## Journal of Applied Mathematics

### The Center Conditions and Bifurcation of Limit Cycles at the Degenerate Singularity of a Three-Dimensional System

#### Abstract

We investigate multiple limit cycles bifurcation and center-focus problem of the degenerate equilibrium for a three-dimensional system. By applying the method of symbolic computation, we obtain the first four quasi-Lyapunov constants. It is proved that the system can generate 3 small limit cycles from nilpotent critical point on center manifold. Furthermore, the center conditions are found and as weak foci the highest order is proved to be the fourth; thus we obtain at most 3 small limit cycles from the origin via local bifurcation. To our knowledge, it is the first example of multiple limit cycles bifurcating from a nilpotent singularity for the flow of a high-dimensional system restricted to the center manifold.

#### Article information

Source
J. Appl. Math., Volume 2014 (2014), Article ID 546243, 8 pages.

Dates
First available in Project Euclid: 2 March 2015

https://projecteuclid.org/euclid.jam/1425305889

Digital Object Identifier
doi:10.1155/2014/546243

Mathematical Reviews number (MathSciNet)
MR3232919

#### Citation

Song, Shugang; Feng, Jingjing; Wang, Qinlong. The Center Conditions and Bifurcation of Limit Cycles at the Degenerate Singularity of a Three-Dimensional System. J. Appl. Math. 2014 (2014), Article ID 546243, 8 pages. doi:10.1155/2014/546243. https://projecteuclid.org/euclid.jam/1425305889

#### References

• C. Sparrow, The Lorenz Equations: Bifcations, Chaos, and Strange Attractors, Springer, New York, NY, USA, 1982.
• T. Li, G. Chen, and Y. Tang, “On stability and bifurcation of Chen's system,” Chaos, Solitons and Fractals, vol. 19, no. 5, pp. 1269–1282, 2004.
• L. F. Mello and S. F. Coelho, “Degenerate Hopf bifurcations in the Lü system,” Physics Letters A, vol. 373, no. 12-13, pp. 1116–1120, 2009.
• M. Messias, D. D. C. Braga, and L. F. Mello, “Degenerate Hopf bifurcations in Chua's system,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 19, no. 2, pp. 497–515, 2009.
• J. Hofbauer and J. W.-H. So, “Multiple limit cycles for three-dimensional Lotka-Volterra equations,” Applied Mathematics Letters, vol. 7, no. 6, pp. 65–70, 1994.
• Z. Lu and Y. Luo, “Two limit cycles in three-dimensional Lotka-Volterra systems,” Computers & Mathematics with Applications, vol. 44, no. 1-2, pp. 51–66, 2002.
• M. Gyllenberg and M. Yan, “Four limit cycles for a three-dimensional competitive Lotka–Volterra system with a heteroclinic cycle,” Computers & Mathematics with Applications, vol. 58, no. 4, pp. 649–669, 2009.
• Q. Wang, W. Huang, and B. Li, “Limit cycles and singular point quantities for a 3D Lotka-Volterra system,” Applied Mathematics and Computation, vol. 217, no. 21, pp. 8856–8859, 2011.
• Q. Wang, Y. Liu, and H. Chen, “Hopf bifurcation for a class of three-dimensional nonlinear dynamic systems,” Bulletin des Sciences Mathématiques, vol. 134, no. 7, pp. 786–798, 2010.
• J. Carr, Applications of Centre Manifold Theory, vol. 35 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1981.
• M. J. Alvarez and A. Gasull, “Generating limit cycles from a nilpotent critical point via normal forms,” Journal of Mathematical Analysis and Applications, vol. 318, no. 1, pp. 271–287, 2006.
• M. J. Álvarez and A. Gasull, “Monodromy and stability for nilpotent critical points,” International Journal of Bifurcation and Chaos, vol. 15, no. 4, pp. 1253–1265, 2005.
• J. Chavarriga, H. Giacomin, J. Gine, and J. Llibre, “Local analytic integrability for nilpotent centers,” Ergodic Theory and Dynamical Systems, vol. 23, no. 2, pp. 417–428, 2003.
• R. Moussu, “Symetrie et forme normale des centres et foyers degeneres,” Ergodic Theory and Dynamical Systems, vol. 2, no. 2, pp. 241–251, 1982.
• A. F. Andreev, A. P. Sadovskii, and V. A. Tsikalyuk, “The center-focus problem for a system with homogeneous nonlinearities in the case of zero eigenvalues of the linear pact,” Differential Equations, vol. 39, no. 2, pp. 155–164, 2003.
• H. Giacomini, J. Giné, and J. Llibre, “The problem of distinguishing between a center and a focus for nilpotent and degenerate analytic systems,” Journal of Differential Equations, vol. 227, no. 2, pp. 406–426, 2006.
• Y. Liu and J. Li, “Bifurcations of limit cycles and center problem for a class of cubic nilpotent system,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 20, no. 8, pp. 2579–2584, 2010.
• Y. Liu and J. Li, “On third-order nilpotent critical points: integral factor method,” International Journal of Bifurcation and Chaos, vol. 21, no. 5, pp. 1293–1309, 2011.
• 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.
• A. Buică, I. A. García, and S. Maza, “Existence of inverse Jacobi multipliers around Hopf points in ${\mathbb{R}}^{3}$: emphasis on the center problem,” Journal of Differential Equations, vol. 252, no. 12, pp. 6324–6336, 2012.
• V. F. Edneral, A. Mahdi, V. G. Romanovski, and D. S. Shafer, “The center problem on a center manifold in ${\mathbb{R}}^{3}$,” Nonlinear Analysis, vol. 75, no. 4, pp. 2614–2622, 2012.
• F. S. Dias and L. F. Mello, “Analysis of a quadratic system obtained from a scalar third order differential equation,” Electronic Journal of Differential Equations, vol. 2010, 6 pages, 2010.
• J. Llibre, “On the integrability of the differential systems in dimension two and of the polynomial differential systems in arbitrary dimension,” The Journal of Applied Analysis and Computation, vol. 1, no. 1, pp. 33–52, 2011. \endinput