Limit Cycle Bifurcations Near a Cuspidal Loop

: In this paper, we study limit cycle bifurcation near a cuspidal loop for a general near-Hamiltonian system by using expansions of the ﬁrst order Melnikov functions. We give a method to compute more coefﬁcients of the expansions to ﬁnd more limit cycles near the cuspidal loop. As an application example, we considered a polynomial near-Hamiltonian system and found 12 limit cycles near the cuspidal loop and the center.


Introduction
As we know, in the qualitative theory of planar differential systems, one of the most important problems is to study the number of limit cycles for a near-Hamiltonian system, which is closely related to the Hilbert's 16th problem. Many studies focused on the number of limit cycles for the following near-Hamiltonian system:ẋ = H y + ε f (x, y, δ), where H, f and g are analytic functions, ε > 0 is a small parameter, and δ ∈ D ⊂ R m with m ∈ Z + and D compact; see [1][2][3][4][5]. When ε = 0, system (1) becomes the following Hamiltonian system: We suppose that the equation H(x, y) = h defines a family periodic orbits L h of system (2), where h ∈ J with J an open interval. The boundary of the family of periodic orbits L h may be a center, a homoclinic loop or a heteroclinic loop, among other possibilities. To study limit cycle bifurcations of system (1), the following Melnikov function plays an important role. In order to study the maximal number of limit cycles of system (1), we convert to study the maximal number of isolated zeros of M which is called the weak Hilbert's 16th problem posed by Arnold. Some interesting advances can be found in Li et al. [6] around the weak Hilbert's 16th problem.
Recently, there are many new results have been obtained about the problem; see [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21] for example. Suppose that the Hamiltonian system (2) has a homoclinic loop L 0 defined by the equation H(x, y) = 0, passing through the origin O which is a cusp, and two families of periodic orbits For the case of cuspidal loop, Han et al. [15] studied the property of M ± for general near-Hamiltonian system and obtained the expansions of M ± near h = 0. Later, based on Han et al. [15], Atabigi et al. [16] and Xiong [17] gave the formulas of the first eight and eleven coefficients in the expansions of M ± as the cusp has order 2 and order 3, respectively.
As we know, for the purpose of estimating the maximal number of limit cycles, we need to find the maximal number of zeros of the first order Melnikov function. We can do this by using the expansion of it. In [15], the authors gave formulas of the first five coefficients in the expansions of the functions M ± as the cusp point has order 1. Then using these formulas one can find up to six limit cycles. If one wants to find more limit cycles near the cuspidal loop, one needs to establish formulas of more coefficients in the expansions of the functions M ± than [15]. For that purpose, in this paper we develop the idea for computing coefficients in the the expansion of M near a homoclinic loop passing through a hyperbolic saddle used in [18] to the case of a cuspidal loop. We give some conditions under which formulas of the first 11 coefficients in the expansions of M ± can be obtained, as the cusp has order 1; see Theorem 1. Thus, in our case we can obtain up to 19 limit cycles near the cuspidal loop; see Theorems 2 and 3. As an application example, we prove that the following Liénard systeṁ a i x i y can have 12 limit cycles. We can see that this paper is a continuation of [15].
We organize the paper as follows. In Section 2, we present some preliminary lemmas. In Section 3, we give the formulas for some coefficients in the expansions of M ± and the conditions to obtain limit cycles near the cuspidal loop of order 1. In Section 4, as an application example, we consider a Liénard system and find 12 limit cycles near the cuspidal loop and the center.

Preliminary
Suppose that system (2) has a nilpotent singular point and it is at the origin. In other words, the function H(x, y) satisfies H x (0, 0) = 0, H y (0, 0) = 0, and Further, we can suppose that Then, the function H at the origin has the following form H(x, y) = 1 2 Applying the implicit function theorem, there is a unique C ∞ function ϕ(x) = ∑ j≥2 e j x j exists such that H y (x, ϕ(x)) = 0 for 0 < |x| 1. By (5), we can write Let k ≥ 3 be an integer such that Regarding the order of the nilpotent singular point, Han et al. [15] gave the following definition.
We assume that the equation H(x, y) = 0 defines a cuspidal loop L 0 of the unperturbed system (2), and L 0 passes through a cusp of order 1 at the origin, and surrounds an elementary center C(x c , y c ). We further suppose that the cuspidal loop L 0 is clockwise oriented. Then, two families of periodic orbits L + h and L − h are defined by the equation H(x, y) = h for 0 < h < h + and h c < h < 0, respectively, where h c = H(x c , y c ) and h + is a positive constant as given before. The phase portrait is shown in Figure 1. The aim of our paper is to estimate the number of limit cycles of system (1) in the region V, where and Cl.(V 1 V 2 ) denotes the closure of V 1 V 2 . Under the above assumptions, two Melnikov functions of system (1) have the following form By Han et al. [15] we have the following lemma.

