LIMIT CYCLE BIFURCATION FOR A NILPOTENT SYSTEM IN <i>Z</i><sub>3</sub>-EQUIVARIANT VECTOR FIELD

Chaoxiong Du; Qinlong Wang; Yirong Liu; Qi Zhang

doi:10.11948/2017089

2017 Volume 7 Issue 4

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Chaoxiong Du, Qinlong Wang, Yirong Liu, Qi Zhang. LIMIT CYCLE BIFURCATION FOR A NILPOTENT SYSTEM IN Z3-EQUIVARIANT VECTOR FIELD[J]. Journal of Applied Analysis & Computation, 2017, 7(4): 1463-1477. doi: 10.11948/2017089

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Chaoxiong Du, Qinlong Wang, Yirong Liu, Qi Zhang. LIMIT CYCLE BIFURCATION FOR A NILPOTENT SYSTEM IN Z₃-EQUIVARIANT VECTOR FIELD[J]. Journal of Applied Analysis & Computation, 2017, 7(4): 1463-1477. doi: 10.11948/2017089

LIMIT CYCLE BIFURCATION FOR A NILPOTENT SYSTEM IN Z₃-EQUIVARIANT VECTOR FIELD

1 Department of Mathematics, Changsha Normal University, Changsha, Hunan, 410100, China;
2 Department of Mathematics, Hezhou University, Hezhou, Guangxi 542800, China;
3 School of Mathematics, Central South University, ChangSha, Hunan 410083, China

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Abstract

Abstract

Our work is concerned with the problem on limit cycle bifurcation for a class of Z₃-equivariant Lyapunov system of five degrees with three third-order nilpotent critical points which lie in a Z₃-equivariant vector field. With the help of computer algebra system-MATHEMATICA, the first 5 quasiLyapunov constants are deduced. The fact of existing 12 small amplitude limit cycles created from the three third-order nilpotent critical points is also proved. Our proof is algebraic and symbolic, obtained result is new and interesting in terms of nilpotent critical points' Hilbert number in equivariant vector field.
- Third-order nilpotent critical point /
- Z₃-equivariant /
- limit cycle bifurcation /
- Quasi-Lyapunov constant
MSC: 34C07

