Sain Received 25 October 2011 Available online 16 November 2011 Abstract We consider the class of comlex lanar olynomial differential systems having a olynomial first integral. Inside this class the systems having minimal olynomial first integrals without critical remarkable values are the Hamiltonian ones. Here we mainly study the subclass of olynomial differential systems such that their minimal olynomial first integrals have a unique critical remarkable value. In articular we characterize all the Liénard olynomial differential systems ẋ = y, ẏ = f(x)y g(x), with f(x)and g(x) comlex olynomials in the variable x, having a minimal olynomial first integral with a unique critical remarkable value Elsevier Masson SAS. All rights reserved. MSC: 34C05; 34A34; 34C14 Keywords: Polynomial differential system; Liénard differential system; Polynomial first integral; Critical remarkable value 1. Introduction and statement of the main results The nonlinear ordinary differential equations or simle the differential systems aear in many branches of alied mathematics, hysics, and in general in alied sciences. In general the differential systems cannot be solved exlicitly, so the qualitative information rovided by the theory of dynamical systems is the best that one can exect to obtain. * Corresonding author. addresses: belen.garcia@uniovi.es (B. García), llibre@mat.uab.cat (J. Llibre), sr@uniovi.es (J.S. Pérez del Río) /$ see front matter 2011 Elsevier Masson SAS. All rights reserved. doi: /.bulsci

2 310 B. García et al. / Bull. Sci. math. 136 (2012) For a lanar differential system the existence of a first integral determines comletely its hase ortrait, i.e. the descrition of the domain of definition of the differential system as union of all the orbits or traectories of the system. To rovide the hase ortrait of a differential system is the main obective of the qualitative theory of the differential systems. Thus for lanar differential systems one of the main questions is: How to recognize if a given lanar differential system has a first integral? In this aer we study the existence of olynomial first integrals in lanar olynomial differential systems. More recisely, we want to study the olynomial first integrals of the differential systems ẋ = P(x,y), ẏ = Q(x,y), (1) where P and Q are comlex olynomials in the variables x and y, and where the dot denotes derivative with resect to the variable t that can be consider real or comlex. The search of first integrals is a classical tool for classifying all traectories of a lanar differential system (1). Polynomial first integrals are a articular case of the Darboux first integrals. In 1878 Darboux [7] showed how the first integrals of lanar olynomial systems ossessing sufficient invariant algebraic curves can be constructed. The best imrovements to Darboux s results for lanar olynomial systems are due to Poincaré [14] in 1897, to Jouanolou [10] in 1979, to Prelle and Singer [15] in 1983, and to Singer [16] in But the results of the Darboux theory of integrability rovide sufficient conditions for finding in general Liouvillian first integrals, and in articular rational first integrals, but do not rovide neither sufficient nor necessary conditions for the existence of olynomial first integrals. As usual C[x,y] denotes the ring of all comlex olynomials in the variables x and y. We say that H C[x,y]\C is a olynomial first integral of system (1) on C 2 if H (x(t), y(t)) is constant for all values of t such that (x(t), y(t)) is defined on C 2. Obviously, H is a first integral of system (1) if and only if P H x + Q H y = 0 (2) in C 2. Polynomial first integrals for the following 3-dimensional quadratic olynomial differential system of Lotka Volterra kind x = x(cy + z), y = y(x + Az), z = z(bx + y), have been characterized by Moulin-Ollagnier [13] and Labrunie [12]. Cairó and Llibre [3] classify the olynomial first integrals for the 2-dimensional quadratic olynomial differential system of Lotka Volterra kind x = x(a 1 + b 11 x + b 12 y), y = y(a 2 + b 21 x + b 22 y). In fact, both results on Lotka Volterra systems are related, see the relationshi between both systems in [2]. Recently in [5] the authors obtain all quadratic olynomial differential systems having a olynomial first integral and do the toological classification of the hase ortraits of such quadratic systems in [9]. This classification has been imroved using invariant theory in the 12-dimensional arameter sace of all quadratic olynomial differential systems, see [1]. Also in [4] the olynomial first integrals have been studied for weight-homogeneous lanar olynomial differential systems of weight degree 3.

3 B. García et al. / Bull. Sci. math. 136 (2012) From now on we write the olynomial differential system (1) as follows ẋ = P(x,y) = k (x)y, ẏ = Q(x,y) = =0 and we assume in all this aer that (i) k (x) 0, (ii) q l (x) 0, and (iii) P and Q corime. l q i (x)y i, (3) Note that we always can assume conditions (i) and (ii). If P and Q are not corime in the ring of all olynomials C[x,y], and R is their greatest common divisor, then doing a rescaling by R of the indeendent variable we get the olynomial differential system ẋ = P = P/R and ẏ = Q = Q/R,forwhichP and Q are corime. In short the three conditions (i), (ii) and (iii) are essentially working conditions that simlify the statement of the results and their roofs. Our first result is the next one roved in Section 2. Theorem 1. The following statements hold. (a) If k>l 1 and H = H(x,y) is a olynomial first integral of system (3) then, excet by the multilication for a nonzero constant, we have that H = y s + s 1 i=0 H i(x)y i with s>0. (b) If k = l 1 and system (3) has a olynomial first integral, then the degree of the olynomial k (x) is equal to the degree of the olynomial q l (x) lus one. (c) If k<l 1, then system (3) has no olynomial first integrals. A olynomial first integral H of system (3) is called minimal if for any other olynomial first integral H of(3)wehavethatthedegreeofh is smaller than or equal to the degree of H. Now we introduce the concet of remarkable value due to Poincaré (see [14]). Poincaré used these values in order to study the olynomial differential systems having a rational first integral, and in articular a olynomial first integral. Let H be a minimal olynomial first integral of the differential system (3). We say that c C is a remarkable value of H if the olynomial H + c is not irreducible in C[x,y], i.e.ifthere exist values 1,..., q N such that H + c = u u q q, where u i are irreducible olynomials in C[x,y] called remarkable factors associated to c with exonent i. Furthermore if there exists i such that i > 1, then the remarkable value is called critical and the corresonding factor u i is called critical remarkable factor. Note that every curve u i = 0 is an invariant algebraic curve of system (3), see Section 2 for the definition. In [6] the authors have roved that the number of remarkable values of a minimal olynomial first integral of a olynomial differential system is finite. For additional information on the remarkable values see [8]. System (3) is called Hamiltonian if there exists a olynomial H = H(x,y) such that P = H/ y and Q = H/ x. Clearly H is a first integral of this system. Javier Chavarriga roved that a olynomial differential system having a olynomial first integral without critical remarkable values is Hamiltonian, see the roof in [8]. Our main interest is the study of the olynomial differential systems (3) having minimal olynomial first integrals with a unique critical remarkable value. We note that the minimal olynomial first integrals having a unique critical remarkable value can have other non-critical remarkable values. i=0

4 312 B. García et al. / Bull. Sci. math. 136 (2012) Our first result on the minimal olynomial first integrals having a unique critical remarkable value is the following one roved in Section 2. Theorem 2. Let H = s i=0 H i (x)y i be with H s (x) 0 a minimal olynomial first integral of system (3). Assume that H has a unique critical remarkable value c and that q H + c = u, (4) with ositive integers and with some > 1. Let n be the degree of the olynomial u in the variable y. Then k + 1 = n. Now we restrict our attention to the olynomial differential systems (3) having minimal first integrals with a unique critical remarkable value and such that the degree k of the olynomial P in the variable y is one. As we shall see this class of olynomial differential systems contain the relevant class of Liénard olynomial differential systems. If k = 1 in systems (3) then, using Theorem 1, all these systems having a olynomial first integral can be written as ẋ = a(x)y + b(x), ẏ = c(x)y 2 + d(x)y + e(x), (5) with a(x) 0. Theorem 3. Assume that the olynomial differential system (5) has a minimal olynomial first integral H with a unique critical remarkable value. Then H = F(x) ( A(x)y + B(x) ) ( ) q, C(x)y + D(x) (6) where A, B, C, D and F are olynomials in the variable x, and and q are distinct ositive integers, or H = F(x) ( A(x)y 2 + B(x)y + C(x) ), (7) where A, B, C and F are olynomials in the variable x, and is a ositive integer. is Theorem 3 is roved in Section 2. In the study of dynamical systems and differential equations, a Liénard differential equation ẍ + f(x)ẋ + g(x) = 0, (8) where f(x) and g(x) are real C 1 functions. Here the dot denotes differentiation with resect to the time t. These equations aeared in the works of the French hysicist Alfred-Marie Liénard when he studied the develoment of radio and vacuum tubes. Liénard differential equations were intensely studied as they can be used to model oscillating circuits. Under certain additional assumtions Liénard s theorem guarantees the existence of a limit cycle for Eq. (8), see [11]. Instead of working with the differential equations of second order (8) we shall work with the following equivalent lanar differential system with two equations of first order ẋ = y, ẏ = f(x)y g(x). (9)

