� EXTRACTA MATHEMATICAE Volumen 33, Número 2, 2018 instituto de investigación de matemáticas de la universidad de extremadura EXTRACTA MATHEMATICAE Vol. 38, Num. 1 (2023), 85 – 104 doi:10.17398/2605-5686.38.1.85 Available online May 4, 2023 Estimating the number of limit cycles for one step perturbed homogeneous degenerate centers M. MolaeiDerakhtenjani, O. RabieiMotlagh, H.M. MohammadiNejad@ Department of Applied Mathematics, University of Birjand, Birjand, Iran m.molaei@birjand.ac.ir , orabieimotlagh@birjand.ac.ir , hmohammadin@birjand.ac.ir Received January 15, 2023 Presented by R. Prohens Accepted April 3, 2023 Abstract: We consider a homogeneous degenerate center of order 2m + 1 and perturb it by a homogeneous polynomial of order 2m. We study the Lyapunov constants around the origin to estimate the number of limit cycles. To do it, we classify the parameters and study their effect on the number of limit cycles. Finally, we find that the perturbed degenerate center without any condition has at least two limit cycles, and the number of the bifurcated limit cycles could reach 2m + 3. Key words: Degenerate Center, Limit cycle, Lyapunov constant. MSC (2020): 34C07, 34D10, 34D08. 1. Introduction The 16th Hilbert problem is one of the 23 mathematical problems proposed by D. Hilbert in 1900 at the Second International Congress of Mathematical, cf. [10]. The second part of this problem is to find an upper bound for the number of limit cycles that bifurcates from planar polynomial ordinary differ- ential systems. Since then, this problem has been studied by many authors, cf. [6, 11, 15, 16]. The weakened 16th Hilbert problem is a weaker version of this problem which was proposed by Arnold in 1977, cf. [1]. This problem is to find an up- per bound for the number of bifurcated limit cycles from the period annulus of systems near Hamiltonian ones. D. Hilbert conjectured that his 16th problem could approach by perturbation techniques. Since then, some authors have been studying the number of limit cycles by perturbing the periodic orbits of a center. We remind that a fixed point of a system is called a center if it is surrounded by a neighborhood filled with periodic orbits. When the cen- ter is perturbed, the system may have limit cycles that bifurcate from some @ Corresponding author ISSN: 0213-8743 (print), 2605-5686 (online) c© The author(s) - Released under a Creative Commons Attribution License (CC BY-NC 3.0) https://doi.org/10.17398/2605-5686.38.1.85 mailto:m.molaei@birjand.ac.ir mailto:orabieimotlagh@birjand.ac.ir mailto:hmohammadin@birjand.ac.ir mailto:hmohammadin@birjand.ac.ir https://publicaciones.unex.es/index.php/EM https://creativecommons.org/licenses/by-nc/3.0/ 86 m. molaeiderakhtenjani et al. of the periodic orbits of the center. Therefore, these studies could approach mathematicians to solve the 16th Hilbert problem. To this aim, some authors studied the perturbed center by applying the methods such as averaging the- ory, Melnikov function, or Poincaré map, cf. [4, 8, 14, 18]. There exist studies in which the authors used the Lyapunov constant to estimate the number of limit cycles, cf. [5, 9, 17]. Investigating the Taylor expansion of the corresponding Poincaré map is one way to compute the Lya- punov constant. In this case, the coefficients of Taylor expansion are the Lyapunov constants, cf. [13, 19]. The idea to obtain k small amplitude limit cycles is based on imposing conditions on Lyapunov constants Vi such that V0 = · · · = Vk−1 = 0 and Vk 6= 0, and then performing a suitable perturbation to have k small limit cycles. Few papers