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References

[1]	A. Algaba, C. Garca and M. Reyes, Local bifurcation of limit cycles and integrability of a class of nilpotent systems of differential equations, Applied Mathematics and Computation, 2009, 215, 314-323. Google Scholar
[2]	C. Du and W. Huang, Center-focus problem and limit cycles bifurcations for a class of cubic Kolmogorov model, Nonlinear Dynamics, 2013, 72, 197-206. Google Scholar
[3]	C. Du, Q. Wang and W. Huang, Three-Dimensional Hopf bifurcation for a class of cubic Kolmogorov model, International Journal of Bifurcation and Chaos, 2014, 24(3), 1450036. Google Scholar
[4]	C. Du, Y. Liu and W. Huang, Limit cycles bifurcations for a class of Kolmogorov model in symmetrical vector field, International Journal of Bifurcation and Chaos, 2014, 24(3), 1450040. Google Scholar
[5]	C. Du, W. Huang and Q. Zhang, Center problem and the bifurcation of limit cycles for a cubic polynomial system, Applied Mathematical Modelling, 2015, 39, 5200-5215. Google Scholar
[6]	C. Du, Y. Liu and Q. Zhang, Limit cycles in a class of quartic Kolmogorov model with three positive equilibrium points, International Journal of Bifurcation and Chaos, 2015, 25(6), 1550080. Google Scholar
[7]	C. Du, Y. Liu and W. Huang, Limit cycles bifurcations behavior for a class of quartic Kolmogorov model in symmetrical vector field, Applied Mathematical Modelling, 2016, 40, 4094-4108. Google Scholar
[8]	C. Du and Y. Liu, Limit cycle bifurcation of the infinity and degenerate singular point in 3-dimensional vector field,International Journal of Bifurcation and Chaos, 2016, 26(9), 1650152. Google Scholar
[9]	C. Du, Y. Liu and W. Huang, A Class of 3-Dimensional Quadratic Systems with 10 Limit Cycles, International Journal of Bifurcation and Chaos, 2016, 26(9), 1650149. Google Scholar
[10]	C. Du and Y. Liu, Isochronicity for a Z₂-equivariant cubic system, Nonlinear Dynamics, 2016. DOI:10.1007/s11071/016/3112/7. Google Scholar
[11]	M. Han, J. Jiang and H. Zhu, Limit cycle bifurcations in near-hamiltonian systems by perturbing a nilpotent center, International Journal of Bifurcation and Chaos, 2008, 18, 3013-3027. Google Scholar
[12]	M. Han, C. Shu, J. Yang and A. C.-L. Chian, Polynomial Hamiltonian systems with a nilpotent critical point, Advances in Space Research, 2010, 46, 521-525. Google Scholar
[13]	J. Jiang, Bifurcation of limit cycles for a quartic near-Hamiltonian system by perturbing a nilpotent center, Journal of Mathematical Analysis and Applications, 2010, 365, 376-384. Google Scholar
[14]	J. Li, Hilbert's 16th problem and bifurcation of Planar polynomial vector fields, International Journal of Bifurcation and Chaos, 2003, 13, 47-106. Google Scholar
[15]	J. Llibre, R. Ramírez and N. Sadovskaia, On the 16th Hilbert problem for limit cycles on nonsingular algebraic curves, Journal of Differential Equations, 2011, 250(2), 983-999. Google Scholar
[16]	J. Li and Y. Liu, New results on the study of Z_q-equivariant planar polynomial vector fields, Qualitative Theory of Dynamical Systems, 2010, 9, 167-219. Google Scholar
[17]	Y. Liu, Theory of center-focus for a class of higher-degree critical points and infinite points, Science in China (Series A), 2001, 44, 37-48. Google Scholar
[18]	Y. Liu and J. Li, Theory of values of singular point in complex autonomous differential system, Science in China (Series A), 1990, 33, 10-24. Google Scholar
[19]	Y. Liu, J. Li and W. Huang, Singular point values, center problem and bifurcations of limit cycles of two dimensional differential autonomous systems, Science Press, Beijing, China, 2008. Google Scholar
[20]	Y. Liu and J. Li, On third-order nilpotent critical points:integral factor method, International Journal of Bifurcation and Chaos, 2011, 21(5), 497-504. Google Scholar
[21]	Y. Liu and J. Li, Bifurcations of limit cycles created by a multiple nilpotent critical point of planar dynamical systems, International Journal of Bifurcation and Chaos, 2011, 21(2), 293-1309. Google Scholar
[22]	F. Li and M. Wang, Bifurcation of limit cycles in a quintic system with ten parameters, Nonlinear Dynamics, 2013, 71, 213-222. Google Scholar
[23]	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, 2011, 16, 2598-2608. Google Scholar
[24]	X. Pan, Bifurcation of limit cycles for a class of cubic polynomial system having a nilpotent singular point, Applied Mathematics and Computation, 2011, 218, 1161-1165. Google Scholar
[25]	S. Shi, A concrete example of the existence of four limit cycles for quadratic systems, Science in China, Ser A, 1980, 23, 16-21. Google Scholar
[26]	Y. Wu, F. Li and H. Chen, Center conditions and bifurcation of limit cycles at three-order nilpotent critical point in a cubic Lyapunov system, Communications in Nonlinear Science and Numerical Simulation, 2012, 17, 292-304. Google Scholar
[27]	L. Yirong and L. Feng, Double bifurcation of nilpotent focus, International Journal of Bifurcation and Chaos, 2015, 25, 1550036. Google Scholar
[28]	P. Yu and M. Han, Small limit cycles bifurcation from fine focus points in cubic order z₂-equivariant vector fields, Chaos,Solitons and Fractals, 2005, 24, 329-348. Google Scholar
[29]	W. Zhang and H. Hu, New Development of Theory and Applications on Nonlinear Dynamics, Science Press, Beijing, 2009. Google Scholar