5 B. García et al. / Bull. Sci. math. 136 (2012) The Liénard differential systems (9) with f(x) and g(x) comlex olynomials in the variable x are called olynomial Liénard differential systems. One of our main goals of this aer is to characterize the olynomial Liénard differential systems (8) having a olynomial first integral with a unique critical remarkable value, and to rovide an exlicit exression of these systems and of their olynomial first integrals. Note that the olynomial Liénard differential systems (9) are a articular subclass of systems (5), and consequently of systems (3). Theorem 4. For the comlex olynomial Liénard differential system (9) the following statements hold. (a) If g(x) = 0, then a olynomial first integral of system (9) is H = y + F(x) where F(x)= f(x)dx. (b) If f(x)= 0, then a olynomial first integral of system (9) is H = y 2 /2 + g(x)dx. (c) Assume that f(x)g(x) 0 and let H be a minimal olynomial first integral of system (9) having a unique remarkable value. Then there exist ositive integers and q, q such that g(x) = c C, and ( H = y + c + ) ( q F(x) y q ( c + excet by the multilication for a nonzero constant. )) q q F(x), q q f (x)(c + q F(x))with Theorem 4 is roved in Section 2. We have some numerical evidences that the following oen question can have a ositive answer. Oen question. All the olynomial Liénard differential systems (9) with f(x)g(x) 0 and with a olynomial first integral are the ones described in the statement (c) of Theorem Proof of the results In this section we rove Theorems 1, 2, 3 and 4. But first we recall two basic definitions that we shall use. A non-constant function R = R(x,y) is called an integrating factor of system (3) if (RP) x + (RQ) y = 0. So the differential system ẋ = RP, ẏ = RQ is Hamiltonian, and consequently there exists a first integral H such that ẋ = RP = H y, ẏ = RQ = H x. Then we say that R is the integrating factor associated to the first integral H, and vice versa. Let u = u(x,y) C[x,y], i.e.u is a comlex olynomial in the variables x and y. Then we say that the algebraic curve u = 0isinvariant for the system (3) if P u x + Q u y = Ku, (10) for some olynomial K C[x,y].

6 314 B. García et al. / Bull. Sci. math. 136 (2012) Proof of Theorem 1. We write the olynomial H as a olynomial in the variable y with coefficients olynomials in the variable x,i.e. s H = H i (x)y i. i=0 We claim that s>0. For roving the claim we suose that H = H(x). Then, from the definition of first integral (2) we get that P(x,y)H (x) = 0. Here as usual H (x) denotes the derivative of H with resect to the variable x. Since P(x,y) 0wehavethatH is constant, in contradiction with the fact that H is a first integral. Consequently the claim is roved. From the definition of the first integral we get that ( k )( s ) ( l i)( s ) (x)y H i (x)yi + q i (x)y ih i (x)y i 1 = 0. (11) =0 i=0 i=0 i=1 If k>l 1 the degree of (11) in the variable y is s + k. The coefficient of y s+k in (11) is k (x)h s (x) = 0. Therefore, since k(x) 0 we get that H s (x) is a constant α C. So, in what follows we shall work with the first integral H/α instead of H. Hence H s (x) = 1. Then statement (a)isroved. Assume k = l 1 again the degree of (11) in the variable y is s + k. The coefficient of y s+k in (11) is k (x)h s (x) + sq l(x)h s (x) = 0. So, clearly in order that this differential equation has a olynomial solution it is necessary that the degree of k (x) must be equal to the degree of q l (x) lus one. Hence statement (b) is roved. If k<l 1 the degree of (11) in the variable y is s + l 1. The coefficient of y s+l 1 in (11) is sq l (x)h s (x). Since s>0, and q l (x) and H s (x) are nonzero, we get that system (3) has no olynomial first integral. Proosition 5. Assume that system (3) has a minimal olynomial first integral H = s i=0 H i (x)y i with s>0(see Theorem 1). Suose that H has only one critical remarkable value c. Let u 1,...,u q be all the distinct remarkable factors of H + c with exonents 1,..., q, resectively; i.e. H + c = q u. Then α q u 1 is the olynomial integrating factor of system (3) associated to H for some α C. Proof. We have that H/ x = S q and H/ y = T q, and every u does not divide both olynomials S and T.IfR is the integrating factor associated to H we have that H/ y = PR and H/ x = QR. So, since the olynomials P and Q are corime, we obtain that R = U q u 1, and for all = 1,...,q the olynomial u does not divide U. We claim that U is a constant. Otherwise each irreducible factor w of U in C[x,y] divides to H/ x and H/ y, so it would exist another critical remarkable value d such that w 2 divides H + d, in contradiction with the assumtion that we have a unique critical remarkable value. u 1 u 1 Proof of Theorem 2. From Proosition 5 we know that the integrating factor R associated to the first integral H is q R = α u 1, (12) with α C.

