一个三次等时中心在非光滑扰动下的极限环分支
|
摘要
本文研究了一个三次等时中心在非光滑扰动下的极限环分支问题.利用非光滑系统的一阶平均方法,获得了在任意小的分段三次多项式扰动下,从未扰动系统的周期环域中至多分支出7个极限环,而且此上界可以达到,推广了光滑扰动下的结果.
This paper is devoted to study the bifurcation of limit cycles from a cubic isochronous center under any small non-smooth perturbations. By using the averaging theory for discontinuous differential systems, it proves that under any small piecewise cubic polynomial perturbations, at most seven limit cycles bifurcate from the period annulus sounding the center of the unperturbed system, and this upper bound can be reached, which extends the resultant under smooth perturbations.
This paper is devoted to study the bifurcation of limit cycles from a cubic isochronous center under any small non-smooth perturbations. By using the averaging theory for discontinuous differential systems, it proves that under any small piecewise cubic polynomial perturbations, at most seven limit cycles bifurcate from the period annulus sounding the center of the unperturbed system, and this upper bound can be reached, which extends the resultant under smooth perturbations.
引文
[1] Li S M, Zhao Y L. Limit cycles of perturbed cubic isochronous center via the second order averaging method[J]. Intern. J. Bifur. Chaos, 2014, 24(3):1450035.
[2] Coll B, Gasull A, Prohens R. Bifurcation of limit cycles from two family of centers[J]. Dyn. Contin.Discrete Impuls. Syst., 2005, 12(2):275-287.
[3] Han M A, Li J B. Lower bounds for the Hilbert number of polynomial systems[J]. J. Diff. Equ.,2012, 252(4):3278-3304.
[4] Han M A, Yu P. Normal forms, Melnikov functions and bifurcations of limit cycles[M]. New York:Springer, 2012.
[5] Iliev I D. Perturbations of quadratic centers[J]. Bull. Sci. Math., 1998, 122(2):107-161.
[6] Chen F D, Li C Z, Llibre J, Zhang Z H. A unified proof on the weak Hilbert 16th problem for n=2[J]. J. Diff. Equ., 2006, 221(2):309-342.
[7] Wang J H, Xiao D M. On the number of limit cycles in small perturbations of a class of hyper-elliptic Hamiltonian systems with one nilpotent saddle[J]. J. Diff. Equ., 2011, 250(4):2227-2243.
[8] Zoladek H. Quadratic systems with center and their perturbations[J]. J. Diff. Equ., 1994, 109(2):223-273.
[9] Zhang P G. On the distribution and number of limit cycles for quadratic systems with two foci[J].Qual. The. Dyn. Syst., 2002, 3(2):437-463.
[10] Zhang Z F, Li C Z. On the number of limit cycles of quadratic Hamiltonian systems under quadratic perturbations[J]. Adv. Math., 1997, 26(5):445-460.
[11] Liu X, Han M A. Bifurcation of limit cycles by perturbing piecewise Hamiltonian systems[J]. Intern.J. Bifur. Chaos, 2010, 20(5):1-12.
[12] Llibre J, Novaes D D, Teixeira M A. On the birth of limit cycles for non-smooth dynamical systems[J]. Bull. Sci. Math., 2015, 139(3):229-244.
[13] Liang F, Han M A. Limit cycles near generalized homoclinic and double homoclinic loops in piecewise smooth systems[J]. Chaos, Soli. Prac., 2012, 45(4):454-464.
[14] Li S M, Liu C J. A linear estimate of the number of limit cycles for some planar piecewise smooth quadratic differential systems[J]. J. Math. Anal. Appl., 2015, 428(2):1354-1367.
[15] Cen X L, Li S M,Zhao Y L. On the number of limit cycles for a class of discontinuous quadratic differential systems[J]. Math. Anal. Appl., 2017, 449(1):314-342.
[16] Buica A, Libre J. Averaging methods for finding periodic orbits via brouwer degree[J]. Bull. Sci.Math., 2004, 128(1):7-22.
[17] Sanders J A, Verbulst F. Averaging methods in nonlinear dynamic systems[M]. New York:SpringVerlag, 1985.
[2] Coll B, Gasull A, Prohens R. Bifurcation of limit cycles from two family of centers[J]. Dyn. Contin.Discrete Impuls. Syst., 2005, 12(2):275-287.
[3] Han M A, Li J B. Lower bounds for the Hilbert number of polynomial systems[J]. J. Diff. Equ.,2012, 252(4):3278-3304.
[4] Han M A, Yu P. Normal forms, Melnikov functions and bifurcations of limit cycles[M]. New York:Springer, 2012.
[5] Iliev I D. Perturbations of quadratic centers[J]. Bull. Sci. Math., 1998, 122(2):107-161.
[6] Chen F D, Li C Z, Llibre J, Zhang Z H. A unified proof on the weak Hilbert 16th problem for n=2[J]. J. Diff. Equ., 2006, 221(2):309-342.
[7] Wang J H, Xiao D M. On the number of limit cycles in small perturbations of a class of hyper-elliptic Hamiltonian systems with one nilpotent saddle[J]. J. Diff. Equ., 2011, 250(4):2227-2243.
[8] Zoladek H. Quadratic systems with center and their perturbations[J]. J. Diff. Equ., 1994, 109(2):223-273.
[9] Zhang P G. On the distribution and number of limit cycles for quadratic systems with two foci[J].Qual. The. Dyn. Syst., 2002, 3(2):437-463.
[10] Zhang Z F, Li C Z. On the number of limit cycles of quadratic Hamiltonian systems under quadratic perturbations[J]. Adv. Math., 1997, 26(5):445-460.
[11] Liu X, Han M A. Bifurcation of limit cycles by perturbing piecewise Hamiltonian systems[J]. Intern.J. Bifur. Chaos, 2010, 20(5):1-12.
[12] Llibre J, Novaes D D, Teixeira M A. On the birth of limit cycles for non-smooth dynamical systems[J]. Bull. Sci. Math., 2015, 139(3):229-244.
[13] Liang F, Han M A. Limit cycles near generalized homoclinic and double homoclinic loops in piecewise smooth systems[J]. Chaos, Soli. Prac., 2012, 45(4):454-464.
[14] Li S M, Liu C J. A linear estimate of the number of limit cycles for some planar piecewise smooth quadratic differential systems[J]. J. Math. Anal. Appl., 2015, 428(2):1354-1367.
[15] Cen X L, Li S M,Zhao Y L. On the number of limit cycles for a class of discontinuous quadratic differential systems[J]. Math. Anal. Appl., 2017, 449(1):314-342.
[16] Buica A, Libre J. Averaging methods for finding periodic orbits via brouwer degree[J]. Bull. Sci.Math., 2004, 128(1):7-22.
[17] Sanders J A, Verbulst F. Averaging methods in nonlinear dynamic systems[M]. New York:SpringVerlag, 1985.