几类平面微分系统的广义中心、等时中心与极限环分枝
|
摘要
本篇博士论文以计算机代数系统为工具对几类平面微分系统的极限环分枝、中心与广义中心问题、等时中心问题进行了一些研究,同时将已有理论应用到生态系统中去研究一些生态模型的定性性质.全文共分七章.
第一章对平面多项式微分系统的极限环分枝、中心与等时中心以及定性与分枝理论在生态系统中的应用等问题的历史背景和研究现状进行了综述,并归纳了本文的特色工作.
第二章给出了一种寻找Zn等变系统的简便方法,并对Zn等变向量场中的微分系统的一些性质做了一些归纳总结.同时,以一类Z8-等变对称的七次系统为研究示例,在个人计算机上推导出八个拓扑结构相同的焦点中其中一个的前5个奇点量,进而得出其前5阶焦点量,并得出由八个拓扑结构相同的焦点共可在一定条件下分支出40个极限环的好的实例,同时找出了它的分枝条件及极限环稳定性的判断条件.
第三章主要是关于三次系统等变对称结构的分析和几类具有等变对称性的三次系统的极限环分支的研究,一种寻找等变对称系统的方法被给出.并对未曾研究过的具有不同等变对称结构的三次系统(即z。等变对称的三次系统、Z_3,等变对称的三次系统、Z_4等变对称的三次系统)的定性与分支行为进行了一一的研究.本章的工作是现有研究结论的有益的补充.(这一结论已经提交给《湘潭大学学报》)
第四章主要研究了一类广义对称的平面九次微分自治系统的中心焦点问题与极限环分枝行为.通过作Bendixson倒径变换与时间变换,以及广义焦点量的仔细计算,我们得出了该系统的无穷远点与三个初等奇点能够同时成为广义中心的条件,并进行了严格的证明.同时我们得出在一定条件下,这四个广义焦点能够分枝出20个极限环的结果,其中包括5个大振幅极限环和15个小振幅极限环.这种结论到目前为止是新的.
第五章研究一类广义等变对称的九次系统,通过周期常数(或称为等时常数)的计算与分析,找出了无穷远点与三个初等奇点同时成为等时中心(或称为可线性化中心)的必要条件,进一步证明了这些必要条件也是充分条件.无穷远点与几个初等奇点同时成为等时中心的结论是第一次给出,在以往的文献中未有发现,我们的结论有一定意义.
第六章研究一类拟对称七次系统的广义中心、等时中心与极限环分枝问题,得出了该系统的无穷远点与初等奇点同时成为广义中心与等时中心的条件并进行了严格的证明,进一步给出该系统可以由无穷远点分枝出5个赤道环的同时还可以由初等奇点分枝出5个小振幅极限环.这种结论是罕见的,本章的结论是新的.
第七章利用微分方程定性与分支理论研究了一类食饵具有常投放的稀疏效应捕食系统,得到了存在唯一极限环和不存在极限环及系统全局渐近稳定的充要条件.此外,还研究了一类三次Kolmogorov捕食系统,算出了其正平衡点(1,1)的前五阶焦点量,同时找到了5个极限环(或少于5个极限环)的分枝条件与这些极限环的相对位置.特别地,文中还显示该系统可以有3个稳定的极限环,就三次Kolmogorov系统的稳定环的数目而言,这一结论是最好的.
This Ph.D. Thesis is devoted to study the bifurcations of limit cycle and the problem of center and ischoronous center for several classes of planar differential systems with the help of computer's algebra system, at the same time, the qualitative theory and bifurcation theory are applied to investigate the qualitative nature of two class of ecological systems or ecological models. It is composed of seven chapters.
In Chapter 1, we introduce the historical background and the present progress of problems which are concerned with centers, isochronous centers, and bifurcations of limit cycles for planar polynomial differential systems. And the qualitative nature of ecological model is considered. The main works of this paper are concluded as well.
In Chapter 2, a kind of simple method to find Z_n -equivariant systems is given, the quality of differential systems in Z_n-equivariant vector field is summarized. At the same time,as an example ,a class of seventh degree Z_8-equivariant systems is investigated, and the first five focal values are given. Moreover, it is shown that 40 limit cycles can bifurcate from this class of system. The conditions of bifurcation and stability are obtained.
