This is a preprint of: “A new approach for the study of limit cycles”, Johanna Denise Garcı́a Saldaña, Armengol Gasull, Hector Giacomini, J. Differential Equations, vol. 269, 6269–6292, 2020. A NEW APPROACH FOR THE STUDY OF LIMIT CYCLES J.D. GARCÍA-SALDAÑA, A. GASULL, AND H. GIACOMINI Abstract. We prove that star-like limit cycles of any planar polynomial system can also be seen either as solutions defined on a given interval of a new associated planar non-autonomous polynomial system or as heteroclinic solutions of a 3- dimensional polynomial system. We illustrate these points of view with several examples. One of the key ideas in our approach is to decompose the periodic solutions as the sum of two suitable functions. As a first application we use these new approaches to prove that all star-like reversible limit cycles are algebraic. As a second application we introduce a function whose zeroes control the periodic orbits that persist as limit cycles when we perturb a star-like reversible center. As far as we know this is the first time that this question is solved in full generality. Somehow, this function plays a similar role that an Abelian integral for studying perturbations of Hamiltonian systems. 1. Introduction and results Consider a planar differential system ẋ = X(x, y), ẏ = Y (x, y), (1) where X and Y are polynomial functions vanishing at the origin. The most elusive problem about this system is to know its number of limit cycles. Many efforts have been dedicated to this objective and the reader can consult several books, and their references, where this question is addressed, see [5, 9, 16, 17]. Of course, it is also very related with the celebrated XVIth Hilbert’s problem, see [7, 12]. The goal of this work is to present a new approach to study the so called star-like limit cycles. To give a first description of our approach we start introducing some notation. By applying the polar coordinates transformation x = r cos θ, y = r sin θ to sys- tem (1) and by eliminating the variable t, we obtain the associated non-autonomous differential equation  dr R r cos θ, r sin θ = , (2) dθ T r cos θ, r sin θ where   R(x, y) = xX(x, y) + yY (x, y) /r and T (x, y) = xY (x, y) − yX(x, y) /r2 . Notice that (2) is well defined in the region R2 \ T , where T = {(x, y) : T (x, y) = 0}. Moreover, the periodic solutions of (1) that do not intersect T correspond to smooth 2π-periodic solutions of (2). These periodic orbits are usually called star- like periodic orbits. In particular when these 2π-periodic solutions are isolated in the set of periodic solutions they will correspond to star-like limit cycles. 2010 Mathematics Subject Classification. Primary 34C25, 34C07; Secondary 34C37. Key words and phrases. Periodic orbits, Limit cycle, Abelian integral, Heteroclinic solution, Reversible center, Algebraic limit cycle. 1 2 J.D. GARCÍA-SALDAÑA, A. GASULL, AND H. GIACOMINI As we will prove in Theorem 2.4 and Corollary 2.5 each smooth star-like periodic orbit r = F (θ) can be written uniquely as r = F (θ) = f (sin θ) + g(sin θ) cos θ, (3) for some f, g ∈ C ∞ ([−1, 1]), that is, both functions are C ∞ in an open neighbor- hood of [−1, 1]. A similar decomposition, but involving the functions sin(2πt/T ) and cos(2πt/T ), can also be applied to each of the components of any T -periodic solution of a k-dimensional differential system, parameterized by the time t. We believe that this point of view will help to a better understanding of this type of solutions. In this paper we will concentrate on the planar case. The starting point of this work is to use equation (3) and an associated non- autonomous polynomial differential system involving u, f (u) and g(u) to characterize the star-like limit cycles of system (1) as the solutions with interval of definition including all u in [−1, 1]. These solutions can also be seen as heteroclinic solutions of a 3d-autonomous system joining singularities contained in the planes u = −1 and u = 1, see Theorems 2.6 and 2.7 in next section. As applications of this approach: • We give a systematic procedure to find reversible star-like limit cycles with respect to a straight line and prove that all of them are algebraic, see Theo- rem 1.1. • We introduce a function, similar to an Abelian integral, that controls which of the periodic orbits of a reversible star-like center can persist after a per- turbation of the system, see Theorem 1.3. Let us describe with more detail our main results given in the above two items. A planar system is called reversible (with respect to a straight line) if after a rotation it is invariant by the change of variables and time (x, y, t) → (−x, y, −t). These vector fields can be written as ẋ = A(x2 , y), ẏ = xB(x2 , y), for some smooth functions A and B. Similarly, a limit cycle is called reversible if after a translation and a rotation writes as (x(t), y(t)) and it satisfies that (x(t), y(t)) = (−x(−t), y(−t)) for all t ∈ R. Recall also that a limit cycle is called algebraic if it is included in the zero set of a real polynomial in two variables, that is in G(x, y) = 0, for some polynomial G. In Section 4 we prove: Theorem 1.1. Star-like reversible limit cycles of polynomial systems are algebraic. An auxiliary result that we use to prove the above theorem is given in the self- contained Section 3, where we develop a method for detecting whether two different polynomial systems share a trajectory. About our second application, consider any perturbed reversible system: ẋ = A(x2 , y) + εC(x, y), ẏ = xB(x2 , y) + εD(x, y), (4) where we assume that in the punctured neighborhood U of