SIXTEEN LARGE-AMPLITUDE LIMIT CYCLES IN A SEPTIC SYSTEM

Lina Zhang; Feng Li; Ahmed Alsaedi

doi:10.11948/2018.1821

2018 Volume 8 Issue 6

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Lina Zhang, Feng Li, Ahmed Alsaedi. SIXTEEN LARGE-AMPLITUDE LIMIT CYCLES IN A SEPTIC SYSTEM[J]. Journal of Applied Analysis & Computation, 2018, 8(6): 1821-1832. doi: 10.11948/2018.1821

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Lina Zhang, Feng Li, Ahmed Alsaedi. SIXTEEN LARGE-AMPLITUDE LIMIT CYCLES IN A SEPTIC SYSTEM[J]. Journal of Applied Analysis & Computation, 2018, 8(6): 1821-1832. doi: 10.11948/2018.1821

SIXTEEN LARGE-AMPLITUDE LIMIT CYCLES IN A SEPTIC SYSTEM

1 Department of Mathematics, Huzhou University, Huzhou, Zhejiang 313000, China;
2 School of Mathematics and Statistics, Linyi University, Linyi, Shandong 276000, China;
3 Nonlinear Analysis and Applied Mathematics(NAAM) Research Group, Faculty of Science King Abdulaziz University, Jeddah 21589, Saudi Arabia

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Abstract

Abstract

In this paper, bifurcation of limit cycles from the infinity of a twodimensional septic polynomial differential system is investigated. Sufficient and necessary conditions for the infinity to be a center are derived and the fact that there exist 16 large amplitude limit cycles bifurcated from the infinity is proved as well. The study relays on making use of a recursive formula for computing the singular point quantities of the infinity. As far as we know, this is the first example of a septic system with 16 limit cycles bifurcated from the infinity.
- Septic system /
- Infinity /
- Singular point quantities /
- Limit cycles
MSC: 34C05;34C07

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References

[1]	V. V. Amel'kin, N. A. Lukashevich, A. P. Sadovskii, Nonlinear oscillations in second order systems (in Russian), Belarusian State University, Minsk, 1982. Google Scholar
[2]	N. N. Bautin, On the number of limit cycles which appear with the variaton of coefficients from an equilibrium position of focus or center type, Amer. Math. Soc. Trans., 1954, 100, 397-413. Google Scholar
[3]	A. Gasull, A. Guillamon, V. Manosa, An explicit expression of the first Liapunov and period constants with applications, J. Math. Anal. Appl., 1997, 211(1), 190-212. Google Scholar
[4]	J. Giné, J. Llibre, C. Valls, Centers for the Kukles homogeneous systems with even degree, J. Appl. Anal. Comp., 2017, 7(4), 1534-1548. Google Scholar
[5]	M. Han, L. Sheng, X. Zhang, Bifurcation theory for finitely smooth planar autonomous differential systems, J. Diff. Eqs., 2018, 264(5), 3596-3618. Google Scholar
[6]	M. Han, X. Hou, L. Sheng, C. Wang, Theory of rotated equations and applications to a population model, Disc. Cont. Dyn. Syst.-A, 2018, 38(4), 2171-2185. Google Scholar
[7]	M. Han, T. Petek, V. G. Romanovski, Reversibility in polynomial systems of ODEs, Appl. Math. Comp., 2018, 338, 55-71. Google Scholar
[8]	W. Huang, V.G. Romanovski, W. Zhang, Weak centers and local bifurcations of critical periods at infinity for a class of rational systems, Acta Math. Appl. Sin., 2013, 29(2), 377-390. Google Scholar
[9]	N. V. Kuznetsov, G. A. Leonov, Computation of Lyapunov quantities, Proceedings of the 6th EUROMECH Nonlineaar Dynamics Conference, 2008, 1-10. Google Scholar
[10]	F. Li, Y. Liu, Y. Liu, P. Yu, Bi-center problem and bifurcation of limit cycles from nilpotent singular points in Z2-equivariant cubic vector fields, J. Diff. Eqs., 2018, 265(10):4965-4992. Google Scholar
[11]	T. Liu, Y. Liu, F. Li, A kind of bifurcation of limit cycle from a nilpotent critical point, J. Appl. Anal. Comp., 2018, 8(1), 10-18. Google Scholar
[12]	Y. Liu, J. Li, Theory of values of singular point in complex autonomous differential systems, Sci. China Ser. A,1989, 33(4), 10-23. Google Scholar
[13]	Y. Liu, Theory of center-focus for a class of higher-degree critical points and infinite points, Sci. China Ser. A, 2001, 44(3), 37-48. Google Scholar
[14]	Y. Liu, H. Chen, Stability and bifurcations of limit cycles of the equator in a class of cubic polynomial systems, Comput. Math. Appl., 2002, 44(8-9), 997-1005. Google Scholar
[15]	Y. Liu, W. Huang, Seven large-amplitude limit cycles in a cubic polynomial system, Int. J. Bifurcation Chaos, 2006,16(2), 473-485. Google Scholar
[16]	Y. Liu, J. Li, Some classical problems about planar vector fields (in Chinese), Science Press, Beijing, 2010. Google Scholar
[17]	V. G. Romanovski, D.S. Shafer, On the Center Problem for p:-q Resonant Polynomial Vector Fields, Bull. Belg. Math. Soc. Simon Stevin, 2018, 15(5), 871-887. Google Scholar
[18]	V. G. Romanovski, Y.-H. Xia, X. Zhang, Varieties of local integrability of analytic differential systems and their applications, J. Diff. Eqs., 2014, 257(9), 3079-3101. Google Scholar
[19]	V. G. Romanovski, M. Han, W. Huang, Bifurcation of critical periods of a quintic system, Elec. J. Diff. Eqs., 2018. Google Scholar
[20]	K. S. Sibirskii. Algebraic Invariants of Differential Equations and Matrices, (Russian), Kishinev:Shtiintsa, 1976. Google Scholar
[21]	P. Yu, G. Chen, Computation of focus values with applications, Nonlinear Dynam., 2008, 51(3), 409-427. Google Scholar
[22]	P. Yu, M. Han, J. Li, An Improvement on the Number of Limit Cycles Bifurcating from a Nondegenerate Center of Homogeneous Polynomial Systems, Int. J. Bifurcation Chaos, 2018, 28(6), 1850078. Google Scholar
[23]	Q. Zhang, Y. Liu, A cubic polynomial system with seven limit cycles at infinity, Appl. Math. Comput., 2006, 177(1), 319-329. Google Scholar
[24]	Q. Zhang, Y. Liu, A quintic polynomial differential system with eleven limit cycles at the infinity, Comput. Math. Appl., 2007, 53(10), 1518-1526. Google Scholar
[25]	Q. Zhang, W. Gui, Y. Liu, Bifurcation of limit cycles at the equator for a class of polynomial differential system, Nonlinear Anal. 2009, 10(2), 1042-1047. Google Scholar