THE CENTER-FOCUS PROBLEM AND BIFURCATION OF LIMIT CYCLES IN A CLASS OF 7TH-DEGREE POLYNOMIAL SYSTEMS

Bo Sang; Qinlong Wang

doi:10.11948/2016052

2016 Volume 6 Issue 3

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Bo Sang, Qinlong Wang. THE CENTER-FOCUS PROBLEM AND BIFURCATION OF LIMIT CYCLES IN A CLASS OF 7TH-DEGREE POLYNOMIAL SYSTEMS[J]. Journal of Applied Analysis & Computation, 2016, 6(3): 817-826. doi: 10.11948/2016052

Citation:

Bo Sang, Qinlong Wang. THE CENTER-FOCUS PROBLEM AND BIFURCATION OF LIMIT CYCLES IN A CLASS OF 7TH-DEGREE POLYNOMIAL SYSTEMS[J]. Journal of Applied Analysis & Computation, 2016, 6(3): 817-826. doi: 10.11948/2016052

THE CENTER-FOCUS PROBLEM AND BIFURCATION OF LIMIT CYCLES IN A CLASS OF 7TH-DEGREE POLYNOMIAL SYSTEMS

Bo Sang^1,2,
Qinlong Wang^2,3

1 School of Mathematical Sciences, Liaocheng University, Liaocheng 252059, P. R. China;
2 Guangxi Education Department Key Laboratory of Symbolic Computation and Engineering Data Processing, Hezhou University, Hezhou 542899, Guangxi, P. R. China;
3 School of Science, Hezhou University, Hezhou 542899, P. R. China

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Abstract

Abstract

By computing singular point values, the center conditions are established for a class of 7th-degree planar polynomial systems with 15 parameters. It is proved that such systems can have 13 small-amplitude limit cycles in the neighborhood of the origin. To the best of our knowledge, this is the first example of a 7th-degree system having non-homogeneous nonlinearities with thirteen limit cycles bifurcated from a fine focus.
- Limit cycle /
- center variety /
- singular point value /
- time-reversibility
MSC: 34C05;34C07