In Chapter 3, our work is concerned with the analysis of cubic system's equivariant symmetric structure and the investigation of limit cycles bifurcation for several classes of equivariant cubic systems. A kind of method to find equivariant symmetric systems is given, which is significative for researching the bifurcation behave -or of Z_n -equivariant systems.In terms of equivariant symmetric cubic system, we investigate other three cases (i.e., Z_∞-equivariant case , Z_3 -equivariant case and Z_4 -equivariant case) except studied Z_2 -equivariant case and obtain center condition and bifurcation results of each case .These results are supplements for some studied references about cubic system.
In Chapter 4, our researches are concerned with the center-focus problem and the limit cycle bifurcation problem for a class of planar general equivariant system of nine degrees. By making Bendixson transformation and time transformation of system and calculating general focal values carefully, we obtain the conditions that the infinity and three elementary focuses become four general centers at the same time. More -over, 20 limit cycles including 15 small limit cycles from three element -ary foci and 5 large limit cycles from the infinity can occur under a certain condition. What is worth mentioning is that similar conclusions are hardly seen in published papers up till now and our work is completely new.
In Chapter 5, a class of general equivariant system of nine degrees is investigated. Through the calculation and the analysis of periodic constants (or called ischoronous constants ), we obtain the conditions that the infinity and three elementary singular points become isochronous centers at the same time, and these conditions are also proved to be sufficient conditions of isochronicity. It is hardly seen in published articles for the result that the the infinity and elementary singular points become isochronous centers at the same time, our work is significant.
In Chapter 6, general center and isochronous centers and limit cycles bifurcation for a class of quasi-symmetric seventh degree system are considered. we obtain the conditions that the infinity and the elementary singular point become isochronous centers at the same time, and prove them strictly. Moreover, we give the conclusions that the infinity can bifurcate 5 large limit cycles and the elementary focus can bifurcate 5 small limit cycles under the same condition. This kind of work is new.
In Chapter 7, firstly, a class of predator-prey system with scanty effect of which the prey species possesses a constant invest is studied by making use of qualitative and bifurcation theory, and we obtain sufficient and necessary condition for the existence and uniqueness of limit cycle surrounding the positive equilibrium point and for the global stability of the system. Secondly, A class of cubic Kolmogorov system are considered and we obtain 5 focal values of the positive equilibrium point (1,1),at the same time the condition for the existence of 5 limit cycles or less surrounding the positive equilibrium point (1,1) and the locality of 5 limit cycles is given.Especially,3 stable limit cycles are shown in this paper ,which is the best result in terms of the number of stable limit cycles for cubic Kolmogorov system.
第一章对平面多项式微分系统的极限环分枝、中心与等时中心以及定性与分枝理论在生态系统中的应用等问题的历史背景和研究现状进行了综述,并归纳了本文的特色工作.
第二章给出了一种寻找Zn等变系统的简便方法,并对Zn等变向量场中的微分系统的一些性质做了一些归纳总结.同时,以一类Z8-等变对称的七次系统为研究示例,在个人计算机上推导出八个拓扑结构相同的焦点中其中一个的前5个奇点量,进而得出其前5阶焦点量,并得出由八个拓扑结构相同的焦点共可在一定条件下分支出40个极限环的好的实例,同时找出了它的分枝条件及极限环稳定性的判断条件.
第三章主要是关于三次系统等变对称结构的分析和几类具有等变对称性的三次系统的极限环分支的研究,一种寻找等变对称系统的方法被给出.并对未曾研究过的具有不同等变对称结构的三次系统(即z。等变对称的三次系统、Z_3,等变对称的三次系统、Z_4等变对称的三次系统)的定性与分支行为进行了一一的研究.本章的工作是现有研究结论的有益的补充.(这一结论已经提交给《湘潭大学学报》)
第四章主要研究了一类广义对称的平面九次微分自治系统的中心焦点问题与极限环分枝行为.通过作Bendixson倒径变换与时间变换,以及广义焦点量的仔细计算,我们得出了该系统的无穷远点与三个初等奇点能够同时成为广义中心的条件,并进行了严格的证明.同时我们得出在一定条件下,这四个广义焦点能够分枝出20个极限环的结果,其中包括5个大振幅极限环和15个小振幅极限环.这种结论到目前为止是新的.