consider this problem for degenerate centers. We remind that a center of a polynomial differential system is a degenerate center if, after applying a linear change of variables and a suitable time rescale, the system can be written as ẋ = F1(x,y) and ẏ = F2(x,y), where F1(x,y) and F2(x,y) are nonlinear polynomials. Authors in [2] used Melnikov functions to study the number of limit cycles for the perturbation of the degenerate center ẋ = −y ( x2 + y2 2 )m , ẏ = x ( x2 + y2 2 )m , m ≥ 1. Authors in [12] used the averaging method of second order to study the per- turbation of the cubic degenerate center ẋ = −y ( 3x2 + y2 ) , ẏ = x ( x2 −y2 ) , and prove the existence of at most three limit cycles. In this paper, we will study the number of limit cycles for the perturbed system ξ̇ = X2m+1(ξ) + �X2m(ξ), ξ = (x,y), 0 < � � 1, (1.1) where X2m+1 = (P2m+1(x,y),Q2m+1(x,y)) is the homogeneous polynomial system with P2m+1(x,y) = m∑ i=0 aix 2iy2(m−i)+1, Q2m+1(x,y) = m∑ i=0 bix 2i+1y2(m−i), such that aibi < 0 for all i, and X2m = (P2m(x,y),Q2m(x,y)) with P2m(x,y) = 2m∑ k=0 αkx ky2m−k, Q2m(x,y) = 2m∑ k=0 βkx ky2m−k. estimating the number of limit cycles 87 The origin is a symmetric degenerate center for the unperturbed homogeneous polynomial system (1.1), cf. [13, Theorem 1(II)]. Such systems are known as one step polynomial perturbations. In these systems, a homogeneous polynomial system is perturbed by another homoge- neous polynomial term with one step bigger or smaller order. Such systems receive attention from mathematicians and physicists due to their role in non- linear mechanics, cf. [3, 7]. Here, we aim to estimate the number of limit cycles for the perturbed system (1.1) by studying the Lyapunov constants. We compute the Lyapunov constants through the Taylor expansion of the Poincaré map P(r0,�) = r(2π,r0,�) = n∑ j=0 1 j! �j ∂jr ∂�j (2π,r0, 0) + O(� n+1) of the system (1.1). Here, (r,θ) shows the polar coordinates around the origin. The origin is assumed to be a perturbed symmetric degenerate center, and r = r(θ,r0,�) is the solution of equation (1.1) such that r(0,r0,�) = r0 and r(2π,r0, 0) = r0. Then, we will study the Lyapunov constants by considering the coefficients of X2m, αks and βks, as the parameters. The paper is oriented as follows. In Section 2, we will compute( ∂jr/∂�j ) (θ,r0, 0) for the perturbed system (1.1) and prove the equality (2.2). In Section 3, we will simplify ( ∂jr/∂�j ) (θ,r0, 0) for θ = 2π and classify the parameters into two types α2k+1, β2k and α2k, β2k+1. In Section 4, we will present our main results and estimate the number of limit cycles for the per- turbed degenerate center (1.1) in Theorem 4.5. We will see that the one step perturbed degenerate center has at least two limit cycles, and the number of limit cycles can reach 2m + 3. At last, we will consider our results for a perturbed degenerate center of order 3. 2. Preliminaries Consider the perturbed system (1.1). As [13, Section 3], the corresponding polar perturbed system is dr dθ = rS(θ) + �r C A(rA + �B) , (2.1) where S(θ) = 〈nθ,X2m+1(θ)〉 (nθ ∧X2m+1(θ)) , A(θ) = (nθ ∧X2m+1(θ)), 88 m. molaeiderakhtenjani et al. B(θ) = (nθ ∧X2m(θ)), C(θ) = (X2m(θ) ∧X2m+1(θ)), such that nθ = (cos(θ), sin(θ)), Xi(θ) = (Pi(cos(θ), sin(θ)),Qi(cos(θ), sin(θ)) for i = 2m, 2m+ 1. Also, the notations 〈 , 〉 and (∧) are respectively the inner and the wedge product of vectors in R2, i.e., 〈(a,b) , (c,d)〉 = ac + bd, ( (a,b) ∧ (c,d) ) = ad− bc. Then by applying the relation [13, (20)], we obtain ( ∂jr/∂�j ) (θ,r0, 0) for the polar perturbed system (2.1) as ∂jr ∂�j (θ,r0, 0) = j χ(θ) j−1∑ k=0 