This is a preprint of: “On the limit cycles surrounding a diagonalizable linear node with homoge- neous nonlinearities”, Jaume Llibre, Tayeb Salhi, Appl. Math. Lett., vol. 98, 427–431, 2019. DOI: [10.1016/j.aml.2019.07.002] ON THE LIMIT CYCLES SURROUNDING A DIAGONALIZABLE LINEAR NODE WITH HOMOGENEOUS NONLINEARITIES JAUME LLIBRE1 AND TAYEB SALHI2 Abstract. In this paper we study the existence and non–existence of limit cycles for the class of polynomial differential systems of the form ẋ = λx + Pn (x, y), ẏ = µy + Qn (x, y), where Pn and Qn are homogeneous polynomials of degree n. 1. Introduction and statement of the main results A polynomial differential system in R2 is a differential system of the form dx dy (1) = ẋ = P (x, y), = ẏ = Q(x, y), dt dt where P (x, y) and Q(x, y) are polynomials in the variables x and y with real coefficients. Then m = max{deg P, deg Q} is the degree of the polynomial system. As usual a limit cycle of a system (1) is an isolated periodic solution in the set of all periodic solutions of system (1). Limit cycles of planar differential systems were defined by Poincaré [21] and started to be studied intensively at the end of the 1920s by van der Pol [22], Liénard [12] and Andronov [1]. In the qualitative theory of the polynomial differential equations in the plane R2 one of the more difficult problems is the study of their limit cycles. Thus the second part of the unsolved 16–th Hilbert problem [13] asked for an upper bound on the maximum number of limit cycles for the polynomial differential systems of a given degree in function of this degree, see for more details the surveys [14] and [11]. 2010 Mathematics Subject Classification. Primary 34C05, 34A34. Key words and phrases. Polynomial differential systems, polynomial vector fields, limit cycles. 1 2 J. LLIBRE AND T. SALHI In this paper for the class of polynomial differential systems in R2 of the form (2) ẋ = P1 (x, y) + Pn (x, y), ẏ = Q1 (x, y) + Qn (x, y), where n > 1, and Pk (x, y) and Qk (x, y) are homogeneous polynomials of degree k, we want to study the existence and non–existence of limit cycles. For the polynomial differential systems (2) having a linear focus at the origin of coordinates of the form ẋ = λx − y + Pn (x, y), ẏ = x + λy + Qn (x, y), their limit cycles have been studied intensively, see for instance [3, 4, 5, 6, 8, 9, 10, 15, 17, 18, 20]. But there are very few results on the limit cycles of the polynomial differential systems having a linear node at the origin of coordinates of the form (3) ẋ = λx + Pn (x, y), ẏ = µy + Qn (x, y), with λµ > 0. Recently in [2] the polynomial differential systems (3) with λ = µ and n > 1 have been analyzed, proving that if n is odd such systems have at most one limit cycle, and if n is even then they have no limit cycles. On the other hand, in Proposition 6.3 and Remark 6.4 of the paper [7] are examples of systems (3) having two, one or zero limit cycles surrounding the origin. Finally, when λ 6= µ and λµ > 0 in [16] the authors provide sufficient conditions for the non–existence of limit cycles, or for the existence of one or two limit cycles. Using polar coordinate x = r cos(θ) and y = sin θ system (3) becomes (4) ṙ = f0 (θ)r + f (θ)rn , θ̇ = g0 (θ) + g(θ)rn−1 , and in the region R = {(r, θ) : g0 (θ) + g(θ)rn−1 > 0} it can be studied using the differential equation dr f0 (θ)r + f (θ)rn (5) = , dθ g0 (θ) + g(θ)rn−1 LIMIT CYCLES OF A CLASS OF POLYNOMIAL DIFFERENTIAL SYSTEMS 3 where f0 (θ) λ cos2 θ + µ sin2 θ, = g0 (θ) (µ − λ) cos θ sin θ, = f (θ) = cos θPn (cos θ, sin θ) + sin θQn (cos θ, sin θ), g(θ) cos θQn (cos θ, sin θ) − sin θPn (cos θ, sin θ), = i=n X Pn (x, y) = a(n−i)i xn−i y i , i=0 i=n X Qn (x, y) = b(n−i)i xn−i y i . i=0 Theorem 1. The polynomial differential system (3) with n ≥ 2 has no limit cycles surrounding the origin in the region R if one of the following conditions holds. (a) f = 0. (b) g = 0. (c) a0n = 0. (d) bn0 = 0. (e) g = g0 f /f0 . (f) (nf g + (2 − n)f0 g)2 − 4f gf0 g0 ≤ 0. (g) ((2n − 1)f g0 − (2n − 3)f0 g)2 − 4f0 g0 f g ≤ 0. The polynomial differential system (3) has at most one limit cycle sur- rounding the origin if the following condition holds (h) ((2n − 1)f0 g − (2n − 3)f g0 )2 − 4f0 g0 f g ≤ 0. Theorem 1 is proved in the next section. 