7 B. García et al. / Bull. Sci. math. 136 (2012) Since R is the integrating factor associated to H we have that H y = PR. Comuting the degrees of the both sides of the revious equality in the variable y we obtain that s 1 = k + ( 1)n. By (4) we know that n = s. Consequently we get that k + 1 = n. Hence the theorem is roved. Proof of Theorem 3. Using the notations introduced in Theorem 2 we get that k + 1 = n. If we denote r the number of remarkable factors that deends on y, since k = 1wehavetwo ossibilities, either r = 2 and n 1 = n 2 = 1, or r = 1 and n 1 = 2. Hence the corresonding first integrals can be written in the form (6) or (7). Proof of Theorem 4. Statements (a) and (b) of Theorem 4 follows easily. Now we rove statement (c). Since system (9) is a articular case of (3) with k = 1 and l = 1, clearly k>l 1. Then, by Theorem 1 we know that the olynomial first integral of system (9) begins with y s and then, using Theorem 3, can be written as in the form (6) with F = A = C = 1 or (7) with F = A = 1. In the second case taking in account that the first integral is minimal, we have that = 1 and the first integral has no remarkable critical values. In short, the first integral can be written as H = ( y + L(x) ) ( y + M(x) ) q, (13) where and q are different ositive integers. Substituting H in the definition of the first integral (2) we get that where a(x)y 2 + b(x)y + c(x) = 0, a(x) = f (x) qf (x) + L (x) + qm (x), b(x) = ( + q)g(x) f(x) ( ql(x)+ M(x) ) + M(x)L (x) + ql(x)m (x), c(x) = g(x) ( ql(x)+ M(x) ).

8 316 B. García et al. / Bull. Sci. math. 136 (2012) From c(x) = 0 we obtain M(x) = ql(x)/. Then the equation b(x) = 0 becomes ( + q) ( g(x) + ql(x)l (x) ) = 0. Consequently g(x) = ql(x)l (x)/. Simlifying the equation a(x) = 0 we get that ( + q) ( f (x) + (q )L (x) ) = 0. Therefore L(x) = c + q F(x). Substituting L(x) in (13) and (14) the roof of the theorem is comleted. (14) Acknowledgements The first and third authors are artially suorted by a MEC/FEDER grant number MTM The second author is artially suorted by a MEC/FEDER grant number MTM and by a CICYT grant number 2009SGR References [1] J.C. Artés, J. Llibre, N. Vule, Quadratic systems with a olynomial first integral: A comlete classification in the coefficient sace R 12, J. Differential Equations 246 (2009) [2] L. Cairó, H. Giacomini, J. Llibre, Liouvillian first integrals for the lanar Lotka Volterra systems, Rend. Circ. Mat. Palermo 52 (2003) [3] L. Cairó, J. Llibre, Integrability of the 2-dimensional Lotka Volterra system via olynomial (inverse) integrating factors, J. Phys. A 33 (2000) [4] L. Cairó, J. Llibre, Polynomial first integrals for weight-homogeneous lanar olynomial differential systems of weight degree 3, J. Math. Anal. Al. 331 (2007) [5] J. Chavarriga, B. García, J. Llibre, J.S. Pérez del Río, J.A. Rodríguez, Polynomial first integrals of quadratic vector fields, J. Differential Equations 230 (2006) [6] J. Chavarriga, H. Giacomini, J. Giné, J. Llibre, Darboux integrability and the inverse integrating factor, J. Differential Equations 194 (2003) [7] G. Darboux, Mémoire sur les équations différentielles algébriques du remier ordre et du remier degré (Mélanges), Bull. Sci. Math. 2ème Série 2 (1878) 60 96, , [8] A. Ferragut, J. Llibre, On the critical remarkable values of the rational first integrals of olynomial vector fields, J. Differential Equations 241 (2007) [9] B. García, J. Llibre, J.S. Pérez del Río, Phase ortraits of quadratic vector fields with a olynomial first integral, Rend. Circ. Mat. Palermo 55 (2006) [10] J.P. Jouanolou, Equations de Pfaff Algébriques, Lecture Notes in Math., vol. 708, Sringer-Verlag, New York Berlin, [11] A. Liénard, Étude des oscillations entrenues, Rev. Génerale de l Électricité 23 (1928) [12] S. Labrunie, On the olynomial first integrals of the (abc) Lotka Volterra system, J. Math. Phys. 37 (1996) [13] J. Moulin-Ollagnier, Polynomial first integrals of the Lotka Volterra system, Bull. Sci. Math. 121 (1997) [14] H. Poincaré, Sur l intégration des équations différentielles du remier ordre et du remier degré I and II, Rend. Circ. Mat. Palermo 5 (1891) , Rend. Circ. Mat. Palermo 11 (1897) [15] M.J. Prelle, M.F. Singer, Elementary first integrals of differential equations, Trans. Amer. Math. Soc. 279 (1983) [16] M.F. Singer, Liouvillian first integrals of differential equations, Trans. Amer. Math. Soc. 333 (1992)