Ten Limit Cycles in a Quintic Lyapunov System
Authors: Li Feng
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.
Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1334726Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 1255
 M.J. A' lvarez, A. Gasull, Momodromy and stability for nilpotent critical points, Internat. J. Bifur. Chaos 15 (2005) 1253-1265.
 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.
 V.V. Amelkin, N.A Lukashevich and A.N. Sadovskii, Nonlinear Oscillations in the Second Order Systems, BGU Publ., Minsk (in Russian).
 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.
 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.
 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.
 J. Chavarriga, H. Giacomini, J. Gin'e, J. Llibre, Local analytic integrability for nilpotent centers, Ergodic Theory Dynam. Systems 23 (2003) 417-428.
 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.
 Y. Liu, J. Li, some classical problems about planar vector fileds (in chinese), Science press (China), 2010 pp.279-316.
 R. Moussu, Sym'etrie et forme normale des centres et foyers d'eg'en'er'es, Ergodic Theory Dynam. Systems 2 (1982) 241-251.
 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.
 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.
 F. Takens, Singularities of vector fields, Inst. Hautes ' Etudes Sci. Publ. Math. 43 (1974) 47-100.
 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.
 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.