Local bifurcation of limit cycles and integrability of a class of nilpotent systems
© Li et al.; licensee Springer. 2012
Received: 21 June 2011
Accepted: 28 February 2012
Published: 28 February 2012
In this article, center conditions and bifurcation of limit cycles at the nilpotent critical point in a class of seventh-degree systems are investigated. With the help of computer algebra system MATHEMATICA, the first 13 quasi-Lyapunov constants are deduced. As a result, sufficient and necessary conditions in order to have a center are obtained. The result that there exist 13 small amplitude limit cycles created from the three-order nilpotent critical point is also proved. Henceforth, we give a lower bound of cyclicity of three-order nilpotent critical point for seventh-degree nilpotent systems.
MSC: 34C05; 34C07.
Computation of Lyapunov quantities is a hot topic with large number of articles per year, but it is very difficult to obtain general results. The methods of computation of Lyapunov quantities when the critical points are non-degenerate have been greatly developed by many mathematicians. The method in [1, 2] is based on the sequential construction of Lyapunov functions. Furthermore, computing Lyapunov quantities using the reduction of system to normal form could be seen in [3–5]. Another approach to numerical computation of Lyapunov quantities which uses the passage to the polar coordinates and the procedure of sequential construction of solution approximations is related with the obtaining of approximations of system solution, see , they also could be seen in [6, 7]. But computations of Lyapunov quantities become difficult when the critical points are degenerate because the method of the Poincaré formal series cannot be used in order to compute Lyapunov constants in a neighborhood of the critical point.
The nilpotent center problem was investigated by Moussu  and Stróżyna and Żołądek . In , Takens proved that Lyapunov system can be formally transformed into a generalized Liénard system. Furthermore, in , Álvarez and Gasull proved that the generalized Lienard system could be simplified even more by a reparametrization of the time. At the same time, Giacomini et al. [12, 13] proved that the analytic nilpotent systems with a center can be expressed as limit of non-degenerate systems with a center.
As far as we know, there are essentially three differential ways of obtaining Lyapunov constant for nilpotent critical points in theory: by using normal form theory , by computing the Poincaré return map  or by using Lyapunov functions . Álvarez investigated the momodromy and stability for nilpotent critical points with the method of computing the Poincaré return map, see for instance ; Chavarriga et al. investigated the local analytic integrability for nilpotent centers by using Lyapunov functions, see for instance ; Moussu investigated the center-focus problem of nilpotent critical points with the method of normal form theory, see for instance .
where F k and G k are (1, n)-quasi-homogeneous functions of degree k. Chavarriga et al. investigated the integrability of centers perturbed by (p, q)-quasi-homogeneous polynomials in . Fortunately, Yirong Liu and Jibin Li  found that there always exists a formal inverse integrating factor for three-order nilpotent critical points in 2009, and they gave a new definition of the focal values under the generalized triangle polar coordinates and the method of commuting Lyapunov constants using the inverse integral factors for the three-order nilpotent critical point.
We will prove N(7) ≥ 13. To the best of authors' knowledge, their result on the lower bounds of cyclicity of three-order nilpotent critical points for septic systems is new. It is helpful to Hilbert's 16th problem.
The rest of the article is organized as follows. In Section 2, some preliminary knowledge given in  which is useful throughout the article are introduced. In Section 3, using the linear recursive formulae in  to do direct computation, the first 13 quasi-Lyapunov constants and the sufficient and necessary conditions of center are obtained. This article is ended with Section 4 in which the 13-order weak focus conditions and the result that there exist 13 limit cycles in the neighborhood of the three-order nilpotent critical point is proved.
2 Preliminary knowledge
We have the following result.
whereis a polynomial of ν j (π), ν j (2π), ν j (-2π), (j = 2,3,..., 2m) with rational coefficients.
It is different from the center-focus problem for the elementary critical points, we know from Lemma 2.1 that when k > 1 for the first non-zero ν k (-2π), k is an even integer.
Definition 2.1. 1. For any positive integer m, ν2m(-2π) is called the m-order focal value of system (2.2) in the origin.
2. If ν2(-2π) ≠ 0, the origin of system (2.2) is called 1-order weak focus. If there is an integer m > 1, such that ν2(-2π) = ν4(-2π) = ⋯ = ν2m-2(-2π) = 0, ν2m(-2π) ≠ 0, then, the origin of system (2.2) is called m-order weak focus.
3. If for all positive integer m, we have ν2m(-2π) = 0, then, the origin of system (2.2) is called a center.
For an integer k, letting ν2k(-2π, γ) be the k-order focal value of the origin of system (2.7)δ = 0.
then, there exist two positive numbers δ* and γ*, such that for 0 < |δ| < δ*, 0 < ||γ-γ0|| < γ*, in a neighborhood of the origin, system (2.7) has at most m limit cycles which enclose the origin (an elementary node) O(0,0). In addition, under the above conditions, there exist, such that when, there exist exact m limit cycles of (2.7) in a small neighborhood of the origin.
The following key results which define the quasi-Lyapunov constants and provide a way of computing them were also given by Liu and Li .
Definition 2.2. If there exists a natural number s and a formal series M(x, y) = x4 + y2 + o(r4), such that (2.10) holds, then, λ m is called the m-th quasi-Lyapunov constants of the origin of system (2.2).
where for all k, M k (x, y) is a k-homogeneous polynomial of x, y and sμ = 0.
It is easy to see that (2.15) is linear with respect to the function M, so that we can easily find the following recursive formulae for the calculation of c αβ and ω m (s, μ).
Clearly, the recursive formulae by Theorem 2.5 is linear with respect to all c αβ . Therefore, it is convenient to perform the computations by using computer algebraic system like MATHEMATICA.
3 Quasi-Lyapunov constants and center conditions
Furthermore, the following conclusion holds.
where λ m is the mth quasi-Lyapunov constant at the origin of system (1.2), m = 1, 2,..., 13.
In the above expressions of λ k , we have already, let λ1 = λ2 = · · · = λk-1= 0, k = 2,..., 13.
It follows from Theorem 3.1 that
Proposition 3.2 implies that
Proposition 3.3. The origin of systems (3.6) is a center.
From Propositions 3.2 and 3.3, we further have
Theorem 3.2. The origin of system (1.2) is a center if and only if the first 13 quasi-Lyapunov constants are zero, that is, the condition in Proposition 3.2 is satisfied.
4 Multiple bifurcation of limit cycles
It is very interesting to study the number of limit cycles which could be bifurcated from critical point O(0,0), because it is closely related with 16th problem of 23 problems of Hilbert. In this section, we will prove that the perturbed system of (1.2) can generate 13 limit cycles enclosing an elementary node at the origin when the three-order nilpotent critical point O(0,0) is a 13-order weak focus.
we have that
So λ13 ≠ 0, the origin of system (1.2) is a 13-order weak focus.
When conditions of (4.1) hold, by the relationships λ 1 = λ2 = λ3 = λ4 = λ5 = λ6 = λ7 = λ8 = λ9 = λ10 = λ11 = λ 12 = 0, the values of b21, a12, b23, a14, b03a32, b05, a03, b14, a05, a23, μ could be determined. Notice that μ = ± -0.00917916 are the simple zeros of λ12 = 0. Hence, when conditions in (4.1) hold, we have
So when μ ≈ ±0.00917916, J ≠ 0.
From the statement mentioned above, Theorem 2.1 follows that
Theorem 4.2. If the three-order nilpotent critical point O(0,0) of system (1.2) is a 13-order weak focus, for 0 < δ ≪ 1, making a small perturbation to the coefficients of system (1.2), then, for system (4.3), in a small neighborhood of the origin, there exist exactly 13 small amplitude limit cycles enclosing the origin O(0,0), which is an elementary node.
when k < 0, or j < 0, c[k, j] = 0;
This research was partially supported by the Nature Science Foundation of Shandong Province (Y2007A17, ZR2010AL005) and the National Nature Science Foundation of China (11071222, 11101126).
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