k∑ l=0 ( j − 1 k )( k l )∫ θ 0 χ−1(ψ) ∂j−1−kr ∂�j−1−k Dldk−l ( C A(rA + �B) ) dψ, where χ(θ) = Exp (∫ θ 0 S(t)dt ) is the fundamental matrix of the homogeneous part of (2.1). Also, D and d are the linear functional operators DF(g(x),x) = (∂F/∂g) (g(x),x)g′(x) and dF(g(x),x) = (∂F/∂x) (g(x),x), see [13, page 10] for more details. Next, we consider the above relation more precisely. For l = 0, we have j χ(θ) ∫ θ 0 χ−1(ψ) ∂j−1r ∂�j−1 ( C A(rA + �B) ) dψ + j χ(θ) j−1∑ k=1 ( j − 1 k )( k 0 )∫ θ 0 χ−1(ψ) ∂j−1−kr ∂�j−1−k dk ( C A(rA + �B) ) dψ = j χ(θ) ∫ θ 0 χ−1(ψ) ∂j−1−kr ∂�j−1−k ( C A(rA + �B) ) dψ + j χ(θ) j−1∑ k=1 ( j − 1 k )∫ θ 0 χ−1(ψ) ∂j−1−kr ∂�j−1−k C AB (−1)kk!( r A B + � )k+1 dψ = j χ(θ) j−1∑ k=0 ( j − 1 k )∫ θ 0 χ−1(ψ) ∂j−1−kr ∂�j−1−k C AB (−1)kk!( r A B + � )k+1 dψ. So for � = 0, we have j−1∑ k=0 (−1)kj χ(θ) ( j − 1 k ) k! rk+10 ∫ θ 0 ∂j−1−kr ∂�j−1−k C A2 ( B A )k 1 χ(ψ)k+2 dψ. estimating the number of limit cycles 89 For the remaining (when l ≥ 1), we have j χ(θ) j−1∑ k=1 k∑ l=1 ( j − 1 k )( k l )∫ θ 0 χ−1(ψ) ∂j−1−kr ∂�j−1−k Dldk−l ( C A(rA + �B) ) dψ = j χ(θ) j−1∑ k=1 k∑ l=1 ( j − 1 k )( k l )∫ θ 0 χ−1(ψ) ∂j−1−kr ∂�j−1−k C AB Dl ( (−1)k−l(k − l)!( r A B + � )k−l+1 ) dψ. From the formula of Faà di Bruno, we have Dl ( 1( r A B + � )k−l+1 ) = l=1∑ t=1 ( 1( r A B + � )k−l+1 )(t) Bl,t ( ∂r ∂� , ∂2r ∂�2 , . . . , ∂l−t+1r ∂�l−t+1 ) = l=1∑ t=1 (−1)t(k − l + t)! (k − l)! ( r + � B A )k−l+t+1 Bl,t ( ∂r ∂� , ∂2r ∂�2 , . . . , ∂l−t+1r ∂�l−t+1 ) . So for l ≥ 1, we get j−1∑ k=1 k∑ l=1 l∑ t=1 j χ(θ) ( j − 1 k )( k l ) (−1)k−l+t(k − l + t)! ∫ θ 0 χ−1(ψ) ∂j−1−kr ∂�j−1−k C AB 1( r + � B A )k−l+t+1 Bl,t ( ∂r ∂� , ∂2r ∂�2 , . . . , ∂l−t+1r ∂�l−t+1 ) dψ. Thus for � = 0, we have j−1∑ k=1 k∑ l=1 l∑ t=1 j χ(θ) ( j − 1 k )( k l ) (−1)k−l+t (k − l + t)! rk−l+t+20 ∫ θ 0 χ−1(ψ) ∂j−1−kr ∂�j−1−k C A2 ( B A )k−l 1 χ(ψ)k−l+t+2 Bl,t ( ∂r ∂� , ∂2r ∂�2 , . . . , ∂l−t+1r ∂�l−t+1 ) dψ. Finally, the following proposition is immediate. Proposition 2.1. Consider the system (2.1). We have ∂jr ∂�j (θ,r0, 0) = j−1∑ k=0 I(j,k)(θ) + j−1∑ k=1 k∑ l=1 l∑ t=1 M(j,k,l,t)(θ), (2.2) 90 m. molaeiderakhtenjani et al. where I(j,k)(θ) = (−1)kj χ(θ) ( j − 1 k ) k! rk+10 ∫ θ 0 ∂j−1−kr ∂�j−1−k C A2 ( B A )k 1 χ(ψ)k+2 dψ, (2.3) M(j,k,l,t)(θ) = (−1) k−l+tj χ(θ) ( j − 1 k )( k l ) (k − l + t)! rk−l+t+10∫ θ 0 ∂j−1−kr ∂�j−1−k C A2 ( B A )k−l 1 χ(ψ)k−l+t+2 Bl,t ( ∂r ∂� , . . . , ∂l−t+1r ∂�l−t+1 ) dψ. (2.4) Lemma 2.2. Consider (2.3) and (2.4). We have R1: I(j, 0) + M(j,j−1,j−1, 1) = 0 for all j; R2: M(j, j−1, j−1, i + 1) + ∑j−2 k=i M(j, k, k, i) = 0 for all j, 1 ≤ i ≤ j−2. Furthermore, ( ∂jr/∂�j ) (θ,r0, 0) is a j-th order homogeneous polynomial w.r.t. αks or βks. Proof. The R1 and R2 prove by substituting indexes in (2.3) and (2.4), and proportional use of Bl,t in appendix 5.1. For the remain, consider (2.2). The proof is obvious by induction on j and the second condition of Faà di Bruno’s formula. We note that the power of the parameters in B(θ) and C(θ) is one. In the following remark, we indicate the ( ∂jr/∂�j ) (θ,r0, 0) for j = 1, 2, 3, 4. Remark 2.3. ∂r ∂� (θ,r0, 0) = χ(θ) ∫ θ 0 C χA2 dψ, ∂2r ∂�2 (θ,r0, 0) = − 2 χ(θ) r0 ∫ θ 0 C B χ2 A3 dψ, ∂3r ∂�3 (θ,r0, 0) = 6 χ(θ) r20 ∫ θ 0 C B2 χ3 A4 dψ + 6 χ(θ) r20 ∫ θ 0 ∂r ∂� C B χ3 A3 dψ, ∂4r ∂�4 (θ,r0, 0) = − 24χ(θ) r30 ∫ θ 0 C B3 χ4 A5 dψ − 48χ(θ) r30 ∫ θ 0 ∂r ∂� C B2 χ4A4 dψ − 24χ(θ) r30 ∫ θ 0 ( ∂r ∂� )2 C B χ4A3 dψ + 12χ(θ) r20 ∫ θ 0 ∂2r ∂�2 C B χ3A3 dψ. estimating the number of limit cycles 91 3. Lyapunov constant In the following, we consider ( ∂jr/∂�j ) (θ,r0, 0) for θ = 2π. We prove some points to simplify them and apply for some j. Consider the