第五章研究一类广义等变对称的九次系统,通过周期常数(或称为等时常数)的计算与分析,找出了无穷远点与三个初等奇点同时成为等时中心(或称为可线性化中心)的必要条件,进一步证明了这些必要条件也是充分条件.无穷远点与几个初等奇点同时成为等时中心的结论是第一次给出,在以往的文献中未有发现,我们的结论有一定意义.
第六章研究一类拟对称七次系统的广义中心、等时中心与极限环分枝问题,得出了该系统的无穷远点与初等奇点同时成为广义中心与等时中心的条件并进行了严格的证明,进一步给出该系统可以由无穷远点分枝出5个赤道环的同时还可以由初等奇点分枝出5个小振幅极限环.这种结论是罕见的,本章的结论是新的.
第七章利用微分方程定性与分支理论研究了一类食饵具有常投放的稀疏效应捕食系统,得到了存在唯一极限环和不存在极限环及系统全局渐近稳定的充要条件.此外,还研究了一类三次Kolmogorov捕食系统,算出了其正平衡点(1,1)的前五阶焦点量,同时找到了5个极限环(或少于5个极限环)的分枝条件与这些极限环的相对位置.特别地,文中还显示该系统可以有3个稳定的极限环,就三次Kolmogorov系统的稳定环的数目而言,这一结论是最好的.
This Ph.D. Thesis is devoted to study the bifurcations of limit cycle and the problem of center and ischoronous center for several classes of planar differential systems with the help of computer's algebra system, at the same time, the qualitative theory and bifurcation theory are applied to investigate the qualitative nature of two class of ecological systems or ecological models. It is composed of seven chapters.
In Chapter 1, we introduce the historical background and the present progress of problems which are concerned with centers, isochronous centers, and bifurcations of limit cycles for planar polynomial differential systems. And the qualitative nature of ecological model is considered. The main works of this paper are concluded as well.
In Chapter 2, a kind of simple method to find Z_n -equivariant systems is given, the quality of differential systems in Z_n-equivariant vector field is summarized. At the same time,as an example ,a class of seventh degree Z_8-equivariant systems is investigated, and the first five focal values are given. Moreover, it is shown that 40 limit cycles can bifurcate from this class of system. The conditions of bifurcation and stability are obtained.
In Chapter 3, our work is concerned with the analysis of cubic system's equivariant symmetric structure and the investigation of limit cycles bifurcation for several classes of equivariant cubic systems. A kind of method to find equivariant symmetric systems is given, which is significative for researching the bifurcation behave -or of Z_n -equivariant systems.In terms of equivariant symmetric cubic system, we investigate other three cases (i.e., Z_∞-equivariant case , Z_3 -equivariant case and Z_4 -equivariant case) except studied Z_2 -equivariant case and obtain center condition and bifurcation results of each case .These results are supplements for some studied references about cubic system.
In Chapter 4, our researches are concerned with the center-focus problem and the limit cycle bifurcation problem for a class of planar general equivariant system of nine degrees. By making Bendixson transformation and time transformation of system and calculating general focal values carefully, we obtain the conditions that the infinity and three elementary focuses become four general centers at the same time. More -over, 20 limit cycles including 15 small limit cycles from three element -ary foci and 5 large limit cycles from the infinity can occur under a certain condition. What is worth mentioning is that similar conclusions are hardly seen in published papers up till now and our work is completely new.
In Chapter 5, a class of general equivariant system of nine degrees is investigated. Through the calculation and the analysis of periodic constants (or called ischoronous constants ), we obtain the conditions that the infinity and three elementary singular points become isochronous centers at the same time, and these conditions are also proved to be sufficient conditions of isochronicity. It is hardly seen in published articles for the result that the the infinity and elementary singular points become isochronous centers at the same time, our work is significant.