the origin, the function x2 B(x2 , y) − yA(x2 , y), that gives the sign of θ̇, does not vanish and C(0, 0) = D(0, 0) = 0. This set U is non empty for instance when A(0, y) = ay + O2 (y), where O2 (y) denotes terms of order at least 2 in y, and B(0, 0) = b, ab < 0, because then the origin is a monodromic non-degenerate critical point. Notice that under these hypotheses and in this region, when ε = 0 the origin is a star-like reversible center. To state the results of our second application we also need the notation given in the following lemma. A NEW APPROACH FOR THE STUDY OF LIMIT CYCLES 3 Lemma 1.2. Introduce the variables f, g, u, v, with v 2 = 1 − u2 , and consider x = (f + vg)v, y = (f + vg)u. Then, any polynomial H(x, y) writes, using that v 2k+1 = v(1 − u2 )k , as  H(x, y) = H (f + vg)v, (f + vg)u) = H0 (f, g, u) + vH1 (f, g, u). Moreover, when H(x, y) = K(x2 , y) for some polynomial K, H0 (f, g, u) = K0,0 (f, u) + (1 − u2 )g 2 K0,1 (f, g, u) and H1 (f, g, u) = gK1,0 (f, g, u),  for some polynomials K0,0 , K0,1 and K1,0 , where K0,0 (f, u) = K (1 − u2 )f 2 , uf . As we will see in Lemma 5.1 the solutions of the unperturbed system (4)ε=0 with initial condition r(0) = ρ are r = fρ (sin θ) where fρ (u) is the solution of the first order Cauchy initial value problem (A0,0 + uB0,0 f ) f f0 = , f (0) = ρ, (5) (1 − u2 )B0,0 f − uA0,0   where A0,0 = A (1 − u2 )f 2 , uf and B0,0 = B (1 − u2 )f 2 , uf . Our main result is: Theorem 1.3. For each ρ > 0, such that r = fρ (sin θ) is periodic orbit of the unperturbed system (4)ε=0 , we define Z 1 Ru N (ρ) = Mρ (u)e− 0 Lρ (s)ds du, (6) −1 where Sρ (u) Tρ (u) Lρ (u) = , Mρ (u) = , Wρ (u) Wρ (u) with  2 Sρ (u) =(1 − u2 ) A1,0 B0,0 − A0,0 B1,0 + 2u(1 − u2 )B0,0 2 fρ (u) − 4u2 (1 − u2 )A0,0 B0,0 fρ (u) − u(1 − 2u2 )A20,0 ,  Tρ (u) = − fρ (u) A0,0 D0 − (1 − u2 )B0,0 C1 fρ (u) , 2 Wρ (u) =(1 − u2 ) uA0,0 − (1 − u2 )B0,0 fρ (u) , where all the functions D0 , C1 , A0,0 , A1,0 , B0,0 and B1,0 , introduced in Lemma 1.2, are evaluated either at (f, g, u) = (fρ (u), 0, u) or at (f, u) = (fρ (u), u). If for ε > 0 small enough, there is a continuous family r = r(θ, ε) of limit cycles of (4) such that r(0, ε) → ρ0 as ε goes to zero, then N (ρ0 ) = 0. Recall that Abelian integrals are used to study the same problem but for pertur- bations of Hamiltonian systems and that this approach can be easily extended to study the perturbations of systems with an integrating factor, see for instance [3]. Theorem 1.3 is similar but for perturbations of reversible centers. As far as we know, this is the first time that this question is solved in full generality for systems having a star-like reversible center. We are studying if simple zeroes of N always give rise to limit cycles of the perturbed system. In Section 5 we apply Theorem 1.3 to a couple of examples that can also be approached with the classical theory of Abelian integrals because the respective unperturbed reversible systems have also explicit integrating factor. 4 J.D. GARCÍA-SALDAÑA, A. GASULL, AND H. GIACOMINI Notice that our point of view allows to treat perturbations of general reversible centers, independently whether they have or not explicit integrating factors. More- over, as we will explain in Remark 5.2, our result allows to approach the question of the existence of persistent periodic orbits as a two point connection problem for singular linear differential equations which turns out to be a very classical problem, see [8]. We believe that this fact provides a promising new way for studying limit cycles arising from perturbed reversible centers. 2. General results We start proving the general decomposition result stated in (3) for analytic 2π- periodic solutions that, as we have already said, is one of the key points of our approach. Next lemma shows that the functions f and g are analytic in (−1, 1). Lemma 2.1. Let F (θ) be a real analytic 2π-periodic function. Then: (i) There exist two unique analytic functions f and g defined in (−1, 1) such that F (θ) = f (sin θ) + g(sin θ) cos θ. (ii) There exist two unique analytic functions f¯ and ḡ defined in (−1, 1) such that F (θ) = f¯(cos θ) + ḡ(cos θ) sin θ. (7) Proof. We will prove item (ii). It is easy to see that both are equivalent, simply by changing θ by θ + π/2. Notice that evaluating (7) at −θ we get that   F (−θ) = f¯ cos(−θ) + ḡ cos(−θ) sin(−θ) = f¯(cos θ) − ḡ(cos θ) sin θ. Joining this equation and (7) we obtain that F (θ) + F (−θ) F (θ) − F (−θ) f¯(cos θ) = and ḡ(cos θ) = . 2 2 sin θ For θ ∈ (0, π), cos θ takes all values in (−1, 1) and doing the change of variables u = cos θ we get that F (arccos u) + F (− arccos u) F (arccos u) − F (− arccos u) f¯(u) = , ḡ(u) = √ . 2 2 1 − u2 Clearly, both functions are analytic in (−1, 1).  Next, we introduce some preliminary results that will be used to show that the functions f, g, f¯ and ḡ in the above decompositions are C ∞ ([−1, 1]). Recall that the Chebyshev polynomials of first and second kind, are respectively, Tn and Un where cos(nθ) = Tn (cos θ), sin(nθ) = Un−1 (cos θ) sin θ, (8) and the polynomials Tn and Un both satisfy the recurrence Pn+1 (x) = 2xPn (x) − Pn−1 (x), (9) with initial conditions P0 = 1 and P1 = x, (first kind and Tn = Pn ) or P0 = 1 and P1 = 2x, (second kind and Un = Pn ). Given any smooth function F , F (k) denotes its k-th derivative. A NEW APPROACH FOR THE STUDY OF LIMIT CYCLES 5 Given any polynomial P, define kP k := maxx∈[−1,1] |P (x)|. Let Pn denote the set of polynomials of degree ≤ n. Markov inequality for polynomials says that for any 0 < k ∈ N, kP (k) k ≤ kTn(k) k kP k, P ∈ Pn (10) and n2 (n2 − 12 )(n2 − 22 ) · · · (n2 − (k − 1)2 ) kTn(k) k = Tn(k) (1) = . 