following lemma. Lemma 3.1. ([13, Lemma 6]) Let f(θ) > 0 (< 0) be an even π-periodic function and i,j be nonnegative integers such that at least one of them is odd. Then ∫ 2π 0 cosi(θ) sinj(θ) f(θ) dθ = 0. So we conclude the following lemma. Lemma 3.2. Let f(θ) > 0 (< 0) be an even π-periodic function and i,j be odd nonnegative integers. Then∫ π 0 cosi(θ) sinj(θ) f(θ) dθ = 0. Proof. We have∫ 2π 0 cosi(θ) sinj(θ) f(θ) dθ = ∫ π 0 cosi(θ) sinj(θ) f(θ) dθ + ∫ 2π π cosi(θ) sinj(θ) f(θ) dθ = ∫ π 0 cosi(θ) sinj(θ) f(θ) dθ + ∫ π 0 (−1)i+j cosi(ψ) sinj(ψ) f(ψ) dψ. Then by considering Lemma 3.1, we get 0 = ∫ 2π 0 cosi(θ) sinj(θ) f(θ) dθ = 2 ∫ π 0 cosi(θ) sinj(θ) f(θ) dθ. Next, we consider X2m(θ) = (P2m(θ),Q2m(θ)) as P2m(θ) = m∑ k=0 α2k cos 2k(θ) sin2m−2k(θ) + m−1∑ k=0 α2k+1 cos 2k+1(θ) sin2m−2k−1(θ), Q2m(θ) = m∑ k=0 β2k cos 2k(θ) sin2m−2k(θ) + m−1∑ k=0 β2k+1 cos 2k+1(θ) sin2m−2k−1(θ). 92 m. molaeiderakhtenjani et al. Now, by defining D2m and D ⊥ 2m as D2m(θ) = ( m−1∑ k=0 α2k+1 cos 2k+1(θ) sin2m−2k−1(θ), m∑ k=0 β2k cos 2k(θ) sin2m−2k(θ) ) , D⊥2m(θ) = ( m∑ k=0 α2k cos 2k(θ) sin2m−2k(θ), m−1∑ k=0 β2k+1 cos 2k+1(θ) sin2m−2k−1(θ) ) , we have X2m = D2m +D ⊥ 2m. This helps us to separate the parameters into two collections. One collection contains α2k+1s and β2ks, and the other collection contains α2ks and β2k+1s. Thus, we can decompose B(θ) and C(θ) as the following, B(θ) : (nθ ∧D2m(θ)) + (nθ ∧D⊥2m(θ)) = B(θ) + B ⊥(θ), C(θ) : (D2m(θ) ∧X2m+1(θ)) + (D⊥2m(θ) ∧X2m+1(θ)) = C(θ) + C ⊥(θ). By considering B(θ) and C⊥(θ), we can see that the power of sin(θ) is even, and the power of cos(θ) is odd. Also by considering B⊥(θ) and C(θ), we can see that the power of sin(θ) is odd, and the power of cos(θ) is even. This classification helps us to use Lemma 3.1 and Lemma 3.2. We note that we only use this classification when the relations could simplify. Next, we introduce some particular cases. Remark 3.3. Let K be a nonnegative integer and consider C BK. Accord- ing to C(θ) and B(θ), we can see that the multiplication of sin(θ) and cos(θ) exists in C BK, for all K. By applying an induction on K, we have the following results: - If K is even, then the sin(θ) or cos(θ) in C BK has odd order. - If K is odd, then the sin(θ) and cos(θ) in C BK are both have an even or both have an odd order. Remark 3.4. Consider (2.2) and note that (∂0r/∂�0)(θ,r0, 0) = χ(θ)r0. By substituting the suitable conditions, one of the terms of (∂jr/∂�j)(2π,r0, 0) for k = j − 1 is (−1)j−1 j! r j−1 0 ∫ 2π 0 C Bj−1 Aj+1χj(ψ) dψ. By applying an induction on j and according to Lemma 3.1 and the above classification, this term is equal to zero when j is an odd integer number. estimating the number of limit cycles 93 In the following proposition, we use these remarks to investigate( ∂jr/∂�j ) (2π,r0, 0) for j = 1, 2, 3, 4. Proposition 3.5. Consider ( ∂jr/∂�j ) (θ,r0, 0) for j = 1, 2, 3, 4. We have ∂r ∂� (2π,r0, 0) = ∫ 2π 0 C + C⊥ χA2 dψ = 0, (3.1) ∂2r ∂�2 (2π,r0, 0) = −2 r0 ∫ 2π 0 C⊥B + CB⊥ χ2 A3 dψ, (3.2) ∂3r ∂�3 (2π,r0, 0) = 6 r20χ(π) ∂r ∂� (π,r0, 0) ∫ π 0 CB + C⊥B⊥ χ2 A3 dψ, (3.3) and also, ∂4r ∂�4 (2π,r0, 0) = − 24 r30 ∫ 2π 0 (CB + C⊥B⊥)(2BB⊥) + (CB⊥ + C⊥B)(B2 + B⊥ 2 ) χ4 A5 dψ + ( −24 r30χ 2(π) ( ∂r ∂� (π) )2 + 12 r20χ(π) ∂2r ∂�2 (π) )∫ π 0 CB + C⊥B⊥ χ2 A3 dψ + 48 r30χ(π) ∂r ∂� (π) ∫ π 0 C B2 χ3A4 dψ − 96 r30 ∫ π 0 ∂r ∂� C B2 χ4A4 dψ + 48 r30χ(π) ∂r ∂� (π) ∫ π 0 ∂r ∂� C B χ3A3 dψ − 48 r30 ∫ π 0 ( ∂r ∂� )2 C B χ4A3 dψ + 24 r20 ∫ π 0 ∂2r ∂�2 C B χ3A3 dψ. (3.4) Where the relations (3.2), (3.3), and (3.4) are not generally equal to zero. Proof. First, we note that A(θ) and χ(θ) are π- periodic functions, and also