Fourteen Limit Cycles in a Seven-Degree Nilpotent System
Center conditions and the bifurcation of limit cycles for a seven-degree polynomial differential system in which the origin is a nilpotent critical point are studied. Using the computer algebra system Mathematica, the first 14 quasi-Lyapunov constants of the origin are obtained, and then the conditions for the origin to be a center and the 14th-order fine focus are derived, respectively. Finally, we prove that the system has 14 limit cycles bifurcated from the origin under a small perturbation. As far as we know, this is the first example of a seven-degree system with 14 limit cycles bifurcated from a nilpotent critical point.
In the qualitative theory of planar differential equations, the center-focus problem and bifurcation of limit cycles for nilpotent system are known as a difficult problem. Some advance of this problem can be dated back to [1–3]. In recent years, due to the improvement of research method and development of computer symbolic computation, the problem has attracted more and more scholars’ attention and has received a lot of results. For instance, in [4, 5], the center conditions of the nilpotent critical points were obtained for several systems. In  the center conditions and the bifurcations of limit cycles were investigated for a quintic and a nine-degree nilpotent systems. The center and the limit cycles problems of a quintic nilpotent system were also solved in . And in , the authors gave a recursive method to calculate quasi-Lyapunov constants of the nilpotent critical point. The nilpotent center problem and limit cycles bifurcations were performed also in . It is interesting how many limit cycles can be bifurcated from the nilpotent critical point. Let be the maximum possible number of limit cycles bifurcated from a nilpotent critical point of system (1) when and are of degree at most . The known results of are: Andreev et al. given have , , , see . Y. Liu and J. Li showed , , , see [8, 10–12]. Li et al. found in . Recently, Li et al.  obtained .
In this paper, we study the bifurcation of limit cycles for a seven-degree nilpotent system with the following form: By the computation of the quasi-Lyapunov constants, we prove that its perturbed system has 14 small-amplitude limit cycles bifurcated from the origin, namely, which improves the result in .
In Section 2, we give some preliminary knowledge concerning the nilpotent critical point. In Section 3, we obtain the first 14 quasi-Lyapunov constants and derive the sufficient and necessary conditions of the origin to be a center and a 14th-order fine focus. At the end, it is proved that there exist 14 limit cycles in the neighborhood of the origin of the system.
2. Focal Values and Quasi-Lyapunov Constants
In order to discuss limit cycles of the system, we state some preliminary results given by .
According to , the origin of system is a 3th-order monodromic critical point and a center or a focus if and only if , . Without loss of generality, we assume that , , , , otherwise let , .
Under the substitutions system (1) becomes
By the transformation of the generalized polar coordinates, system (4) is transformed into where
For sufficiently small , let be a solution of (6) satisfying the initial value condition , where
Because for all sufficiently small , there is , in a small neighborhood; we obtain the Poincaré return map of (6) in a small neighborhood of the origin as follows:
Lemma 1. For any positive integer , has the form where is a polynomial of , , , with rational coefficients.
Definition 2. For any positive integer , is called the th-order focal value of system (4) at the origin; if , the origin of system (4) is called an 1th-order weak focus; if there is an integer such that , , then the origin of system (4) is called a th-order weak focus; if for all positive integer , we have , the origin of system (4) is called a center.
Lemma 3. For system (4), one can derive successively the formal series such that
Lemma 4. If there exists a natural number and formal series such that (13) holds, then where In (15), is the symbol of algebraic equivalence, meaning that there exists , polynomial functions of the coefficients of system (4), such that
Lemma 6. For system (4), one can derive successively the formal series such that where , . For , , , and are determined by the following recursive formulas: where By choosing such that one has
One considers the perturbed system of system (4)
For system (24), from Lemma 4, we know that the first nonvanishing quasi-Lyapunov constant is positive constant times as much as the first nonvanishing focal value, so the former shows the same effect as the latter in the study of bifurcation of limit cycles. From [10, Theorem 4.7], we have the following.
Theorem 7. For the system (27), assume that the quasi-Lyapunov constants of the origin have independent parameters ; that is, . If , the origin of the system (4) is an th-order weak focus (), and the Jacobian determinant then, the perturbed system (24) exists small amplitude limit cycles bifurcated from the origin.
3. Criterion of Center Focus and Bifurcation of Limit Cycles
Theorem 8. For system (2), the first 14 quasi-Lyapunov constants are as follows:
Here, every () was computed under the assumption .
It is easy to obtain the following Theorem.
Theorem 9. For system (2), the first 14 quasi-Lyapunov constants at the origin are all zero if and only if the following condition is satisfied:
If and the condition (27) holds, system (2) becomes which is symmetric with respect to the -axis, one has the following.
Theorem 11. The origin of system (2) is a 14th-order weak focus if and only if
By computing carefully, we obtain that the Jacobian determinant
This paper is partly supported by the Nature Science Foundation of China Grants 11261013 and 11361017 and the Nature Science Foundation of Guangxi (2012GXNSFAA053003).
V. V. Amel’kin, N. A. Lukashevich, and A. P. Sadovskiĭ, Nonlinear Oscillations in Second Order Systems, Belarusian State University, Minsk, Russia, 1982, (Russian).View at: MathSciNet
V. G. Romanovskii, “On the cyclicity of the equilibrium position of the center or focus type of a certain system,” Vestnik St. Petersburg University: Mathematics, vol. 19, pp. 51–56, 1986.View at: Google Scholar
A. F. Andreev, A. P. Sadovskiĭ, and V. A. Tsikalyuk, “The center-focus problem for a system with homogeneous nonlinearities in the case of zero eigenvalues of the linear part,” Differential Equations, vol. 39, no. 2, pp. 155–164, 2003.View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
F. Li, Y. Liu, and Y. Wu, “Center conditions and bifurcation of limit cycles at three-order nilpotent critical point in a seventh degree Lyapunov system,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 6, pp. 2598–2608, 2011.View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet