Electronic Journal of Differential Equations, Vol. 2019 (2019), No. 71, pp. 1–8. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu CHIELLINI HAMILTONIAN LIÉNARD DIFFERENTIAL SYSTEMS JAUME GINÉ, JAUME LLIBRE, CLAUDIA VALLS Abstract. We characterize the centers of the Chiellini Hamiltonian Liénard second-order differential R equations x0 = y, y 0 = −f (x)y − g(x) where g(x) = f (x)(k − α(1 + α) f (x)dx) with α, k ∈ R. Moreover we study the phase portraits in the Poincaré disk of these systems when f (x) is linear. 1. Introduction In the study of differential equations, a Liénard equation in R is a second-order differential equation of the form x00 + f (x)x0 + g(x) = 0, named in honor of the French physicist Alfred-Marie Liénard [13], who during the development of radio and vacuum tube technology, introduced such equations to model oscillating circuits. These equations have been intensively studied, Math- SciNet reports more than 1480 articles under the keywords “Liénard” and “equa- tion”. Recent papers on this topic are found in [2, 6, 7, 10, 12, 14] and the references therein. Here we deal with Liénard second-order differential equations x00 + f (x)x0 + g(x) = 0, where f and g are polynomials, or equivalently with first-order differential system of equations ẋ = y, ẏ = −f (x)y − g(x), (1.1) in R2 . In 1931 Chiellini [3] proved that system (1.1) is integrable if the functions f (x) and g(x) satisfy the condition d  g(x)  = sf (x), (1.2) dx f (x) where s is a constant. This condition is now known as the Chiellini condition. Recently Ghose Choudhury and Guha in [9] studied when the Liénard sys- tems (1.1) satisfying Chiellini condition admit a Hamiltonian formulation and they proved that this holds if in (1.2) we have s = −α(1+α) with α ∈ R. More precisely, if  Z  g(x) = f (x) k − α(1 + α) f (x)dx , (1.3) 2010 Mathematics Subject Classification. 34C05, 34A34, 34C14. Key words and phrases. Liénard system; center-focus problem; first integrals. c 2019 Texas State University. Submitted January 26, 2019. Published May 17, 2019. 1 2 J. GINÉ, J. LLIBRE, C. VALLS EJDE-2019/71 where k ∈ R, then the Liénard systems (1.1) satisfying (1.3) admit an integrating factor R(x, y) such that the differential system ∂H ∂H ẋ = R(x, y)y = − , ẏ = R(x, y)(−f (x)y − g(x)) = , ∂y ∂x is Hamiltonian with the Hamiltonian function  α(y(1+α)f (x)+g(x)) α g(x) 1+α   (1+2α)f (x) y − αf (x) if α ∈ / {−1, −1/2, 0},  y+F (x) −k H= e (k + y) if α ∈ {−1, 0},   4 exp F4(F (x)+4k)   F (x) + 4k + 2y if α = −1/2,  (x)+4k+2y R where F (x) = f (x)dx. Note that H is a Darboux first integral, for more details on Darboux integrability see Chapter 8 of [5]. We remark that the first integrals H are not given in [9]. First we are interested in the center problem for the Liénard systems (1.1) sat- isfying (1.3) and we prove the following result. Theorem 1.1. The singular point (x0 , 0) of system (1.1) is a center if it satisfies (1.3), g(x0 ) = 0 and g 0 (x0 ) > 0. The above theorem is proved in section 2. Now we consider system (1.1) satis- fying condition (1.3) in the case in which f (x) is linear that is, f (x) = ax + b with a, b ∈ R, a 6= 0. The second main result is concerned with the phase portraits in the Poincaré disk of these systems, i.e. 1 ẋ = y, ẏ = − (b + ax)(2k + 2y − 2α(1 + α)bx − α(1 + α)ax2 ), (1.4) 2 with a 6= 0 and α2 (α + 1)2 + k 2 6= 0 (otherwise g(x) ≡ 0). See [5, Chapter 5] for the definition and results concerning the Poincaré disk and the Poincaré compactification, and [5, Chapter 1] for the definition