χ(2π) = 1, cf. [13]. Now by considering Lemma 3.1 and the above classification, we obviously obtain (3.2) and (3.3). Next, we consider( ∂3r/∂�3 ) (2π,r0, 0). According to remark 3.4, we get it as ∂3r ∂�3 (2π,r0, 0) = 6 r20 ∫ 2π 0 C B χ3 A3 ∂r ∂� dψ, 94 m. molaeiderakhtenjani et al. where∫ 2π 0 ∂r ∂� C B χ3 A3 dψ = ∫ π 0 ∂r ∂� C B χ3 A3 dψ + ∫ π 0 ∂r ∂� (φ + π,r0, 0) C(φ + π) B(φ + π) χ3(φ + π) A3(φ + π) dφ. By considering C(θ), B(θ), and (∂r/∂�)(θ,r0, 0), we get C(θ + π) = −C(θ), B(θ + π) = −B(θ), and ∂r ∂� (θ + π,r0, 0) = χ(θ) χ(π) ∂r ∂� (π,r0, 0) − ∂r ∂� (θ,r0, 0). Thus∫ 2π 0 ∂r ∂� C B χ3 A3 dψ = ∫ π 0 ∂r ∂� C B χ3 A3 dψ + 1 χ(π) ∂r ∂� (π,r0, 0) ∫ π 0 C B χ2 A3 dφ − ∫ π 0 ∂r ∂� (φ,r0, 0) C B χ3 A3 dφ = 1 χ(π) ∂r ∂� (π,r0, 0) ∫ π 0 (C + C⊥)(B + B⊥) χ2 A3 dφ. Now by applying Lemma 3.2, we have∫ 2π 0 C B χ3 A3 ∂r ∂� dψ = 1 χ(π) ∂r ∂� (π,r0, 0) ∫ π 0 CB + C⊥B⊥ χ2 A3 dψ. So, (3.3) is immediate. Finally, by following the same process as above and also by considering ∂2r ∂�2 (θ + π,r0, 0) = χ(θ) χ(π) ∂2r ∂�2 (π,r0, 0) + ∂2r ∂�2 (θ,r0, 0), and Remark 2.3, we obtain (3.4). Finally, we note that by considering (3.2), (3.3) and (3.4), one easily concludes that they are not generally equal to zero. 4. Estimating the number of limit cycles In this section, we represent our main results. Theorem 4.1. Consider the perturbed system (1.1). estimating the number of limit cycles 95 1) The system has at least two limit cycles. 2) The necessary condition that the system has more than two limit cycles is that the perturbed part has the two types of parameters, i.e., the param- eters of D2m, α2k+1 or β2k, and the parameters of D ⊥ 2m, α2k or β2k+1, for arbitrary k. We note that for simplicity, we consider the following notation in proof. Again by considering K as a nonnegative integer, assume CiSj = cos2K+i(θ) sin2m−2K+j(θ). As an example α2t CS = α2t cos 2t(θ) sin2m−2t(θ), β2k C 1S−1 = β2k cos 2k+1(θ) sin2m−2k−1(θ). Proof. 1) It is obvious by considering (3.1) and (3.2). 2) For this part, we consider (3.2) to find that under which necessary condition this relation could be zero. By substituting C, B, C⊥, and B⊥, we get C⊥B + CB⊥ = ( D⊥2m ∧X2m+1 ) (nθ ∧D2m) + (D2m ∧X2m+1) ( nθ ∧D⊥2m ) . Then by considering nθ and X2m+1, we have C⊥B + CB⊥ as P2m+1(θ) (∑m k=0 α2kCS 1 )(∑m k=0 β2kCS ) + P2m+1(θ) (∑m−1 k=0 α2k+1C 1S )(∑m−1 k=0 β2k+1C 1S−1 ) −P2m+1(θ) (∑m k=0 β2kC 1S )(∑m−1 k=0 β2k+1C 1S−1 ) −P2m+1(θ) (∑m k=0 β2kCS )(∑m−1 k=0 β2k+1C 2S−1 ) + Q2m+1(θ) (∑m k=0 α2kCS )(∑m k=0 β2kC 1S ) + Q2m+1(θ) (∑m−1 k=0 α2k+1C 1S−1 )(∑m−1 k=0 β2k+1C 2S−1 ) −Q2m+1(θ) (∑m k=0 α2kCS )(∑m−1 k=0 α2k+1C 1S ) −Q2m+1(θ) (∑m k=0 α2kCS 1 )(∑m−1 k=0 α2k+1C 1S−1 ) . 96 m. molaeiderakhtenjani et al. As Lemma 2.2, we can see that the order of parameters in each term is two. Also, all terms appear as the production of parameters of D2m and D ⊥ 2m. Now let 0 ≤ t ≤ m be a fixed integer number and assume m∑ k=0 α2kCS = α2tCS + m∑ k=0,k 6=t α2kCS. By considering the above relation, we can be zero (3.2) w.r.t. α2t. In this case, we find α2t = 1∫ 2π 0 Γ(ψ) χ2 A3 dψ ∫ 2π 0 Λ(ψ) χ2 A3 dψ, (4.1) where Λ(ψ) is the summation of all terms which are not dependent on the parameter α2t (see appendix 5.2) and Γ(ψ) = (sin(ψ)P2m+1(ψ) + cos(ψ)Q2m+1(ψ)) ( m∑ k=0 β2kC 2tS2m−2t ) − 2Q2m+1(ψ) ( m−1∑ k=0 α2k+1C 2t+1S2m−2t ) . As we can see, (4.1) is well defined if the parameters α2k+1 or β2k are not equal to zero for some k. It is worthwhile to note that we conclude the same result by considering the other parameters, α2t+1, β2t, and β2t+1, to be zero (3.2). In the following, we study the perturbed system (1.1) to estimate the number of limit cycles. For this consideration, we have two points. First, according to Theorem 4.1, the perturbed system (1.1) must have the two types of parameters to study the existence of more than two limit cycles. We can directly conclude the following lemma from Theorem 4.1. Lemma 4.2. Consider the perturbed system (1.1) and assume that the perturbed part has two types of parameters. By considering the parameter α2t as (4.1) for a fixed integer t, the perturbed system (1.1) has at least three limit cycles. In the following remark, we study the effect of assuming the specific pa- rameter α2t as (4.1) in considering the other Lyapunov constants. estimating the number of limit cycles 97 Remark 4.3. Consider (4.1). We obtain that the parameters α2k and β2k+1 appear in the numerator of (4.1) and the parameters α2k+1 and β2k appear in the denominator of (4.1). Again we can see our classification of the pa- rameters. Now by substituting (4.1) in the other Lyapunov constants, we can arrange the Lyapunov constants in the form of algebraic equations. The variables of these algebraic equations are the parameters that appear in the numerator of (4.1). The other parameters, the parameters in the denomina- tor of (4.1), use to study the existence of zero for the algebraic equations. We emphasize that we conclude the same result by considering the other pa- rameters to be zero ( ∂2r/∂�2 ) (2π). The only difference is the change of the two types of parameters from the numerator and denominator. Second, according to Lemma 2.2, ( ∂jr/∂�j ) (2π,r0, 0) is a j-th order ho- mogeneous polynomial w.r.t. αks or βks. In the following lemma, we consider the simplified ( ∂jr/∂�j ) (2π,r0, 0) for j ≥ 3. We can see that these relations have terms in which the order of a specific parameter is equal to j. Lemma 4.4. For j ≥ 3, ( ∂jr/∂�j ) (2π,r0, 0) contains a parameter with an exponent of the order of j. Proof. We note two points: first, the parameters α2k and β2k+1 exist in C⊥(θ) and B⊥(θ); and the parameters α2k+1 and β2k exist in C(θ) and B(θ). Second, for the existence of a term with the coefficient, for example α j 2k, the term must be the j times production of C⊥(θ) or B⊥(θ). For j = 3, (3.3) simplifies as 6 r20χ(π) ∫ π 0 C + C⊥ χA2 dψ ∫ π 0 CB + C⊥B⊥ χ2 A3 dψ. We can see that the relation has CCB and C⊥C⊥B⊥. For j > 3, see the following considerations. Let j be an even integer number and assume (2.3) for k = j − 2, i.e., I(j,j−2)(θ) = j χ(θ) ( j − 1 j − 2 ) (j − 2)! r j−1 0 ∫ θ 0 ∂r ∂� C A2 ( B A )j−2 1 χ(ψ)j dψ. By simplifying the I(j,j−2)(2π) even if (∂r/∂�) (π) comes out of the integral, the CBj−2 does not decompose to C, C⊥, B, and B⊥. We note that the order of B is even. The result is obvious according to remark 3.3. So the relation has terms with C2Bj−2 and also C⊥2B⊥j−2. Now let j be an odd integer number 98 m. molaeiderakhtenjani et al. and assume (2.4) for k = j − 1 and l = t = 2, i.e., M(j,j−1,2,2)(θ) = j χ(θ) ( j − 1 2 ) (j − 1)! r j−1 0 ∫ θ 0 C A2 ( B A )j−3 1 χ(ψ)j ( ∂r ∂� )2 dψ. By following the same consideration as I(j,j−2)(2π), we conclude the result. Now we can conclude the following theorem. Theorem 4.5. Consider the perturbed system (1.1) and assume that the perturbed part has two types of parameters. Then the number of the bifur- cated limit cycles could reach 2m + 3. Proof. According to Theorem 4.1, the perturbed system (1.1) could have more than two limit cycles. We substitute the parameters α2t as (4.1) and study the other Lyapunov constants. From Lemma 2.2, remark 4.3, and Lemma 4.4 each Lyapunov constant, i.e., ( ∂jr/∂�j ) (2π) for