1 · 3 · · · (2k − 1) It was proved by Andrei Markov in 1889 for k = 1 (kP 0 k ≤ n2 kP k) and extended to any k ≥ 1 by his kid brother Vladimir Markov in 1892, see [15]. We will use that n2k kTn(k) k ≤ (11) (2k − 1)!! and, moroever, since Un−1 = Tn0 /n (k) 1 (k+1) n2k+1 kUn−1 k = kTn k≤ . n (2k + 1)!! We will also need the following version of Bernstein theorem for analytic 2π- periodic functions, see Exercise 4 in [10, Sec. 4]. Theorem 2.2. Let F be a 2π-periodic analytic function. Then there exist two positive constants K and a such that its n-th Fourier coefficients Fn , n ∈ Z, satisfies |Fn | ≤ Ke−a|n| . Next proposition is our last preliminary result. Its proof is a direct consequence of the dominated convergence theorem. Proposition 2.3. Fix k ∈ N and let U ⊂ R be an open interval. Assume that for all n ∈ N ∪ {0} the functions Gn : U → R are C k (U), and for each x0 ∈ U there is an open subset of U, Vx0 ,k 3 x0 , such that ∞ X G(k) n (x) {x∈Vx0 ,k } ≤ Mn,k and Mn,k < ∞. n=0 P∞ P∞ (k) Then, if G(x) = n=0 Gn (x), it holds that G ∈ C k (U) and G(k) (x) = n=0 Gn (x). Proof. We will prove the result for k = 1. The result for k > 1 simply follows by (k−1) applying the same proof to Gn . Consider k = 1. Let us prove first that G is derivable. By the mean value theorem, we have that for |h| small enough ∞ ∞ G(x0 + h) − G(x0 ) X Gn (x0 + h) − Gn (x0 ) X 0 = = Gn (sn ), h n=0 h n=0 for some sn ∈ (x − |h|, x + |h|) ⊂ Vx0 ,1 . Hence X∞ X∞ G(x0 + h) − G(x0 ) 0 = Gn (sn ) ≤ Mn,1 < ∞ h n=0 n=0 6 J.D. GARCÍA-SALDAÑA, A. GASULL, AND H. GIACOMINI and by the dominated convergence theorem, X∞ G(x0 + h) − G(x0 ) Gn (x0 + h) − Gn (x0 ) G0 (x0 ) = lim = lim h→0 h h→0 n=0 h ∞ X ∞ Gn (x0 + h) − Gn (x0 ) X 0 = lim = Gn (x0 ). n=0 h→0 h n=0 That G0 is continuous is again a consequence of the dominated P convergence theorem, because the fact that |G0n (x)| ≤ Mn,1 for all x ∈ Vx0 , and ∞n=0 Mn,1 < ∞ implies that ∞ X ∞ X ∞ X 0 lim G (x0 + h) = lim G0n (x0 + h) = lim G0n (x0 + h) = G0n (x0 ) = G0 (x0 ). h→0 h→0 h→0 n=0 n=0 n=0  Next result gives the desired decomposition. Theorem 2.4. Let F (θ) be a real analytic 2π-periodic function. Then: (i) There exist two unique C ∞ ([−1, 1]) functions f and g such that F (θ) = f (sin θ) + g(sin θ) cos θ. (ii) There exist two unique C ∞ ([−1, 1]) functions f¯ and ḡ such that F (θ) = f¯(cos θ) + ḡ(cos θ) sin θ. (12) Proof. As in the proof of Lemma 2.1 it suffices to prove item (ii). The uniqueness is a consequence of this lemma. Let us prove (12). We start decomposing the Fourier series of F as follows ∞ X ∞ X F (θ) = an cos(nθ) + bn sin(nθ) n=0 n=0 ∞ X X∞ = an Tn (cos θ) + bn Un−1 (cos θ) sin θ n=0 n=1 ∞ X ∞ X = an Tn (cos θ) + sin θ bn Un−1 (cos θ), n=0 n=1 where, we have used (8), and since F is analytic, by Theorem 2.2, |an | ≤ M e−an , |bn | ≤ M e−an , (13) for some M > 0 and a > 0. Hence, we introduce the functions ∞ X ∞ X f¯(u) := an Tn (u) and ḡ(u) := bn Un−1 (u), n=0 n=1 and we prove first that they are of class C ∞ (−1, 1). Consider for instance f¯. Although we already know that f¯ is analytic in (−1, 1), see Lemma 2.1, we include a different A NEW APPROACH FOR THE STUDY OF LIMIT CYCLES 7 proof that it is C ∞ (−1, 1) because it gives the idea of how to proceed at −1 and 1. By joining inequalities (11) and (13) it holds that for k ∈ N ∞ X ∞ X ∞ X n2k an Tn(k) (u) ≤ an Tn(k) (u) ≤ M e−an n=0 n=0 n=0 (2k − 1)!! ∞ X 2k M n ≤ < ∞. (2k − 1)!! n=0 ean By applying Proposition 2.3 to Gn = an Tn for all k ∈ N, we get that f¯ ∈ C ∞ (−1, 1). In order to apply Proposition 2.3 to prove that f¯ is C ∞ , at the points ±1 we need to bound all the derivatives of Tn on a fixed neighborhood of each of these points. We consider, for instance u = 1 and define Jε := [1 − ε, 1 + ε], for ε > 0 to be fixed. (k) We will bound maxy∈Jε |Tn (y)|. With this goal, we introduce Qn,ε,k (u) := ε−k Tn (εu + 1). It is clear that (k) Qn,ε,k (u) = Tn(k) (εu + 1) and (k) max |Tn(k) (y)| = max |Tn(k) (εu + 1)| = kTn(k) (εu + 1)k = kQn,ε,k k. y∈Jε u∈[−1,1] By applying Markov inequality (10) to Qn,ε,k we get that for any k ∈ N, (k) kQn,ε,k k ≤ kTn(k) k kQn,ε,k k, or equivalently, (k) kTn k max |Tn(k) (y)| ≤ kTn (εu + 1)k. (14) y∈Jε εk Since kTn k = 1, Tn0 (1) = n2 and all roots of Tn are real and contained in (−1, 1) it holds that |Tn0 (u)|u≥1 > 0 and as a consequence kTn (εu + 1)k = max |Tn (y)| = Tn (1 + ε). (15) y∈Jε Let us compute this last expression. If tn := tn (ε) = Tn (1 + ε), by (9) it holds that tn+1 = 2(1 + ε)tn − tn−1 , t0 = 1 and t1 = 1 + ε. Either by solving the above second order linear difference equation with constant coefficients, or by using the expression of the Chebyshev polynomials in terms of square roots, we get that 1 p n 1  p n tn = Tn (1 + ε) = 1 + ε + ε(2 + ε) + 1 + ε − ε(2 + ε) , 2 2 and as a consequence, p n |Tn (1 + ε)| = |tn | ≤ 1 + ε + ε(2 + ε) . By using the above inequality, (13), (14) and (15) we obtain that n2k p n max |Tn(k) (y)| ≤ k 1 + ε + ε(2 + ε) . y∈Jε ε (2k − 1)!! 8 J.D. GARCÍA-SALDAÑA, A. GASULL, AND H. GIACOMINI Taking y ∈ Jε , by using the above inequality and (13) we get that ∞ X ∞ X max an Tn(k) (y) ≤ max an Tn(k) (y) y∈Jε y∈Jε n=0 n=0 X∞ n2k p n ≤ M e−an 1 + ε + ε(2 + ε) n=0 εk (2k − 1)!! X ∞ M p n ≤ k n2k e−a (1 + ε + ε(2 + ε)) , ε (2k − 1)!! n=0 p which is a convergent series if ε > 0 is so small that e−a (1 + ε + ε(2 + ε)) < 1. Again, as a consequence of Proposition 2.3 applied to Gn = an Tn and all k ∈ N, we get that f¯ is C ∞ at u = 1. The proof at u = −1 is analogous. The proof for ḡ also follows the same steps.  