**Commenced**in January 2007

**Frequency:**Monthly

**Edition:**International

**Paper Count:**30455

##### Ten Limit Cycles in a Quintic Lyapunov System

**Authors:**
Li Feng

**Abstract:**

In this paper, center conditions and bifurcation of limit cycles at the nilpotent critical point in a class of quintic polynomial differential system are investigated.With the help of computer algebra system MATHEMATICA, the first 10 quasi Lyapunov constants are deduced. As a result, sufficient and necessary conditions in order to have a center are obtained. The fact that there exist 10 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 quintic Lyapunov systems. At last, we give an system which could bifurcate 10 limit circles.

**Keywords:**
Three-order nilpotent critical point,
center-focus problem,
bifurcation of limit cycles,
Quasi-Lyapunov constant

**Digital Object Identifier (DOI):**
doi.org/10.5281/zenodo.1334726

**References:**

[1] M.J. A' lvarez, A. Gasull, Momodromy and stability for nilpotent critical points, Internat. J. Bifur. Chaos 15 (2005) 1253-1265.

[2] M.J. A' lvarez, A. Gasull, Generating limit cycles from a nilpotent critical point via normal forms, J. Math. Anal. Appl. 318 (2006) 271-287.

[3] V.V. Amelkin, N.A Lukashevich and A.N. Sadovskii, Nonlinear Oscillations in the Second Order Systems, BGU Publ., Minsk (in Russian).

[4] A.F. Andreev, Investigation of the behavior of the integral curves of a system of two differential equations in the neighbourhood of a singular point, Transl. Amer. Math. Soc. 8 (1958) 183-207.

[5] A.F. Andreev, A.P. Sadovskii, 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 39 (2003) 155-164.

[6] A.A. Andronov, E.A. Leontovich, I.I. Gordon, A.G. Maier, Theory of Bifurcations of Dynamical Systems on a Plane, Wiley, New York, 1973.

[7] J. Chavarriga, H. Giacomini, J. Gin'e, J. Llibre, Local analytic integrability for nilpotent centers, Ergodic Theory Dynam. Systems 23 (2003) 417-428.

[8] W.W. Farr, C. Li, I.S. Labouriau, W.F. Langford, Degenerate Hopf- bifurcation formulas and Hilbert-s 16th problem, SIAM J. Math. Anal. 20 (1989) 13-29.

[9] Y. Liu, J. Li, some classical problems about planar vector fileds (in chinese), Science press (China), 2010 pp.279-316.

[10] R. Moussu, Sym'etrie et forme normale des centres et foyers d'eg'en'er'es, Ergodic Theory Dynam. Systems 2 (1982) 241-251.

[11] S.L. Shi, On the structure of Poincar'e-Lyapunov constants for the weak focus of polynomial vector fields, J. Differential Equations 52 (1984) 52-57.

[12] E. Str'oİzyna, H. İ Zo┼éa┬©dek, The analytic and formal normal form for the nilpotent singularity, J. Differential Equations 179 (2002) 479-537.

[13] F. Takens, Singularities of vector fields, Inst. Hautes ' Etudes Sci. Publ. Math. 43 (1974) 47-100.

[14] H. Giacomini, J. Gin, J. Llibre, The problem of distinguishing between a center and a focus for nilpotent and degenerate analytic systems, J. Diff. Equat. 227 (2)(2006) 406-426.

[15] Y. Liu, J. Li, Bifurcation of limit cycles and center problem for a class of cubic nilpotent system, Int. J. Bifurcation and Chaos. 20(8)(2010)2579- 2584.