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References

[1]	T.R. Blows and N.G. Lloyd, The number of limit cycles of certain polynomial differential equations, Proc. Roy. Soc. Edinburgh Sect. A, 98(1984)(3-4), 215-239. Google Scholar
[2]	J. Carr, Applications of centre manifold theory, Applied Mathematical Sciences, Vol. 35, Springer-Verleg, New York, 1981. Google Scholar
[3]	J. Chavarriga and J. Giné, Integrability of a linear center perturbed by fourth degree homogeneous polynomial, Publ. Mat., 40(1996)(1), 21-39. Google Scholar
[4]	C. Christopher and S. Lynch, Small-amplitude limit cycle bifurcations for Lienard systems with quadratic or cubic damping or restoring forces, Nonlinearity, 12(1999)(4), 1099-1112. Google Scholar
[5]	C. Christopher, P. Mardešić and C. Rousseau, Normalizable, integrable, and linearizable saddle points for complex quadratic systems in C², J. Dyn. Control Sys., 9(2003)(3), 311-363. Google Scholar
[6]	C. Du, Y. Liu and H. Chen, The bifurcation of limit cycles in Z_n-equivariant vector fields, Appl. Math. Comput., 217(2010)(5), 2041-2056. Google Scholar
[7]	J. Giné, Limit cycle bifurcations from a non-degenerate center, Appl. Math. Comput., 218(2012)(9), 4703-4709. Google Scholar
[8]	G.M. Greuel and G. Pfister, A Singular Introduction to Commutative Algebra, second edition, Springer, Heidelberg, 2008. Google Scholar
[9]	M. Han and P. Yu, Normal Forms, Melnikov Functions, and Bifurcation of Limit Cycles, Springer-Verlag, New York, 2012. Google Scholar
[10]	M. Han, Y. Tian and P. Yu, Small-amplitude limit cycles of polynomial Lienard systems, Sci. China Math., 56(2013)(8), 1543-1556. Google Scholar
[11]	M. Han, Bifurcation Theory of Limit Cycles, Mathematics Monograph Series, Vol. 25, Science Press, Beijing, 2013. Google Scholar
[12]	M. Han and P. Yu, Ten limit cycles around a center-type singular point in a 3-d quadratic system with quadratic perturbation, Appl. Math. Lett., 44(2015)(6), 17-20. Google Scholar
[13]	C. Li, C. Liu and J. Yang, A cubic system with thirteen limit cycles, J. Diff. Equ., 246(2009)(9), 3609-3619. Google Scholar
[14]	J. Li, Hilbert's 16th problem and bifurcations of planar polynomial vector fields, Int. J. Bifur. Chaos, 13(2003)(1), 47-106. Google Scholar
[15]	J. Li and Y. Liu, New results on the study of Z_q-equivariant planar polynomial vector fields, Qual. Theory Dyn. Syst., 9(2010)(1-2), 167-219. Google Scholar
[16]	Y. Liu, J. Li and W. Huang, Singular Point Values, Center Problem and Bifurcations of Limit Cycles of Two Dimensional Differential Autonomous Systems, Nonlinear Sciences Series, Vol. 6, Science Press, Beijing, 2008. Google Scholar
[17]	K.E. Malkin, Criteria for the center for a certain differential equation, Volg. Matem. Sbornik, 2(1964), 87-91. Google Scholar
[18]	V.G. Romanovski and D.S. Shafer, On the center problem for p:-q resonant polynomial vector fields, B. Belg. Math. Soc-Sim., 15(2008)(5), 871-887. Google Scholar
[19]	V.G. Romanovski, Time-Reversibility in 2-Dimensional systems, Open Sys., Inf. Dynamics, 15(2008)(4), 359-370. Google Scholar
[20]	V.G. Romanovski and D.S. Shafer, The Center and Cyclicity Problems:A Computational Algebra Approach, Birkhäuser, Boston, 2009. Google Scholar
[21]	B. Sang, Center problem for a class of degenerate quartic systems, Electron. J. Qual. Theory Differ. Equ., 74(2014), 1-17. Google Scholar
[22]	B. Sang, Q. Wang and W. Huang, Computation of focal values and stability analysis of 4-dimensional systems, Electron. J. Diff. Equ., 209(2015), 1-11. Google Scholar
[23]	D. Schlomiuk, Algebraic particular integrals, integrability and the problem of the center, Trans. Amer. Math. Soc., 338(1993)(2), 799-841. Google Scholar
[24]	Y. Shao and K. Wu, The cyclicity of the period annulus of two classes of cubic isochronous systems, J. Appl. Anal. Comput., 3(2013)(3), 279-290. Google Scholar
[25]	Q. Wang, Y. Liu and H. Chen, Hopf bifurcation for a class of three-dimensional nonlinear dynamic systems, Bull. Sci. Math., 134(2010)(7), 786-798. Google Scholar
[26]	J. Yang, M. Han, J. Li and P. Yu, Existence conditions of thirteen limit cycles in a cubic system, Int. J. Bifurcation and Chaos, 20(2010)(8), 2569-2577. Google Scholar
[27]	P. Yu, Computation of normal forms via a perturbation technique, J. Sound Vib., 211(1998)(1), 19-38. Google Scholar
[28]	P. Yu and M. Han, Twelve limit cycles in a 3rd-order planar system with Z₂ symmetry, Commun. Pure Appl. Anal., 3(2004)(3), 515-526. Google Scholar
[29]	P. Yu and J. Lu, Bifurcation control for a class of Lorenz-like systems, Int. J. Bifurcation and Chaos, 21(2011)(9), 2647-2664. Google Scholar
[30]	P. Yu and M. Han, Four limit cycles from perturbing quadratic integrable systems by quadratic polynomials, Int. J. Bifurcation and Chaos, 22(2012)(10), 1250254(28 pages). Google Scholar
[31]	P. Yu and M. Han, Bifurcation of limit cycles in quadratic Hamiltonian systems with various degree polynomial perturbations, Chaos Solitons Fractals, 45(2012)(6), 772-794. Google Scholar
[32]	P. Yu and M. Han, Bifurcation of limit cycles in 3rd-order Z₂ Hamiltonian planar vector fields with 3rd-order perturbations, Commun. Nonl. Sci. Numer. Simulat., 18(2013)(4), 978-988. Google Scholar
[33]	P. Yu and Y. Tian, Twelve limit cycles around a singular point in a planar cubic-degree polynomial system, Commun. Nonl. Sci. Numer. Simulat., 19(2014)(8), 2690-2705. Google Scholar
[34]	H. Żołądek, _The problem of center for resonant singular points of polynomial vector fields, J. Diff. Equ., 137(1997)(1), 94-118. Google Scholar