A consequence of Theorem 2.4 is: Corollary 2.5. Any star-like limit cycle r = F (θ) of (1) can be uniquely written as r = F (θ) = f (sin θ) + g(sin θ) cos θ, where f and g are C ∞ ([−1, 1]). Proof. Since star-like limit cycles are r = F (θ) where this function is a solution of (2) and in a neighborhood of it the denominator of this equation does not vanish, F is a 2π-periodic analytic function. Then the result follows by using Theorem 2.4.  Our first characterization of the periodic orbits of (1) as solutions of a planar non-autonomous system is given in next theorem. Theorem 2.6. Let r = f (sin θ) + g(sin θ) cos θ be a star-like periodic orbit of (2) where f, g ∈ C ∞ ([−1, 1]). Then f (u) and g(u) are solutions of ( (1 − u2 )T1 f 0 (u) + (1 − u2 )T0 g 0 (u) − uT0 g(u) − R0 = 0, (16) T0 f 0 (u) + (1 − u2 )T1 g 0 (u) − uT1 g(u) − R1 = 0, where T and R are given in the polar expression (2) of (1) and Ti = Ti (f (u), g(u), u) and Ri = Ri (f (u), g(u), u), for i = 0, 1, as in Lemma 1.2. Notice that here, and from now on, T0 and T1 do not denote the first two Cheby- shev polynomials, but the polynomials associated to the decomposition of T. Proof of Theorem 2.6. The right-hand side of the polar expression (2) of (1), replac- ing  x = r cos θ = f (sin θ) + g(sin θ) cos θ cos θ = (f (u) + g(u)v)v,  y = r sin θ = f (sin θ) + g(sin θ) cos θ sin θ = (f (u) + g(u)v)u, can be written as R(r cos θ, r sin θ) R((f (u) + g(u)v)v, (f (u) + g(u)v)u) = . T (r cos θ, r sin θ) T ((f (u) + g(u)v)v, (f (u) + g(u)v)u) A NEW APPROACH FOR THE STUDY OF LIMIT CYCLES 9 Any star-like periodic orbit of (1) r = f (sin θ) + g(sin θ) cos θ must satisfy dr = f 0 (sin θ) cos θ + g 0 (sin θ) cos2 θ − g(sin θ) sin θ dθ = f 0 (u)v + g 0 (u)v 2 − g(u)u = f 0 (u)v + g 0 (u)(1 − u2 ) − g(u)u. Hence R0 (f (u), g(u), u) + R1 (f (u), g(u), u)v f 0 (u)v + g 0 (u)(1 − u2 ) − g(u)u = . T0 (f (u), g(u), u) + T1 (f (u), g(u), u)v Eliminating the denominator and using once more that v 2 = 1 − u2 , we arrive to  (1 − u2 ) f 0 (u)T1 + g 0 (u)T0 − ug(u)T0 − R0 +    T0 f 0 (u) + (1 − u2 )g 0 (u) − ug(u) T1 − R1 v = 0, where Ti = Ti (f (u), g(u), u), Ri = Ri (f (u), g(u), u), i = 0, 1. From the above equality and by using the uniqueness of decomposition of (3) proved in Theorem 2.4 both terms must be zero, giving rise to the system of two differential equations (16).  The following theorem gives a new and geometric interpretation of the star-like periodic orbits of system (1). Theorem 2.7. Let r = f (sin θ) + g(sin θ) cos θ be a star-like periodic  orbit of (2) where f, g ∈ C ∞ ([−1, 1]). Then γ(s) = f (u(s)), g(u(s)), u(s) is a heteroclinic solution of the following 3d-polynomial differential system    df   = (1 − u2 )(R1 T0 − R0 T1 ),   ds dg = −(1 − u2 )T1 (R1 + ugT1 ) + T0 (R0 + ugT0 ), (17)   ds    du = (1 − u2 )(T02 − (1 − u2 )T12 ),  ds joining two critical points, each one of them in one of the two invariant planes u = 1 and u = −1, where Ti = Ti (f, g, u) and Ri = Ri (f, g, u) are as in Theorem 2.6. Proof. By isolating f 0 and g 0 from system (16) of Theorem 2.6, we get R1 T0 − R0 T1 f 0 (u) = , T02 − (1 − u2 )T12 0 −(1 − u2 )T1 (R1 + ug(u)T1 ) + T0 (R0 + ug(u)T0 ) g (u) = . (1 − u2 )(T02 − (1 − u2 )T12 ) After changing the time in such a way that the denominators are removed we can write the above system as the differential system of the statement. From Theo- rem 2.6 we also know that γ(s) is a solution of system (17). Finally, since the periodic orbit is star-like it holds that T0 + vT1 (that is θ̇) does not vanish on it. Thus T02 6= v 2 T12 = (1 − u2 )T12 . This fact implies that (17) has no critical points in the strip {(f, g, u) : −1 < u < 1}. As a consequence, γ(s) is a heteroclinic orbit of (17) with the behavior given in the statement.  10 J.D. GARCÍA-SALDAÑA, A. GASULL, AND H. GIACOMINI The above theorem proves that star-like periodic orbits of (2) can also be seen as heteroclinic solutions of (17) connecting a critical point of the invariant plane {u = −1} with a critical point of the invariant plane {u = 1}. Notice that both planes contain continua of critical points of the form (f, g, ±1) with f and g satisfying T0 (R0 ± gT0 ) = 0, with these functions evaluated at (f, g, ±1). 2.1. A couple of examples. Consider the rigid (θ̇ ≡ 1) cubic system ( ẋ = −y + x(3x2 + 2xy + y 2 − 1), (18) ẏ = x + y(3x2 + 2xy + y 2 − 1). Our associated 3d-differential system (17) is df  = (1 − u2 ) 2uf 3 − g − (1 − u2 )(2u2 − 3)g 3 + 6u(1 − u2 )f g 2 − 3(2u2 − 3)f 2 g , ds dg  = −f − (2 u2 − 3)f 3 + ug + (1 − u2 ) 2 u(1 − u2 )g 2 − 3 (2 u2 − 3)f g + 6 uf 2 g, ds du = 1 − u2 . ds It has the invariant curve 1 + (2u2 − 3)f 2 = 0, g = 0, that gives rise to the solution f (u) = (3 − 2u2 )−1/2 and g = 0 of the corresponding system (16) and to the heteroclinic orbit of the above 3d-differential system  −1/2  2 (f (s), g(s), u(s)) = 3 − 2U (s) , 0, U (s) , with U (s) = tanh (s), joining (1, 0, −1) and (1, 0, 1). The corresponding limit cycle of the original planar system, expressed in polar coordinates, is r = F (θ) = (3 − 2 sin2 θ)−1/2 . In fact, system (18) in polar coordinates correspond to the Bernoulli equation dr  = −r + 2 cos2 θ + 2 cos θ sin θ + 1 r3 dθ and all its solutions are r = 0 and r = ±(3 − 2 sin2 θ + ke2θ )−1/2 . The next example ( ẋ = −y + x(x2 − y 2 ), (19) ẏ = x + y(x2 − y 2 ) is also a rigid cubic system, but with a continuum of periodic orbits, because as we will see it has a center. It is one of the systems appearing in [1]. Its associated 3d-differential system (17) is   df    = (1 − u2 )(1 − 2u2 ) 3f 2 + (1 − u 2 2 )g g,   ds   dg = ug + (1 − 2 u2 ) f 2 + 3(1 − u2 )g 2 f,   ds   du   = 1 − u2 . ds Recall that given an n-dimensional polynomial vector field Z it is said that a hypersurface H = 0 is invariant if Z(H) = KH for some polynomial K, called the A NEW APPROACH FOR THE STUDY OF LIMIT CYCLES 11 cofactor of H, see for instance [13]. Clearly these hypersurfaces are invariant by the flow of Z. It is not difficult to see that H(f, g, u) := 2f g − 2u(f 2 − (1 − u2 )g 2 )2 = 0 is an invariant surface with cofactor K(f, g, u) = u + 8(1 − u2 )(1 − 2u2 )f g, see Figure 1. This surface is full of heteroclinic orbits connecting the planes u = −1 and u = 1 that correspond to the periodic orbits of the center. Figure 1. Continua of heteroclinic solutions contained in H(f, g, u) = 0. Notice that system (19) has a center at the origin because it writes in polar coordinates as dr  = cos2 θ − sin2 θ r3 dθ and its solutions are r = 0 and r(θ) = ±(k − sin(2θ))−1/2 , and they are 2π-periodic for k > 1. 3. Common solutions of two polynomial differential systems In this section we describe a method in six steps to find conditions for two families of differential systems ( ( f˙ = P1 (f, u), f˙ = P2 (f, u), a) b) (20) u̇ = Q1 (f, u), u̇ = Q2 (f, u), where Pi (f, u) and Qi (f, u) are polynomial with respect to f , u and some parameters, for i = 1, 2, to have a common solution. 1. Write systems (20a) and (20b) as the rational differential equations P1 (f, u) P2 (f, u) a) f 0 (u) = , and b) f 0 (u) = . (21) Q1 (f, u) Q2 (f, u) 2. Consider the polynomial H(f, u) := P1 (f, u)Q2 (f, u) − P2 (f, u)Q1 (f, u), 12 J.D. GARCÍA-SALDAÑA, A. GASULL, AND H. GIACOMINI P1 (f,u) P2 (f,u) given by the numerator of Q1 (f,u) − Q2 (f,u) . 3. Calculate the derivative of H(f, u) with respect to u where f = f (u) satisfies either equation (21a) or (21b) ∂H ∂H Ji (f, u) := (f, u)Pi (f, u) + (f, u)Qi (f, u), ∂f ∂u where i = 1 if f (u) satisfies (21a) and i = 2 in the other case. 4. Compute the resultant between H(f, u) and Ji (f, u) with respect to u  Gi (f ) := Resu H(f, u), Ji (f, u) , where Gi (f ) is a polynomial that depends on f and the parameters of the equations. We can also compute the resultant with respect to f and, in this case, we would obtain a polynomial that depends on u and the parameters. 5. For some i = 1 or 2 solve the algebraic system obtained by imposing that all the coefficients of Gi (f ) vanish. 6. A necessary condition for systems (20a) and (20b) to have a common solution is that Resu (H(f, u), Ji (f, u)) ≡ 0 and Resf (H(f, u), Ji (f, u)) ≡ 0. So, the solutions of the algebraic system of the step 5 provide the conditions on the parameters that can give rise to common solutions. Let us illustrate our approach with an example. Consider   f˙ = f 3 + λu − u2 + 1, f˙ = ρf 4 u − f u3 + f u + 2u, a) b) u̇ = f 4 − f u2 + 3f 2 + f, u̇ = f 3 + 3f 2 − u2 + σ. We want to find conditions on the parameters λ, ρ and σ in order that the above systems can have a common solution. Following the steps described above, we have the associated rational differential equations f 3 + λu − u2 + 1 ρf 4 u − u3 f + f u + 2u f 0 (u) = , f 0 (u) = , f 4 − f u2 + 3f 2 + f f 3 + 3f 2 − u2 + σ respectively. Then H(f, u) = −ρf 8 u + (1 − 3 ρ u)f 6 + (3 − (ρ + 1)u + (ρ + 1)u3 )f 5 − 2 f 4 u + (1 + σ + (λ − 3)u − 2 u2 + 3 u3 )f 3 + (3 + (3 λ − 7)u − 3 u2 + 2 u3 − u5 )f 2 − 2u(1 − u2 )f + (σ − u2 )(λ u − u2 + 1) A NEW APPROACH FOR THE STUDY OF LIMIT CYCLES 13 and J2 (f, u) = −ρ(1 + 8 ρ u2 )f 11 − 3 ρf 10 − 3 ρ(6 ρ u2 − 2 u + 1)f 9 + ρ (5 ρ + 13)u4 + (3 − 9 ρ − 5 ρ2 )u2 + ρ(15 u − σ − 10) − 1)f 8 − (5 + 3 ρ − (9 − 15 ρ)u2 )f 7 + (27 ρ u4 − 6(ρ + 1)u3 + (3 ρ λ − 24 ρ + 9)u2 + (3 ρ σ + 3 ρ + 2)u − 3 ρ σ + λ − 9)f 6 + (−(7 ρ + 5)u6 + (11 ρ + 2)u4 − (6 ρ + 15)u3 + (6 ρ λ + 3 ρ σ − 54 ρ + 3 σ + 29)u2 + (6 ρ + 9)u − ρ σ + 6 λ − σ − 16)f 5 + ((12 ρ + 3)u4 − 4(3 ρ − 2)u2 + 12 u + 9 λ − 2 σ − 23)f 4 + (−9 u6 + 6 u5 + (9 − 3 λ)u4 − (3 σ + 1)u3 + (9 σ − λ − 4)u2 + (1 − 3 σ)u + 2 λ σ − 3 σ − 6)f 3 + (2 u8 − u6 + 6 u5 − (6 λ + 5 σ − 30)u4 − 6 u3 + (6 σ − 25)u2 − 6( σ − 1)u + σ (6 λ − 7))f 2 + (−6 u6 + 6 u4 − 12 u3 + (12 λ + 6 σ − 28)u2 + 12 u − 2 σ)f − 4 u5 + (3 λ + 4)u4 + (6 σ + 2)u3 − (4 λ σ + 4)u2 − 2 σ (σ + 1)u + λ σ 2 . Computing the resultant between H(f, u) and J2 (f, u) with respect to u we obtain G2 (f ) = −f 2 (f 3 + 3f 2 + σ)(ρf 4 − f 4 − 3f 3 − σf + f + 2)2 G(f ), where G(f ) is a polynomial of degree 64 in f and whose coefficients depend on the parameters λ, ρ and σ. The algebraic system obtained imposing that these coefficients identically vanish is a simple over-determined system. Its unique solution is ρ = 1, λ = 2 and σ = 1. For these values of the parameters,  H(f, u) = (f 3 − u2 + 1) − f 5 u + f 2 u3 − 3 f 3 u + f 3 − f 2 u + 3 f 2 − 2 uf − u2 + 2 u + 1 . Each of the above two factors gives a candidate to be a common solution of both differential systems. Direct computations show that the only common solution is f 3 − u2 + 1 = 0. 4. Reversible limit cycles Proof of Theorem 1.1. By means of a rotation, if necessary, reversible star-like peri- odic orbits are invariant by the change of variables and time (x, y, t) → (−x, y, −t). This fact is equivalent to find solutions r = F (θ) = f (sin θ), that is with g = 0 in system (16) in Theorem 2.6. Both equations write as (1 − u2 )T1 f 0 (u) − R0 = 0, T0 f 0 (u) − R1 = 0. (22) Thus we are looking for a common solution of them, where Ti = Ti (f, 0, u) and Ri = Ri (f, 0, u), for i = 0, 1. Equivalently, R0 R1 f 0 (u) = , f 0 (u) = (23) (1 − u2 )T1 T0 and this question can be tackled with the method that we have introduced above. In particular we have seen that these periodic orbits are contained in H(f, u) := R0 (f, 0, u)T0 (f, 0, u) − (1 − u2 )R1 (f, 0, u)T1 (f, 0, u) = 0. (24) Moreover, when H(f, u) 6≡ 0, they are isolated periodic orbits and, so, they are limit cycles. 