all j, is an al- gebraic equation of order j. Next, we use remark 4.3 to study the existence of zero for the algebraic equations. The number of the parameters which appear in the numerator of (4.1) is 2m + 1. So if we find the other parameters such that these algebraic equations have real roots, then the perturbed system (1.1) could have 2m + 3 limit cycles. In the following proposition, we consider the above results for the homo- geneous degenerate center of order three. Consider the perturbed system (1.1) for m = 1, i.e., ẋ = a0y 3 + a1x 2y + � ( α0y 2 + α1xy + α2x 2 ) , ẏ = b0xy 2 + b1x 3 + � ( β0y 2 + β1xy + β2x 2 ) . (4.2) Proposition 4.6. Assume the perturbed system (4.2) such that αk and βk, k = 0, 1, 2, is not equal to zero and consider the following conditions. I. Let the parameters α1, β0, and β2 exist such that 1) β2L26 + β0L27 + α1L29 6= 0, 2) L311L321 6= 0. II. Define the parameter α0 = β1 (β2L21 + β0L22 + α1L25) + α2 (β2L23 + β0L24 + α1L28) β2L26 + β0L27 + α1L29 . estimating the number of limit cycles 99 Respectively see Appendix (5.3) and Appendix (5.4) for L2is and L3is. If conditions I(1) and II hold, then the perturbed system (4.2) has at least three limit cycles. If the conditions I(1,2) and II hold, then the perturbed system (4.2) has at least four limit cycles. Finally, if the conditions I(1,2) and II hold, and also the parameters α1, β0, and β2 exist such that the quantic algebraic equation α42 (L424 + L434 + L444 + L464 + L4523) + α32 (L413 + L423 + L433 + L443 + L463 + L473 + L4523) + α22 (L412 + L422 + L432 + L442 + L462 + L472 + L4522) + α2 (L411 + L421 + L431 + L441 + L461 + L471 + L4521) + L410 + L420 + L430 + L440 + L460 + L470 + L4520, (4.3) has a real root, then the perturbed system (4.2) has at least five limit cycles, (the L4is are the coefficients of αi2, i = 0, 1, 2, 3, 4). Proof. By applying (3.2) for the perturbed system (4.2) and simplifying it w.r.t. the parameters, we obtain it as β1β2L21 + β0β1L22 + α2β2L23 + α2β0L24 + α1β1L25 + α0β2L26 + α0β0L27 + α1α2L28 + α0α1L29, (4.4) where L2i, i = 1, . . . , 9, are the coefficients of parameters. Now we consider (4.4) as a quadratic algebraic equation and suppose that I(1) holds. We can easily see that this algebraic equation has a real root α0 = β1 (β2L21 + β0L22 + α1L25) + α2 (β2L23 + β0L24 + α1L28) β2L26 + β0L27 + α1L29 . (4.5) Thus according to this point that (3.3) is not generally equal to zero, the perturbed system (4.2) has at least three limit cycles. Next, we study the third Lyapunov constant, where we have the parameters α0 as (4.5). For simplicity in computation, we consider (4.5) as α0 = β1A + α2B, where A and B obtain by considering (4.5). By computing (3.3) for m = 1 and simplifying it w.r.t. the parameters β1 and α2, we have β31L311L321 + β 2 1 (α2L312L321 + α2L311L322 + L311L320) + β1 ( α22L313L321 + α 2 2L312L322 + α2L312L320 + L310L321 ) + α32L313L322 + α 2 2L313L320 + α2L310L322 + L310L320, (4.6) 100 m. molaeiderakhtenjani et al. where L3is are the coefficients of the parameters. We note that L3is are dependent on the parameters α1, β2, and β0. As we can see (4.6) is the cubic algebraic equation w.r.t. the parameters β1 and the parameters α2. We consider it as an algebraic equation w.r.t. β1. This equation has at least one real root if the coefficient of β1, i.e., L311L321 is not equal to zero. Thus if I(1,2) and II hold, the perturbed system (4.2) has at least four limit cycles. For the last step, we study the fourth Lyapunov constant, (3.4), by substituting the parameters α0 and β1. Then by simplifying it w.r.t. the parameter α2, we obtain (4.3). So if the parameters α1, β2, and β0 exist such that the quantic algebraic equation (4.3) has a real root, then the perturbed system (4.2) has at least five limit cycles. 