of topological equivalence. (a) (b) (c) (d) (e) (f) (g) Figure 1. Phase portraits of system (1.4). EJDE-2019/71 CHIELLINI HAMILTONIAN LIÉNARD DIFFERENTIAL SYSTEMS 3 The second main result in the paper is the following theorem. Theorem 1.2. The phase portraits of system (1.4) in the Poincaré disk are topo- logically equivalent as follows: Figure 1(a) if α(α + 1) = 0 and ak > 0; Figure 1(b) if α(α + 1) = 0 and ak < 0; Figure 1(c) if α(α + 1) > 0 and α(α + 1)(α(α + 1)b2 + 2ak) ≤ 0; Figure 1(d) if α(α + 1) > 0 and α(α + 1)(α(α + 1)b2 + 2ak) > 0; Figure 1(e) if α(α + 1) < 0 and α(α + 1)(α(α + 1)b2 + 2ak) < 0; Figure 1(f ) if 0 > α(α + 1) ≥ −1/4 and α(α + 1)(α(α + 1)b2 + 2ak) = 0; Figure 1(g) if α(α + 1) < 0 and α(α + 1)(α(α + 1)b2 + 2ak) > 0. The above theorem is proved in section 3. 2. Proof of Theorem 1.1 The change of variables x = X + x0 transforms (1.1) into Ẋ = y, ẏ = −f (X + x0 )y − g(X + x0 ) = −f¯(X)y − ḡ(X), (2.1) 0 with ḡ(0) = 0 and ḡ (0) > 0. To prove Theorem 1.1 for (2.1) we recall that this system can be transformed by the change of variables y = Y − F̄ (X) into Ẋ = Y − F̄ (X), Ẏ = −ḡ(X), (2.2) ¯ R where F̄ (X) = f (X)dX. Let us recall a result proved in [8], see also [4, 11]. Theorem 2.1. Consider the differential system ẋ = ϕ(y) − F (x), ẏ = −g(x). (2.3) where all the involved functions are analytic and satisfy ϕ(y) = y 2m−1 + O(y 2m ), F (x) = aj xj + O(xj+1 ), x2` Z G(x) = g(x)dx = + O(x2`+1 ), 2` with m, j and ` nonzero positive integers. Consider F (ξ(u)) = fj uj + O(uj+1 ), where ξ(u) is defined as the inverse function of r 2` G(x) u = x 2` 2` , x and assume that j > `(2m − 1)/m. Then (2.3) has a center at the origin if there exist analytic functions β, γ, and h satisfying β(0) = 0, γ(x) = cx` + O(x`+1 ) with c 6= 0, and such that F (x) = β(h(x)), G(x) = γ(h(x)). Now we apply Theorem 2.1 to our system (2.2) renaming X by x. In this case m = ` = 1 and j = 2, because 0 < ḡ 0 (0) = f¯0 (0)(k − α(1 + α)F̄ (0)), and so f¯0 (0) 6= 0. Now Z Z h  Z i G(x) = ḡ(x)dx = f (x) k − α(1 + α) f¯(x)dx dx ¯ 4 J. GINÉ, J. LLIBRE, C. VALLS EJDE-2019/71 Z Z 1 =k f¯(x)dx − α(1 + α) F̄ (x)f¯(x)dx = k F̄ (x) − α(1 + α) F̄ 2 (x). 2 Hence applying Theorem 2.1 with β(x) = x, h(x) = F (x) and γ(x) = kx − α(1 + α)x2 /2, we have that Theorem 1.1 is proved. 3. Proof of Theorem 1.2 The finite singular points of system (1.4) are  (α(1 + α)b ± pα(1 + α)(α(1 + α)b2 + 2ak)  p1 = (−b/a, 0), p2,3 = − ,0 . α(1 + α)a Note that p2,3 only exist when α(1 + α) 6= 0 and α(1 + α)(α(1 + α)b2 + 2ak) > 0, because when α(1 + α) 6= 0 and α(1 + α)(α(1 + α)b2 + 2ak) = 0, then p1 = p2 = p3 . First we study the finite singular point p1 . The eigenvalues of the Jacobian matrix at p1 are p −α(1 + α)b2 − 2ak ± √ . 2 If α(1 + α) = 0 then p1 is a saddle if ak < 0 and a center if ak > 0. Now we assume that α(1 + α) 6= 0. So, p1 is a center if α(1 + α)b2 + 2ak > 0, a saddle if α(1 + α)b2 + 2ak < 0 and it is nilpotent if α(1 + α)b2 + 2ak = 0 (note that in this last case, that is, when k = −α(1 + α)b2 /(2a)) the Jacobian matrix at p1 is   0 1 . 0 0 Now we take k = −α(1 + α)b2 /(2a) and we introduce the change of variables X = x + b/a, Y = y. With this change of variables system (1.4) becomes 1 X 0 = Y,Y 0 = −aXY + α(1 + α)a2 X 3 . 