14 J.D. GARCÍA-SALDAÑA, A. GASULL, AND H. GIACOMINI Notice that f = r and u = y/r. Hence taking the numerator of H(f, u) = H(r, y/r) = 0 we obtain a polynomial F such that the star-like limit cycles are included in F (r, y) = 0. We can write F (r, y) = rG(r2 , y) + H(r2 , y), for some new polynomials G and H. Hence the limit cycles are contained in the algebraic curve r2 G2 (r2 , y) − H 2 (r2 , y) = 0.  Remark 4.1. Notice that Theorem 1.1 proves that all star-like reversible limit cycles are algebraic. This is no more true for star-like reversible continua of periodic orbits. This is so because for reversible centers it can be seen that R0 = T1 = 0 (see the proof of forthcoming Lemma 5.1) and the first equation in (22) is identically satisfied. For instance, examples of this situation can be found inside the quadratic reversible centers, also called Loud centers. We end this section with a couple of examples. Consider the following system ( ẋ = a1 y + a2 xy 2 , ẏ = b1 x + b2 y + b3 x3 + b4 x2 y + b5 y 3 . By following the proof of Theorem 1.1, its star-like reversible limit cycles are con- tained in (24), that is H(f, u) = R0 T0 − (1 − u2 )R1 T1 , where  R0 = (a2 + b4 )(1 − u2 ) + b5 u2 u2 f 3 + b2 u2 f, R1 = b3 u(1 − u2 )f 3 + u(a1 + b1 )f, T0 = b3 (1 − u2 )2 f 2 − a1 u2 + b1 (1 − u2 ),  T1 = b2 u − a2 u2 − b5 u2 − b4 (1 − u2 ) uf 2 . Thus    H(f, u) = u2 f − a1 b2 + (1 − u2 )(a2 b1 − a1 b4 ) − a1 b5 u2 f 2 + (1 − u2 )2 a2 b3 f 4 . A similar expression to J2 (f, u), but calculating the derivative of function H(f, e u) = 2 H(f, u)/(u f ) with respect to u where f = f (u) satisfies the second differential equation in (23), instead of the derivative of H(f, u), is  Je2 (f, u) = 2a1 uf 2 b3 (1 − u2 )(2a2 − b5 )f 2 − a1 b4 + a2 b1 − b1 b5 . Following the steps of the algorithm described in Section 3 we perform the resultant between He and Je2 /(uf 2 ) with respect to f. We obtain the polynomial G(u) = 16 a41 b23 (1 − u2 )2 H 2 (u), where H(u) = a21 a2 b24 − 2a21 a2 b4 b5 − a21 b24 b5 + a21 b4 b25 − 2a1 a22 b1 b4 − a22 b21 b5 + 2a1 a22 b1 b5 + 4a1 a22 b2 b3 + 2a1 a2 b1 b4 b5 − 3a1 a2 b1 b25 + a1 b1 b35 + a32 b21 − 4a1 a2 b2 b3 b5  − a1 b1 b4 b25 + a1 b2 b3 b25 u2 − a21 a2 b24 + a21 b24 b5 + 2a1 a22 b1 b4 − 4a1 a22 b2 b3 − 2a1 a2 b1 b4 b5 + 4a1 a2 b2 b3 b5 + a1 b1 b4 b25 − a1 b2 b3 b25 − a32 b21 + a22 b21 b5 . Imposing that the coefficients with respect to u of H(u) vanish we obtain several sets of solutions, that might indicate the existence of periodic orbits in the differential system. One of these sets is a2 = b5 /2, b1 = −2a1 b4 /b5 , and a1 , b2 , b4 , b3 , b5 6= 0 free parameters. A NEW APPROACH FOR THE STUDY OF LIMIT CYCLES 15 For instance taking a1 = −1, b2 = 1, b4 = 0, b3 = 1, b5 = −2 we obtain the system ( ẋ = −y(1 + xy), (25) ẏ = y + x3 − 2y 3 . The corresponding differential equations (20) obtained from (16) are ( (  f˙ = −uf (f 2 u2 + f 2 − 1), f˙ = uf (1 − u2 )f 2 − 1 , a) b) u̇ = (1 − u2 )(1 − f 2 u2 ), u̇ = (1 − u2 )2 f 2 + u2 . They share the algebraic curve (1 − u2 )2 f 4 + 2f 2 u2 − 1 = 0 that gives rise to the algebraic star-like reversible limit cycle x4 + 2y 2 − 1 = 0, with cofactor −4y 2 . Moreover, since this cofactor does not change sign we can prove that system (25) has no more limit cycles, algebraic or not. This fact follows by using the ideas introduced in [2, p. 92]. Another set of suitable values of the parameters is a1 = 20, a2 = 2, b1 = 20, b3 = −20, b4 = −2, b5 = 4 and b2 free. This gives ( ẋ = 2y(10 + xy), (26) ẏ = 20x + b2 y − 20x3 − 2x2 y + 4y 3 , and the corresponding differential equations (20) obtained from (16) are (  (  f˙ = uf 4u2 f 2 + b2 , f˙ = uf (1 − u2 )f 2 − 2 , a) b) u̇ = (1 − u2 )(4u2 f 2 − 2f 2 + b2 ), u̇ = (1 − u2 )2 f 2 + 2u2 − 1. By applying our approach we get the common solution to both systems which is given by the curve 2(1 − u2 )2 f 4 + 4(2u2 − 1)f 2 + b2 = 0. In cartesian coordinates it corresponds to the invariant algebraic curve 2x4 + 4y 2 − 4x2 + b2 = 0, with cofactor 8y 2 . It can be seen that for b2 < 0 it gives rise to an algebraic star-like reversible limit cycle, that surrounds three critical points. When b2 = 0 it gives a double heteroclinic loop, while for 0 < b2 < 2 provides two algebraic limit cycles, one being the mirror image of the other with respect to the line x = 0, and each one of them surrounding a unique critical point. Again, as in system (25), the fact that the cofactor does not change sign implies that the only limit cycles of system (26), are the ones contained in this algebraic curve. This example recovers previous results in [14]. 5. Bifurcation of limit cycles from reversible centers The goal of this section is to prove Theorem 1.3 and give some examples of application. Proof of Lemma 1.2. For a general H the proof simply follows using that v 2k = (1 − u2 )k and v 2k+1 = (1 − u2 )k v in each of its monomials. For H(x, y) = K(x2 , y) simply write  H(x2 , y) = H (f + vg)2 v 2 , (f + vg)u  = H (1 − u2 )f 2 + (1 − u2 )2 g 2 + 2(1 − u2 )vf g, uf + uvg  and perform the (finite) Taylor expansion at the point (1 − u2 )f 2 , uf .  16 J.D. GARCÍA-SALDAÑA, A. GASULL, AND H. GIACOMINI For short, we will write the decomposition introduced in Lemma 1.2 as H = H0 + vH1 and K = K0,0 + (1 − u2 )g 2 K0,1 + vgK1,0 . Lemma 5.1. The star-like periodic orbits of ẋ = A(x2 , y), ẏ = xB(x2 , y), passing through the point (0, ρ), ρ > 0, where A and B are polynomials, are given by r = fρ (sin θ) where f = fρ (u) is the solution of the first order Cauchy initial value problem (5): (A0,0 + uB0,0 f (u)) f (u) f 0 (u) = , f (0) = ρ, (1 − u2 )B0,0 f (u) − uA0,0   where A0,0 = A (1 − u2 )f 2 , uf and B0,0 = B (1 − u2 )f 2 , uf . Proof. By using Theorem 2.6 with g(u) = 0, the two differential equations (16) write as ( (1 − u2 )T1 f 0 (u) − R0 = 0, T0 f 0 (u) − R1 = 0. Some computations give that R0 = 0, R1 = A0,0 + uf (u)B0,0 , T0 = (1 − u2 )B0,0 − uA0,0 /f (u) and T1 = 0. Therefore, the first equation of the above system is identically satisfied and the second one gives the desired differential equation.  Proof of Theorem 1.3. Consider system ( ẋ = X(x, y; ε) = c(x, y) + εC(x, y) = A(x2 , y) + εC(x, y), ẏ = Y (x, y; ε) = d(x, y) + εD(x, y) = xB(x2 , y) + εD(x, y). By Theorem 2.6 we have to find solutions of system (16) associated to the above planar system that are well defined in [−1, 1]. After some computations we get that the functions appearing in (16) are: R0 =(1 − u2 )X1 + uY0 , R1 =X0 + uY1 , (u2 Y1 + uX0 − Y1 )f − (1 − u2 )(uX1 − Y0 )g T0 = , (1 − u2 )g 2 − f 2 (uX1 − Y0 )f − (u2 Y1 + uX0 − Y1 )g T1 = , (1 − u2 )g 2 − f 2 where for each ε we have decomposed X and Y as in Lemma 1.2. Equivalently,  R0 =(1 − u2 )c1 + ud0 + ε (1 − u2 )C1 + uD0 , R1 =c0 + ud1 + ε (C0 + uD1 ) A NEW APPROACH FOR THE STUDY OF LIMIT CYCLES 17 and (u2 d1 + uc0 − d1 )f − (1 − u2 )(uc1 − d0 )g T0 = (1 − u2 )g 2 − f 2 (u2 D1 + uC0 − D1 )f − (1 − u2 )(uC1 − D0 )g +ε , (1 − u2 )g 2 − f 2 (uc1 − d0 )f − (u2 d1 + uc0 − d1 )g T1 = (1 − u2 )g 2 − f 2 (uC1 − D0 )f − (u2 D1 + uC0 − D1 )g +ε . (1 − u2 )g 2 − f 2 Using that c(x, y) = A(x2 , y) and d(x, y) = xB(x2 , y) and again the notation of Lemma 1.2 we have that c =c0 + vc1 = A0,0 + (1 − u2 )g 2 A0,1 + vgA1,0 ,  d =d0 + vd1 = (f + vg)v B0,0 + (1 − u2 )g 2 B0,1 + vgB1,0 =  =B0,1 (1 − u2 )2 g 3 + (1 − u2 )(B1,0 f + B0,0 )g + v (1 − u2 )(B0,1 f + B1,0 )g 2 + B0,0 f . Replacing these equalities in the above ones we get that R0 =B0,1 (1 − u2 )2 ug 3 + (1 − u2 )(B1,0 f u + B0,0 u + A1,0 )g  + ε (1 − u2 )C1 + uD0 , R1 =(1 − u2 )(B0,1 f u + B1,0 u + A0,1 )g 2 + B0,0 uf + A0,0  + ε C0 + uD1 and B0,1 (1 − u2 )3 g 4 + (B0,0 − B0,1 f 2 )(1 − u2 )2 g 2 T0 = (1 − u2 )g 2 − f 2 (1 − u2 )(A0,1 f − A1,0 )ug 2 − B0,0 (1 − u2 )f 2 + A0,0 uf + (1 − u2 )g 2 − f 2 (u2 D1 + uC0 − D1 )f − (1 − u2 )(uC1 − D0 )g +ε , (1 − u2 )g 2 − f 2 B1,0 (1 − u2 )2 g 3 − (1 − u2 )(A0,1 ug 3 + B1,0 f 2 g) + ug(A1,0 f − A0,0 ) T1 = (1 − u2 )g 2 − f 2 (uC1 − D0 )f − (u2 D1 + uC0 − D1 )g +ε . (1 − u2 )g 2 − f 2 We look for solutions of the form r = F (θ, ε) = f (u, ε) + vg(u, ε), where f (u, ε) = fρ (u) + εP (u) + O(ε2 ), g(u, ε) = εQ(u) + O(ε2 ) and recall that fρ (u) is given in Lemma 5.1. Next we plug the above expressions of T0 , T1 , R0 , R1 , f (u, ε) and g(u, ε) in sys- tem (16). After some tedious but straightforward computations we get that the coefficient with respect to ε of order one of the equation corresponding to g gives Q0 (u) = Lρ (u)Q(u) + Mρ (u), (27) 18 J.D. GARCÍA-SALDAÑA, A. GASULL, AND H. GIACOMINI where Lρ and Mρ are as in the statement of the theorem. The general solution of this first order linear differential equation is Ru  Z u Rw  L (s)ds − L (s)ds Q(u) = e 0 ρ m+ Mρ (w)e 0 ρ dw , 0 Ru where m is a constant of integration. Since e 0 Lρ (s)ds diverges for u = ±1, due to the singularities at the denominator of Lρ at u = ±1, we need two conditions for Q(u) to be well defined at these two points: Z 1 R Z −1 Ru − 0u Lρ (s)ds m+ Mρ (u)e du = 0 and m + Mρ (u)e− 0 Lρ (s)ds du = 0. 0 0 Joining these equations we obtain that a necessary condition for the persistence of a perturbed periodic orbit is that Z 1 Ru N (ρ) := Mρ (u)e− 0 Lρ (s)ds du = 0, −1 at this value of ρ, as we wanted to prove.  We end the paper with a couple of concrete applications of Theorem 1.3 and some comments about its theoretical interest. Consider the system ( ẋ = y − x2 y + ε(a1 x + a2 xy 2 + a3 x3 ), ẏ = −x − xy 2 + ε(b1 y + b2 x2 y + b3 y 3 ). When ε = 0, it has the first integral H(x, y) = (y 2 + 1)/(x2 − 1) and a center at the origin. We apply Theorem 1.3 with u(1 + 3(1 − u2 )fρ2 (u)) Tρ (u) Lρ (u) = 2 , and Mρ (u) = , 1−u 1 − u2 where  Tρ (u) = u2 (1 − u2 ) b2 − a3 + (a3 − a2 + b3 − b2 )u2 fρ5 (u)  + − b3 u4 + (1 − u2 )(−a3 + (a3 − a1 − a2 + b1 − b2 )u2 fρ3 (u)  + − a1 + (a1 − b1 )u2 fρ (u). 0 3 Moreover, p in this case fρ (u) is the solution of f = uf , with f (0) = ρ. So, fρ (u) = ρ/ 1 − ρ2 u2 . As a consequence, we have that Ru Ru s(4ρ2 s2 −3ρ2 −1) ds e− 0 Lρ (s)ds =e 0 (1−ρ2 s2 )(1−s2 ) = (1 − ρ2 u2 )3/2 (1 − u2 )1/2 , Qρ (u) Mρ (u) = , (1 − ρ u2 )5/2 (1 2 − u2 ) where Qρ (u) =(b3 − b1 − b2 )u4 ρ5 + b2 u2 ρ5 + (a2 − a1 − a3 + b1 + b2 − b3 )u4 ρ3 + (a1 − a2 + 2a3 + b1 − b2 )u2 ρ3 − a3 ρ3 + (a1 − b1 )u2 ρ − a1 ρ. After some more computations we obtain Z 1 Qρ (u) πq(ρ) N (ρ) = 2 2 2 1/2 du = , −1 (1 − ρ u )(1 − u ) 2ρ A NEW APPROACH FOR THE STUDY OF LIMIT CYCLES 19 where q(ρ) =(b2 − b1 + b3 )(1 − ρ2 )2 + 2(a3 − b2 )(1 − ρ2 )3/2 − (a1 − a2 + 3a3 − b1 + 3b3 − b2 )(1 − ρ2 ) − 2(a2 − b3 )(1 − ρ2 )1/2 + a1 + a2 + a3 . Doing the change of variable λ2 = 1 − ρ2 and by using that λ − 1 is a factor of q(ρ) the positive zeroes of N can be found from the solutions of (b2 − b1 + b3 )λ3 +(2a3 − b2 − b1 + b3 )λ2 − (a1 − a2 + a3 + 2b3 )λ − a1 − a2 − a3 = 0. So, by Theorem 1.3, at most 3 periodic orbits can persist. A similar result could be obtained using Abelian integrals. The above system is already studied in [11]. As a second example, consider the system ( ẋ = y − y 2 + ε(ax + bxy), ẏ = −2x. The unperturbed system is hamiltonian, with hamiltonian function y2 y3 