Lemma 1 ([15]
). Consider the analytic system (1). Let (7) hold with h 3 < 0 and Then, with We suppose that for (x, y) near (x c , y c ). About the expansion of M − near the center C(x c , y c ), we have (see [19]) The formulas of b j , j = 0, 1, 2, 3, can be found in [20] and more coefficients can be obtained by using the programs in [21]. The formula whereb 0 (δ) = ( f x + g y )(C, δ) and T 0 > 0 is a constant.

Main Results
In this section, we use Lemma 1 and the method given by Tian and Han in [18] to obtain more coefficients in the expansions of M ± . Then, we will get more limit cycles in the neighborhood of the cuspidal loop L 0 and the center. The following lemma was obtained in Han [22]. Lemma 2 ([22]). For the Melnikov functions M ± defined by (4) we have We assume that there are analytic functions P 1 (x, y, δ) and Q 1 (x, y, δ) defined on V such that for (x, y) near the origin. Differentiating (10) and (13) with respect to h, we obtain Next, applying Lemma 2 and (16) we have By (10), (11), (13) and (14), it can be seen that M − 1 (h, δ) and M + 1 (h, δ) have the form for 0 < −h 1, and and T 0 > 0 is the same as before; µ i and α 01 , α i0 , i = 1, 2, 3, 4 satisfy (12), with a ij and b ij replaced by a ij and b ij , respectively. Let Now by comparing the three expansions of M ± 1 above with (18), we have (25) Applying Lemma 2 to (19) again, we have Suppose there are analytic functions P 2 (x, y, δ) and Q 2 (x, y, δ) defined on V such that forb 0 = c 1 = c 3 =b 1 =c 4 =c 6 = 0, where for (x, y) near the origin. By Lemma 2, we further have Then, by applying formulas (10), (11), (13) and (14) to the functions M ± 2 (h, δ) again, we obtain for 0 < −h 1, and and T 0 > 0 is the constant as given before; µ i and α 01 , α i0 , i = 1, 2, 3, 4 satisfy (12), with a ij and b ij replaced byā ij andb ij , respectively. Let c 8 = c 2,2 ,c 9 = c 2,3 ,c 10 = c 2,4 .
Proof. System (38)| ε=0 has a nilpotent cusp O(0, 0) and an elementary center C(1, 0), and has the following Hamiltionian function It is obvious that From (11) and (14) we havē where and L 0 is given by y = ±x Simple computions give of the two first order Melnikov functions near the loop. This enabled us to find more limit cycles than [15]. By the same method, we can also obtain more coefficients in the expansions of the first order Melnikov functions as the cusp has order 2 and order 3. The method can also be applied to cases of loops with nilpotent saddles. However, we were not able to study an upper bound of the number of limit cycles in Theorems 2 and 3. It should be possible to give an upper bound for the maximal number of zeros of the Melnikov functions M ± under the conditions of Theorems 2 or 3. In Theorem 1 we give formulas for computing the coefficients c 5 , c 6 and c 7 in (10) if the functions P 1 and Q 1 satisfying condition (16) exist. In fact, the formulas of these coefficients do not depend on the existence of P 1 and on Q 1 satisfying condition (16). Then an interesting problem is to give the formulas without assuming (16). For high-dimensional systems, the Melnikov function is a vector function, and it is difficult to study the number of its zeros in this case. However, it is possible to use the expansion of the Melnikov function to study the number of periodic orbits; see [23]. As we knew, the Melnikov method can be also used to predict the occurrence of subharmonic solutions, invariant tori and chaotic orbits in non-autonomous smooth nonlinear systems under periodic perturbation; see [22,24] for example.