2. Proofs For proving Theorem 1 we need the following two lemmas due to Lloyd [19]. Lemma 2. We have in a simply connected open set V containing the origin the differential system in polar coordinates (6) ṙ = S1 (r, θ), θ̇ = S2 (r, θ), where S1 and S2 are C 1 2π-periodic functions such that S1 (0, θ) = 0 for all θ, and S2 (r, θ) > 0 in V . The differential system (6) is equivalent 4 J. LLIBRE AND T. SALHI to the differential equation dr S1 (r, θ) (7) = = S(r, θ). dθ S2 (r, θ) Therefore, if ∂S ∂S ∂S (8) ≡ 0, 6 0, or >0 ∂r ∂r ∂r in V , then the differential system (6) has no limit cycles in V . Lemma 3. Consider the differential system (6) defined in an annular region A that encircles the origin and where S2 (r, θ) > 0. Then in A, the differential system (6) is equivalent to the differential equation (7). If (8) hold in A, then the differential system 6 has at most one limit cycle in A. Proof statement (a) of Theorem 1. If f = 0 equation (5) becomes dr f0 (θ)r = . dθ g0 (θ) + g(θ)rn−1 Since λ > and µ > 0 this last equation does not change sign in the region C. The solution r(θ) of this equation increases or decreases, so these solutions cannot be periodic in the region R, and consequently the polynomial differential system (3) has no limit cycles In R.  Proof statement (b) of Theorem 1. Since g = 0 the differential equa- tion (4) becomes ṙ = f0 (θ)r + f (θ)rn , θ̇ = g0 (θ). The straight lines θ = 0 and θ = π/2 are invariant for system (3). So this system cannot have limit cycles surrounding the origin. This completes the proof of this statement.  Proof statement (c) of Theorem 1. Since a0n = 0 the differential sys- tem (3) has the straight line x = 0 invariant, consequently this system has no limit cycles surrounding the origin.  The same argument used in the proof of statement (c) proves state- ment (d). Proof statement (e) of Theorem 1. Since g = f g0 /f0 system (4) be- comes f (θ) n−1 ṙ = f0 (θ)r + f (θ)rn , θ̇ = g0 (θ)(1 + )r . f0 (θ) LIMIT CYCLES OF A CLASS OF POLYNOMIAL DIFFERENTIAL SYSTEMS 5 So the proof ends following the same argument used in the proof of statement (b).  Proof statement (f ) of Theorem 1. Let f0 r + f r n S(r, θ) = , g0 + grn−1 defined in the simply connected region R. The derivative of S with respect to r is ∂S f0 g0 + (nf g0 + (2 − n)f0 g)rn−1 + f gr2n−2 = . ∂r (g0 + grn−1 )2 Since (nf g+(2−n)f0 g)2 −4f gf0 g0 ≤ 0 the numerator of ∂S/∂r does not change of sign, and we can apply Lemma 2 to the differential equation (5), and the proof of this statement follows.  Proof √ statement (g) of Theorem 1. Doing the change of variables R = r in the region C, the differential equation (5) becomes dR f0 R + f R2n−1 (9) = = S(R, θ). dθ 2(g0 + gR2n−2 ) The derivative of S with respect to R is ∂S f0 g0 + ((2n − 1)f g0 − (2n − 3)f0 g)R2n−2 + f gR4n−4 = . ∂R 2(g0 + gR2n−2 )2 Since ((2n − 1)f g0 − (2n − 3)f0 g)2 − 4f0 g0 f g ≤ 0 the numerator of ∂S/∂R does not change of sign,, and again we can apply Lemma 2 to the differential equation (9), and statement (g) is proved.  Proof √ statement (h) of Theorem 1. Doing the change of variables R = 1/ r in the region R the differential equation (5) becomes dR R(f0 R2n−2 + f ) (10) = = S(R, θ). dθ 2(g0 R2n−2 + g) So the derivative of S with respect to R is ∂S f g + ((2n − 1)f0 g − (2n − 3)f g0 )R2n−2 + f0 g0 R4n−4 =− . ∂R 2(g0 R2n−2 + g)2 √ The image of the region R under the map r → 1/ r is an annular region A, one of the boundaries of this annulus is the infinity. Since ((2n − 1)f0 g − (2n − 3)f g0 )2 − 4f0 g0 f g ≤ 0 the numerator of ∂S/∂R does not change of sign, we can apply Lemma 3 in the annular region A to the differential equation (10), and this completes the proof of this statement.  6 J. LLIBRE AND T. SALHI In short Theorem 1 is proved. 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Poincaré, Mémoire sur les courbes définies par une équation differentielle I, II, J. Math. Pures Appl. 7 (1881), 375–422; 8 (1882), 251–296; Sur les courbes définies pas les équations differentielles III, IV, 1 (1885), 167–244; 2 (1886), 155–217. [22] van der Pol, On relaxation–oscillations, Phil. Mag. 2 (1926), 978–992. 1 Departament de Matemàtiques, Universitat Autònoma de Barce- lona, 08193 Bellaterra, Barcelona, Catalonia, Spain Email address: jllibre@mat.uab.cat 2 Department of Mathematics, University of Mohamed El Bachir El Ibrahimi Bordj Bou Arréridj, El anasser, Algeria Email address: t.salhi@univ-bba.dz