H(x, y) = x2 + − 2 3 and has a center at the origin. In the expression of N (ρ) given in Theorem 1.3 the functions Lρ and Mρ are  u 6 + 2u4 − 7u2 + (4u − 10u3 + 4u5 )fρ (u) + (2u2 − 1)u4 fρ2 (u) Lρ (u) = , (1 − u2 )(u3 fρ (u) + u2 − 2)2 −2fρ (u)(a + bufρ (u)) Mρ (u) = , (u3 fρ (u) + u2 − 2)2 where fρ satisfies −uf (uf + 1) f0 = , f (0) = ρ. u3 f + u2 − 2 Then, Z 1 −2fρ (u)(a + bufρ (u)) − R u Lρ (s)ds N (ρ) = 0 ⇐⇒ e 0 du = 0, −1 (u3 fρ (u) + u2 − 2)2 or equivalently, R1 ufρ2 (u) Ru a −1 (u3 fρ (u)+u2 −2)2 e− 0 Lρ (s)ds du − = R1 R , b fρ (u) − 0u Lρ (s)ds 3 −1 (u fρ (u)+u −2)2 2 e du is a condition for a periodic orbit to persist. After some tricky computations we get that R1 ufρ2 (u) Ru R a −1 (u3 fρ (u)+u2 −2)2 e− 0 Lρ (s)ds du xy dy − = R1 f (u) Ru = RH=h , b 3 ρ 2 2e − 0 Lρ (s)ds du H=h x dy −1 (u fρ (u)+u −2) 2 where h = ρ . Notice that the last fraction corresponds to the one obtained by using the classical Abelian integrals approach to know which periodic orbits of the hamiltonian system persist. Following the results of Drachman, van Gils and Zhang ([4]) the above equation has at most one solution, so, at most one periodic orbit persists. Next remark highlights the theoretical applicability of Theorem 1.3. 20 J.D. GARCÍA-SALDAÑA, A. GASULL, AND H. GIACOMINI Remark 5.2. Notice that equation (27) can also be written as the singular linear differential equation Wρ (u)Q0 (u) = Sρ (u)Q(u) + Tρ (u), 2 with Wρ (u) = (1−u2 ) uA0,0 −(1−u2 )B0,0 fρ (u) . Specifically, the connection problem for such equations is a global problem that consists in finding explicit connection formulas between local solutions and to give conditions for their existence. This question goes back to some works of the celebrated mathematicians Riemann, Stokes or Birkhoff, see [6, 8]. Our necessary condition for the persistence of some of the periodic orbits of the unperturbed system, as limit cycles of the general one, turns out to be a connection problem between u = −1 and u = 1. As a consequence, Theorem 1.3 relates the study of limit cycles with a classical area of research and opens an alternative approach for obtaining new results on this first subject. Acknowledgements. The first author is supported by FONDECyT grant number 11171115. The second author is supported by Ministerio de Ciencia, Innovación y Universidades of the Spanish Government through grants MTM2016-77278-P (MINECO/AEI/FEDER, UE) and by grant 2017-SGR-1617 from AGAUR, Gen- eralitat de Catalunya. References [1] M. A. M. Alwash, The composition conjecture for Abel equation. Expo. Math. 27 (2009), 241–250. [2] J. Chavarriga, H. Giacomini and J. Llibre, Uniqueness of algebraic limit cycles for quadratic systems. J. Math. Anal. Appl. 261 (2001), 85–99. [3] C. Christopher and C. Li, Limit cycles of differential equations. Advanced Courses in Mathe- matics. CRM Barcelona. Birkhäuser Verlag, Basel, 2007. viii+171 pp. [4] B. Drachman, S. A. van Gils and Z. Zhang, Abelian integrals for quadratic vector fields. J. Reine Angew. Math. 382 (1987), 165–180. [5] F. Dumortier, J. Llibre and J. C. Artés, Qualitative theory of planar differential systems. Universitext. Springer-Verlag, Berlin, 2006. xvi+298 pp. [6] F. Naundorf, A connection problem for second order linear differential equations with two irregular singular points. SIAM J. Math. Anal. 7 (1976), 157–175. [7] Y. Ilyashenko, Centennial history of Hilbert’s 16th problem. Bull. Amer. Math. Soc. (N.S.) 39 (2002), 301–354. [8] M. Kohno, A two point connection problem for general linear ordinary differential equations. Hiroshima Math. J. 4 (1974), 293–338. [9] Y. Ilyashenko and S. Yakovenko, Lectures on analytic differential equations. Graduate Studies in Mathematics, 86. American Mathematical Society, Providence, RI, 2008. xiv+625. [10] Y. Katznelson, An introduction to harmonic analysis. Third edition. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 2004. xviii+314 pp. [11] C. Li, W. Li, J. Llibre and Z. Zhang, Linear estimation of the number of zeros of Abelian integrals for some cubic isochronous centers. J. Differ. Equ. 180 (2002), 307–333. [12] J. Li, Hilbert’s 16th problem and bifurcations of planar polynomial vector fields. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 13 (2003), 47–106. [13] J. Llibre and X. Zhang, On the Darboux integrability of the polynomial differential systems. Qual. Theory Dyn. Syst. 11 (2012), 129–144. [14] J. Llibre and Y. Zhao, Algebraic limit cycles in polynomial systems of differential equations. J. Phys. A 40 (2007), no. 47, 14207–14222. [15] A. Shadrin, Twelve proofs of the Markov inequality. Approximation theory: a volume dedicated to Borislav Bojanov, 233–298, Prof. M. Drinov Acad. Publ. House, Sofia, 2004. [16] Y. Ye et others, Theory of limit cycles. Translated from the Chinese. Translations of Mathe- matical Monographs, 66. American Mathematical Society, Providence, RI, 1986. xi+435 pp. A NEW APPROACH FOR THE STUDY OF LIMIT CYCLES 21 [17] Z. Zhang et others, Qualitative theory of differential equations. Translated from the Chinese. Translations of Mathematical Monographs, 101. American Mathematical Society, Providence, RI, 1992. xiv+461 pp. Departamento de Matemática y Fı́sica Aplicadas, Universidad Católica de la Santı́sima Concepción, Alonso de Ribera 2850, Concepción, Chile. Email address: jgarcias@ucsc.cl Departament de Matemàtiques, Universitat Autònoma de Barcelona, Edifici C 08193 Bellaterra, Barcelona, Spain. Email address: gasull@mat.uab.cat Institut Denis Poisson. Université de Tours, C.N.R.S. UMR 7013. 37200 Tours, France. Email address: Hector.Giacomini@lmpt.univ-tours.fr