5. Appendix 5.1. The formula of Faà di Bruno. Given two functions f and g, the generalization of the chain rule is known as Faà di Bruno’s theorem. dn dnx (f(g(x)) = n∑ k=1 f(k)(g(x))Bn,k ( g(1)(x), g(2)(x), . . . ,g(n−k+1)(x) ) , where Bn,k are the Exponential Bell polynomials. The partial or incomplete exponential Bell polynomials are a triangular array of polynomials given by Bn,k(x1,x2, . . . ,xn−k+1) =∑ n! j1! j2! . . . jn−k+1! ( x1 1! )j1(x2 2! )j2 . . . ( xn−k+1 (n−k + 1)! )jn−k+1 , Where the sum is taken over all sequences j1, j2, j3, . . . , jn−k+1 of nonnegative integers such that these two conditions are satisfied: 1. j1 + j2 + j3 + · · · + jn−k+1 = k, 2. j1 + 2 j2 + 3 j3 + · · · + (n−k + 1) jn−k+1 = n. For example Bk,k ( x1 ) : j1 = k, Bk+1,k ( x1,x2 ) : j1 = k − 1, j2 = 1, Bk+2,k ( x1,x2,x3 ) : j1 = k − 1, j2 = 0, j3 = 1, j1 = k − 2, j2 = 2, j3 = 0, estimating the number of limit cycles 101 Bk+3,k ( x1,x2,x3,x4 ) : j1 = k − 1, j2 = 0, j3 = 0, j4 = 1, j1 = k − 2, j2 = 1, j3 = 1, j4 = 0, j1 = k − 3, j2 = 3, j3 = 0, j4 = 0. 5.2. The relation Λ(θ) is m∑ k=0,k 6=t α2kCS { P2m+1(θ) ( m∑ k=0 β2kCS 1 ) − Q2m+1(θ) (m−1∑ k=0 α2k+1C 1S ) −Q2m+1(θ) (m−1∑ k=0 α2k+1C 1S ) + Q2m+1(θ) ( m∑ k=0 β2kC 1S )} + P2m+1(θ) (m−1∑ k=0 α2k+1C 1S )(m−1∑ k=0 β2k+1C 1S−1 ) −P2m+1(θ) ( m∑ k=0 β2kC 1S )(m−1∑ k=0 β2k+1C 1S−1 ) −P2m+1(θ) ( m∑ k=0 β2kCS )(m−1∑ k=0 β2k+1C 2S−1 ) + Q2m+1(θ) (m−1∑ k=0 α2k+1C 1S−1 )(m−1∑ k=0 β2k+1C 2S−1 ) . 5.3. The coefficients in the (4.4): L21 = −2 r0 ∫ 2π 0 −2P3(ψ) sin(ψ) cos4(ψ) χ2 A3 dψ, L22 = −2 r0 ∫ 2π 0 −2P3(ψ) sin3(ψ) cos2(ψ) χ2 A3 dψ, L23 = −2 r0 ∫ 2π 0 P3(ψ) sin(ψ) cos 4(ψ) + Q3(ψ) cos 5(ψ) χ2 A3 dψ, L24 = −2 r0 ∫ 2π 0 P3(ψ) sin 3(ψ) cos2(ψ) + Q3(ψ) sin 2(ψ) cos3(ψ) χ2 A3 dψ, 102 m. molaeiderakhtenjani et al. L25 = −2 r0 ∫ 2π 0 P3(ψ) sin 3(ψ) cos2(ψ) + Q3(ψ) sin 2(ψ) cos3(ψ) χ2 A3 dψ, L26 = −2 r0 ∫ 2π 0 P3(ψ) sin 3(ψ) cos2(ψ) + Q3(ψ) sin 2(ψ) cos3(ψ) χ2 A3 dψ, L27 = −2 r0 ∫ 2π 0 P3(ψ) sin 5(ψ) + Q3(ψ) sin 4(ψ) cos(ψ) χ2 A3 dψ, L28 = −2 r0 ∫ 2π 0 −2Q3(ψ) sin2(ψ) cos3(ψ) χ2 A3 dψ, L29 = −2 r0 ∫ 2π 0 −2Q3(ψ) sin4(ψ) cos(ψ) χ2 A3 dψ. 5.4. The coefficients in the (4.6). Consider α0 = Aβ1 + α2B. We have L310 = 6 r20χ(π) ∫ π 0 ( α1β2P3(ψ) sin 2(ψ) cos3(ψ) + α1β0P3(ψ) sin 4(ψ) cos(ψ) χ2 A3 + β20P3(ψ) sin 4(ψ)(−cos(ψ)) −β22P3(ψ) cos5(ψ) χ2 A3 + −2β0β2P3(ψ) sin2(ψ) cos3(ψ) −α21Q3(ψ) sin 3(ψ) cos2(ψ) χ2 A3 + α1β2Q3(ψ) sin(ψ) cos 4(ψ) + α1β0Q3(ψ) sin 3(ψ) + cos2(ψ) χ2 A3 ) dψ, L311 = 6 r20χ(π) ∫ π 0 ( AP3(ψ) sin4(ψ) cos(ψ) −P3(ψ) sin2(ψ) cos3(ψ) χ2 A3 + +AQ3(ψ) sin3(ψ) cos2(ψ) −A2Q3(ψ) sin5(ψ) χ2 A3 ) dψ, L312 = 6 r20χ(π) ∫ π 0 ( P3(ψ) sin 2(ψ) cos3(ψ) + BP3(ψ) sin4(ψ) cos(ψ) χ2 A3 + Q3(ψ) sin(ψ) cos 4(ψ) − 2AQ3(ψ) sin3(ψ) cos2(ψ) χ2 A3 + −2ABQ3(ψ) sin5(ψ) + BQ3(ψ) sin3(ψ) cos2(ψ) χ2 A3 ) dψ, estimating the number of limit cycles 103 L313 = 6 r20χ(π) ∫ π 0 ( −Q3(ψ) sin(ψ) cos4(ψ) −B2Q3(ψ) sin5(ψ) χ2 A3 + −2BQ3(ψ) sin3(ψ) cos2(ψ) χ2 A3 ) dψ, L320 = ∫ π 0 −β0P3(ψ) sin2(ψ) −β2P3(ψ) cos2(ψ) + α1Q3(ψ) sin(ψ) cos(ψ) χA2 dψ, L321 = ∫ π 0 AQ3(ψ) sin2(ψ) −P3(ψ) sin(ψ) cos(ψ) χA2 dψ, L322 = ∫ π 0 Q3(ψ) cos 2(ψ) + BQ3(ψ) sin2(ψ) χA2 dψ. References [1] V.I. 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Żoladek, Eleven small limit cycles in a cubic vector field, Nonlinearity 8 (1995), 843 – 860. Introduction Preliminaries Lyapunov constant Estimating the number of limit cycles Appendix The formula of Faà di Bruno. The coefficients in the (4.4): The coefficients in the (4.6).