2 Applying [5, Theorem 3.5] and taking into account that α(1 + α) ≥ −1/4 (because α(1 + α) has a unique minimum at α = −1/2 with value −1/4) we get that p1 is a saddle if α(1 + α) > 0 and the union of one hyperbolic and one elliptic sector if α(1 + α) ∈ [−1/4, 0). Now we study the finite singular points p2,3 . In order that they exist we must have α(1 + α) 6= 0 and α(1 + α)(α(1 + α)b2 + 2ak) > 0. The eigenvalues of the Jacobian matrix at p2 are √ p α p α(1 + α)b2 + 2ak −√ 2 α(1 + α)b + 2ak, q , 1+α α 1+α and the eigenvalues of the Jacobian matrix at p3 are √ p α p α(1 + α)b2 + 2ak √ 2 α(1 + α)b + 2ak, − q . 1+α α 1+α Hence, p2 and p3 are both saddles if α(1 + α) > 0 and α(1 + α)b2 + 2ak > 0 and they are both nodes (p2 being unstable and p3 being stable) if α(1 + α) < 0 and α(1 + α)b2 + 2ak < 0. In short we have the following finite singular points: (i) two nodes (p2 unstable and p3 stable) and a saddle (p1 ) if α(1+α)b2 +2ak < 0 and α(1 + α) < 0; EJDE-2019/71 CHIELLINI HAMILTONIAN LIÉNARD DIFFERENTIAL SYSTEMS 5 (ii) two saddles (p2 and p3 ) and one center (p1 ) if α(1 + α)b2 + 2ak > 0 and α(1 + α) > 0; (iii) one saddle p1 if either α(1 + α) = 0 and ak < 0, or α(1 + α) > 0 and α(1 + α)b2 + 2ak ≤ 0; (iv) one center p1 if either α(1 + α) = 0 and ak > 0, or α(1 + α) < 0 and α(1 + α)b2 + 2ak > 0; (v) one singular point formed by one hyperbolic and one elliptic sector if α(1 + α) ∈ [−1/4, 0) and α(1 + α)b2 + 2ak = 0. Now we compute the infinite singular points in the local chart U1 (see again Chapter 5 in [5] for its definition). We have 1 3 u̇ = α(1 + α)a2 + α(1 + α)abv − auv + (α(1 + α)b2 − ak)v 2 − buv 2 2 2 − bkv 3 − u2 v 2 , (3.1) v̇ = −uv 3 . Note that there are no infinite singular points on the local chart U1 if α(1+α)a2 6= 0. In the case α(1 + α) = 0, from system (3.1), after simplifying by the common factor v, we have u̇ = −(au + akv + buv + bkv 2 + u2 v), v̇ = −uv 2 . The unique infinite singular point is (u, v) = (0, 0). In this case the Jacobian matrix at this point is   −a −ak 0 0 and so the point is semi-hyperbolic with eigenvalues −a, 0. Applying [5, Theorem 2.19] we get that it is a saddle if ak > 0, and a node if ak < 0, stable if a > 0 and otherwise it is unstable. Now we study the origin in the local chart U2 (see again [5, Chapter 5] for its definition). We have 1 3 u̇ = v 2 + au2 v + buv 2 − α(1 + α)a2 u4 − α(1 + α)abu3 v, 2 2 − (α(1 + α)b2 − ak)u2 v 2 + bkuv 3 1 3 v̇ = auv 2 + bv 3 − α(1 + α)a2 u3 v − α(1 + α)abu2 v 2 2 2 − (α(1 + α)b2 − ak)uv 3 + bkv 4 . The origin of the local chart U2 is a singular point. The Jacobian matrix at the origin is the 2 × 2 zero matrix and so the origin of U2 is linearly zero. We need to apply blow-up techniques to study it, see for instance [1]. Doing so, we get the following: • if α(1+α) < 0 the origin of the local chart U2 is the union of three hyperbolic and one elliptic sectors. Its local phase portrait on the local chart U2 is topologically equivalent to the one of Figure 2(a) if a > 0, and to the one of Figure 2(b) if a < 0; • if α(1 + α) > 0 the origin of the local chart U2 is the union of two elliptic sectors separated by parabolic sectors. Its local phase portrait on the local chart U2 is topologically equivalent to the one of Figure 2(c); 6 J. GINÉ, J. LLIBRE, C. VALLS EJDE-2019/71 • if α(1+α) = 0 the origin of the local chart U2 is the union of one hyperbolic and one elliptic sectors separated by two parabolic sectors. Its local phase portrait on the local chart U2 is topologically equivalent to the one of Figure 2(d). (a) (b) (c) (d) Figure 2. Local chart U2 . Taking into account the local phase portraits at the finite and infinite singular points we get the following cases: • if α(α + 1) = 0 and ak > 0, there is a unique finite singular point which is a center, the origin of the local chart U1 is a saddle