In Chapter 6, general center and isochronous centers and limit cycles bifurcation for a class of quasi-symmetric seventh degree system are considered. we obtain the conditions that the infinity and the elementary singular point become isochronous centers at the same time, and prove them strictly. Moreover, we give the conclusions that the infinity can bifurcate 5 large limit cycles and the elementary focus can bifurcate 5 small limit cycles under the same condition. This kind of work is new.
In Chapter 7, firstly, a class of predator-prey system with scanty effect of which the prey species possesses a constant invest is studied by making use of qualitative and bifurcation theory, and we obtain sufficient and necessary condition for the existence and uniqueness of limit cycle surrounding the positive equilibrium point and for the global stability of the system. Secondly, A class of cubic Kolmogorov system are considered and we obtain 5 focal values of the positive equilibrium point (1,1),at the same time the condition for the existence of 5 limit cycles or less surrounding the positive equilibrium point (1,1) and the locality of 5 limit cycles is given.Especially,3 stable limit cycles are shown in this paper ,which is the best result in terms of the number of stable limit cycles for cubic Kolmogorov system.
引文
[1] Hilbert D, 1900 Mathematische problem(lecture) :The Second Intern. Congress of Mathematicians Paris 1900 Nachr. Ger. Wiss. Gottingen. Math. Phy.Kl.pp 253-297.
[2] Smale S, Dynamics retrospective: great problems attempts that failed,Physica D, 1991, 51: 261-273.
[3] Smale S, Mathematical problem for the next century, Mathematical Intelligencer, 1998, 20(2): 7-15.
[4] Yu. Ilyashemko, Centennial history of Hilbert's 16th problem, Bull. Amer. Math Soc. (N.S.) 2002, 39 (3) : 301-354.
[5] J. Li, Hilbert's 16th problem and bifurcation of Planar polynomial vector fields, Internat. J. Bifur. Chaos. 2003, 13: 47-106.
[6] J. Gine, On some open problems in planar differential systems and Hilbert's 16th problem, Chaos, Solitons Fractals . 2007, 31 (5): 1118 -1134.
[7] Dulac H, Surles cycles limites, Bull. Soc. Math. France. 1923, 51: 45-188.
[8] Petrovskii I. G.., Landis E. M., On the number of limit cycles of the equation dy/dx=P(x, y)/Q(x, y) where P and Q are polynomials of the second degree (Russian), Mat. Sb. N. S. 1955, 37(79): 209-250.
[9] Petrovskii I. G.., Landis E. M., On the number of limit cycles of the Equation dy/dx=P(x, y)/Q(x, y) where P and Q are polynomials of the second degree (Russian), Mat. Sb. N. S. 1957, 85: 209-250.
[10] Chen L., Wang M., The relative position and number of limit cycles of quadratic differential system, Acta Math. Sinica, 1979,22: 751-758.
[11] Shi S., A concrete example of the existence of four limit cycles for plane quadratic system, Sci. Sinica(A), 1980,23: 153-158.
[12]Ilyashenko Yu. S., Singular points and limit cycles of different -ial equations in the real and the complex planes, NIVTS AN SSSR, Pushchino, 1982, 38.
[13] Ilyashenko Yu. S., Dulac' s memoir" on limit cycles" and related questions of the local theory of differential equations (Russian), Uspekhi Mat. Nauk, 1985, 40(6):41-78.
[14] Ilyashenko Yu.S.,Finiteness for limit cycles, Transl. Math. Monographs, 1991, 94.
[15] Ecalle J., Introduction aux Functions Analysables et Preuve Constructive de la Conjecture de Dulac, Hermann Pulb, Paris, 1992.
[16] J. Li, Q. Huang. Bifurcation set and limit cycles forming compound eyes in the cubic system. Chin. Ann. of Math. 1987, 8B:391-403.
[17] Li J., Liu Z. Bifurcation set and limit cycles forming compound eyes in a perturbed Hamiltonian system. Publ. Math. 1991, 35:487-506.
[18] Viktor Levandovskyy, Valery G. Romanovski, Douglas S. Shafer. The cyclicity of a cubic system with nonradical Bautin ideal, Journal of ifferential Equations, 2009, 246(3): 1274-1287.
[19] T. Zhang, M. Han, H. X. Meng. Bifurcations of limit cycles for a cubic Hamiltonian system under quartic perturbations. Chaos Solitons and Fractals, 2004, 22:1127-1138.
