LIMIT CYCLES FOR TWO CLASSES OF PLANAR POLYNOMIAL DIFFERENTIAL SYSTEMS WITH UNIFORM ISOCHRONOUS CENTERS

Bo Huang; Wei Niu

doi:10.11948/2156-907X.20180189

2019 Volume 9 Issue 3

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2019 > 9(3): 943-961

Next Article Previous Article

Bo Huang, Wei Niu. LIMIT CYCLES FOR TWO CLASSES OF PLANAR POLYNOMIAL DIFFERENTIAL SYSTEMS WITH UNIFORM ISOCHRONOUS CENTERS[J]. Journal of Applied Analysis & Computation, 2019, 9(3): 943-961. doi: 10.11948/2156-907X.20180189

Citation:

Bo Huang, Wei Niu. LIMIT CYCLES FOR TWO CLASSES OF PLANAR POLYNOMIAL DIFFERENTIAL SYSTEMS WITH UNIFORM ISOCHRONOUS CENTERS[J]. Journal of Applied Analysis & Computation, 2019, 9(3): 943-961. doi: 10.11948/2156-907X.20180189

LIMIT CYCLES FOR TWO CLASSES OF PLANAR POLYNOMIAL DIFFERENTIAL SYSTEMS WITH UNIFORM ISOCHRONOUS CENTERS

Bo Huang^1,2,
Wei Niu^3,4, ,

1.
School of Mathematics and Systems Science, Beihang University, LIMB of the Ministry of Education, Beijing, 100191, China
2.
Guangxi Key Laboratory of Hybrid Computation and IC Design Analysis, Nanning, 530006, China
3.
Sino-French Engineer School, Beihang University, Beijing, 100191, China
4.
Beijing Advanced Innovation Center for Big Data and Brain Computing, Beihang University, Beijing, 100191, China

Corresponding author: Email address: Wei.Niu@buaa.edu.cn(W. Niu)
Fund Project: Bo Huang was supported by open fund of Guangxi Key laboratory of hybrid computation and IC design analysis (HCIC 201602) and Wei Niu was supported by National Natural Science Foundation of China (11601023)

Abstract

Abstract

In this article, we study the maximum number of limit cycles for two classes of planar polynomial differential systems with uniform isochronous centers. Using the first-order averaging method, we analyze how many limit cycles can bifurcate from the period solutions surrounding the centers of the considered systems when they are perturbed inside the class of homogeneous polynomial differential systems of the same degree. We show that the maximum number of limit cycles, m and m+1, that can bifurcate from the period solutions surrounding the centers for the two classes of differential systems of degree 2m and degree 2m+1, respectively. Both of the bounds can be reached for all m.
- Averaging method /
- homogeneous polynomial /
- limit cycle /
- period solutions /
- uniform isochronous center
MSC: 34C05, 34C07

FullText(HTML)

References

[1]	A. Buică, J. Françoise and J. Llibre, Periodic solutions of nonlinear periodic differential systems with a small parameter, Comm. Pure Appl. Anal., 2006, 6, 103-111. doi: 10.3934/cpaa CrossRef Google Scholar
[2]	J. Chavarriga and M. Sabatini, A survey of isochronous centers, Qual. Theory Dyn. Syst., 1999, 1, 1-70. doi: 10.1007/BF02969404 CrossRef Google Scholar
[3]	R. Conti, Uniformly isochronous centers of polynomial systems in ℝ², Lect. Notes Pure Appl. Math., 1994, 152, 21-31. Google Scholar
[4]	R. Conti, Centers of planar polynomial systems. A review, Le Matematiche, 1998, 53, 207-240. Google Scholar
[5]	F. Dias and L. Mello, The center-focus problem and small amplitude limit cycles in rigid systems, Disc. Contin. Dyn. Sys., 2012, 32, 1627-1637. doi: 10.3934/dcdsa CrossRef Google Scholar
[6]	A. Gasull, R. Prohens and J. Torregrosa, Limit cycles for rigid cubic systems, J. Math. Anal. Appl., 2005, 303, 391-404. Google Scholar
[7]	S. Gautier, L. Gavrilov and I. Iliev, Perturbations of quadratic center of genus one, Disc. Contin. Dyn. Sys., 2009, 25, 511-535. doi: 10.3934/dcdsa CrossRef Google Scholar
[8]	J. Giné, M. Grau and J. Llibre, Limit cycles bifurcating from plannar polynomial quasi-homogeneous centers, J. Differ. Equ., 2015, 259, 7135-7160. doi: 10.1016/j.jde.2015.08.014 CrossRef Google Scholar
[9]	M. Han and J. Li, Lower bounds for the Hilbert number of polynomial systems, J. Differ. Equ., 2012, 252, 3278-3304. doi: 10.1016/j.jde.2011.11.024 CrossRef Google Scholar
[10]	M. Han, V. Romanovski and X. Zhang, Equivalence of the Melnikov function method and the averaging method, Qual. Theory Dyn. Syst., 2016, 15, 371-479. Google Scholar
[11]	D. Hilbert, Mathematical problems, (M. Newton, Transl.) Bull. Amer. Math., 1902, 8, 437-479. Google Scholar
[12]	B. Huang, Bifurcation of limit cycles from the center of a quintic system via the averaging method, Int. J. Bifur. Chaos, 2017, 27, 1750072-1-16. Google Scholar
[13]	Y. Ilyashenko, Centennial history of Hilbert's 16th problem, Bull. (New Series) Am. Math. Soc., 2002, 39, 301-354. doi: 10.1090/S0273-0979-02-00946-1 CrossRef Google Scholar
[14]	J. Itikawa and J. Llibre, Phase portraits of uniform isochronous quartic centers, J. Comput. Appl. Math., 2015, 287, 98-114. doi: 10.1016/j.cam.2015.02.046 CrossRef Google Scholar
[15]	C. Li, W. Li, J. Llibre and Z. Zhang, Linear estimate of the number of zeros of Abelian integrals for some cubic isochronous centers, J. Differ. Equ., 2002, 180, 307-333. doi: 10.1006/jdeq.2001.4064 CrossRef Google Scholar
[16]	S. Li and Y. Zhao, Limit cycles of perturbed cubic isochronous center via the second order averaging method, Int. J. Bifur. Chaos, 2014, 24, 1450035-1-8. Google Scholar
[17]	H. Liang, J. Llibre and J. Torregrosa, Limit cycles coming from some uniform isochronous centers, Adv. Nonlinear Stud., 2016, 16, 197-220. Google Scholar
[18]	J. Llibre and J. Itikawa, Limit cycles for continuous and discontinuous perturbations of uniform isochronous cubic centers, J. Comput. Appl. Math., 2015, 277, 171-191. doi: 10.1016/j.cam.2014.09.007 CrossRef Google Scholar
[19]	J. Llibre, R. Ramírez and N. Sadovskaia, On the 16th Hilbert problem for algebraic limit cycles, J. Differ. Equ., 2010, 248, 1401-1409. doi: 10.1016/j.jde.2009.11.023 CrossRef Google Scholar
[20]	J. Llibre, R. Ramírez and N. Sadovskaia, On the 16th Hilbert problem for limit cycles on nonsingular algebraic curves, J. Differ. Equ., 2011, 250, 983-999. doi: 10.1016/j.jde.2010.06.009 CrossRef Google Scholar
[21]	J. Llibre and G. Świrszcz, On the limit cycles of polynomial vector fields, Discr. Contin. Dyn. Syst., 2011, 18, 203-214. Google Scholar
[22]	I. Malkin, Some problems of the theory of nonlinear oscillations (in Russian), Gosudarstv. Izdat. Tehn-Teor. Lit., Moscow, 1956. Google Scholar
[23]	L. Peng and Z. Feng, Bifurcation of limit cycles from quartic isochronous systems, Electron. J. Differ. Equations, 2014, 95, 1-14. Google Scholar
[24]	L. Peng and Z. Feng, Bifurcation of limit cycles from a quintic center via the second order averaging method, Int. J. Bifur. Chaos, 2015, 25, 1550047-1-18. Google Scholar
[25]	L. Peng and Z. Feng, Limit cycles from a cubic reversible system via the thirdorder averaging method, Electron. J. Differ. Equations, 2015, 111, 1-27. Google Scholar
[26]	L. Peng and B. Huang, Second-order bifurcation of limit cycles from a quadratic reversible center, Electron. J. Differ. Equations, 2017, 89, 1-17. Google Scholar
[27]	M. Roseau, Vibrations non linéaires et théorie de la stabilité, Springer, New York, 1985. Google Scholar
[28]	J. Sanders, F. Verhulst and J. Murdock, Averaging method in nonlinear dynamical systems, Springer-Verlag, New York, 2007. Google Scholar
[29]	F. Verhulst, Nonlinear differential equations and dynamical Systems, SpringerVerlag, Berlin, 1996. Google Scholar
[30]	D. Wang, Mechanical manipulation for a class of differential systems, J. Symb. Comput., 1991, 12, 233-254. doi: 10.1016/S0747-7171(08)80127-7 CrossRef Google Scholar
[31]	Y. Zhao, On the number of limit cycles in quadratic perturbations of quadratic codimension-four centers, Nonlin. Anal., 2011, 24, 2505-2522. Google Scholar