and the origin of the local chart U2 is the union of one hyperbolic and one elliptic sectors separated by two parabolic sectors see Figure 2(d). Gluing this information together we get the phase portrait of Figure 1(a). • if α(α + 1) = 0 and ak < 0, there is a unique finite singular point which is a saddle, the origin of the local chart U1 is a node and the origin of the local chart U2 is the union of one hyperbolic and one elliptic sectors separated by two parabolic sector as see Figure 2(d). Gluing this information together we get the phase portrait of Figure 1(b). • if α(α + 1) > 0 and α(α + 1)(α(α + 1)b2 + 2ak) ≤ 0, there is a unique finite singular point which is a saddle, no infinite singular points in the local chart U1 and the origin of U2 is the union of two elliptic sector separated by two parabolic sectors see Figure 2(c). Gluing this information together we get the phase portrait of Figure 1(c). • if α(α + 1) > 0 and α(α + 1)(α(α + 1)b2 + 2ak) > 0, there are three finite singular points which are two saddles and one center, no infinite singular points in the local chart U1 and the origin of U2 is the union of two elliptic sectors separated by two parabolic sectors see Figure 2(c). Gluing this information together we get the phase portrait of Figure 1(d). • if α(α + 1) < 0 and α(α + 1)(α(α + 1)b2 + 2ak) < 0, there are three finite singular points which are two nodes and one saddle, no infinite singular points in the local chart U1 and the origin of U2 is the union of three hyperbolic and one elliptic sectors as in Figures 2(a) and 29b). Gluing this information together we get the phase portrait of Figure 1(e). • if 0 > α(α+1) ≥ −1/4 and α(α+1)(α(α+1)b2 +2ak) = 0, there is a unique finite singular point which is the union of one hyperbolic and one elliptic sectors separated by two parabolic sectors, no infinite singular points in the local chart U1 , and the origin of U2 is the union of three hyperbolic and EJDE-2019/71 CHIELLINI HAMILTONIAN LIÉNARD DIFFERENTIAL SYSTEMS 7 one elliptic sectors as in Figures 2(a) and 2(b). Gluing this information together we get the phase portrait of Figure 1(f). • if α(α + 1) < 0 and α(α + 1)(α(α + 1)b2 + 2ak) > 0, there is a unique finite singular point which is a center, no infinite singular points in the local chart U1 , and the origin of U2 is the union of three hyperbolic and one elliptic sectors as in Figures 2(a) and 2(b). Gluing this information together we get the phase portrait of Figure 1(g). Acknowledgements. J. Giné was partially supported by a MINECO/FEDER grant number MTM2017-84383-P and an AGAUR (Generalitat de Catalunya) grant number 2017SGR 1276. J. LLibre was partially supported by a FEDER-MINECO grant number MTM2016-77278-P, and an AGAUR grant number 2017SGR-1617. C. Valls was partially supported by FCT/Portugal through UID/MAT/04459/2013. References [1] M. J. Álvarez, A. Ferragut, X. Jarque; A survey on the blow up technique, Int. J. Bifurcation and Chaos, 21 (2011), 3103–3118. [2] Y. Cao, C. 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Jaume Giné Departament de Matemàtica, Inspires Research Centre, Universitat de Lleida, Avda. Jaume II, 69; 25001 Lleida, Catalonia, Spain Email address: gine@matematica.udl.cat Jaume Llibre Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain Email address: jllibre@mat.uab.cat 8 J. GINÉ, J. LLIBRE, C. VALLS EJDE-2019/71 Claudia Valls Departamento de Matemática, Instituto Superior Técnico, Av. Rovisco Pais 1049-001, Lisboa, Portugal Email address: cvalls@math.ist.utl.pt