[20] P. Yu, M. Han. Twelve limit cycles in cubic case of the Hilbert problem.Int.J. Bifurcation & Chaos, 2005,15(7):2191-2205.
[21] P. Yu, M. Han. Small limit cycles bifurcating from fine focus points in cubic order Z2-equivariant vector fields. Chaos, Solitons & Fractals, 2005, 24: 329-348.
[22] Y. Liu, W. Huang. A cubic system with twelve small amplitude Limit cycles.Bull, des Sci. Math., 2005, 129:83-98.
[23] Q. Wang, Y.Liu, C. Du, Small limit cycles bifurcating from fine focuspoints in quartic order Z3-equivariant vector fields, Journal of Mathematical Analysis and Applications, 2008,337(1): 524-536.
[24] Yuhai Wu , Yongxi Gao , Maoan Han, Bifurcations of the limit cycles in a Z3-equivariant quartic planar vector field, Chaos, Solitons & Fractals, 2008, 38(4): 1177-1186.
[25] J. Li, HSY Chan, KW Chung. Bifurcations of limit cycles in a Z6-equivariant planar vector field of degree 5. Sci. China Ser. A, 2002,45: 817-26.
[26] Yuhai Wu, Maoan Han, New configurations of 24 limit cycles in a quintic system, Computers & Mathematics with Applications, 2008, 55(9): 2064-2075.
[27] Wang S., Yu P., Bifurcation of limit cycles in a quintic Hamiltonian system under a sixth-order perturbation, Chaos, Solitons & Fractals, 2005, 26: 1317-1335.
[28] Li J., Zhang M., Bifurcation of limit cycles in a Z8-equivariant vector field, J.Diff. Eqs. Dyn. Syst., 2004, 16:1123-1139.
[29] Wang S., Yu P., Li J., Bifurcation of limit cycles in a Z10-equivariant vector field of degree 9, Int. J. Bifurcation and chaos, 2006, 16(8): 2309-2324.
[30] Wang S., Yu P., Existence of 121 limit cycles in a perturbed planar polynom -ial Hamiltonian vector field of degree 11, Chaos, Solitons & Fractals, 2006, 30(3):606-621.
[31] C. Christopher, N. G. Lloyd, Polynomial systems: A lower bound for the Hilbert numbers, Proc. Royal Soc. London Ser., 1995 450(A): 219-224.
[32] J. Li, HSY Chan, KW Chung, Some lower bounds for H(n) in Hilbert' s 16 th Problem, Qualitative Theory of Dynamical, 2003, 3(44): 345-360.
[33] Itenberg I., Shustin E., Singular points and limit cycles of planar polynomial vector fields Duke Math. J., 2000, 102(1):1-37.
[34] 秦元勋,蔡隧林,史松林,关于平面二次系统的极限环,中国科学,1981,8:929-938.
[35] N. Bautin. On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center type. Amer. Math. Soc. Trans. 1954,100:397-413.
[36] W. W. Farr, C. Li, I. S. Labouriau, W. F. Langford. Degenerate hopf bifurcation formulas and Hilbert' 16th problem. SIAM. J. Math. Anal. 1989,20:13-30.
[37] F. Gobber, K. Willamowski. Liapunov approach to multiple Hopf bifurcation. J. Math. Anal. Appl. 1979, 71(98):330-350.
[38] S. Shi. A Method of constructing cycles without contact around a weak focus. J. Differential Equations, 1980, 41:301-312.
[39] S. Shi. On the structure of Poincare-Liapunov constants for the focus of polynomial vector fields. J. Differential Equations, 1984, 52:52-57.
[40] J. Chavarriga. Integrable systems in the plane with center type linear part. Application(?)s Math(?)matica(?), 1994, 22:285-309.
[41] 杜乃林,曾宪武,计算焦点量的一类递推式,科学通报,1994,39(19):1742-1745.
[42] J. Gine. Conditions for the existence of a center for the Kukles homogeneous systems. Computers and mathematics with applications. 2002, 43:1261-1269.
[43] N. G. Lloyd, J. M. Pearson. Symmetry in planar dynamical systems. J. Symbolic Computation, 2002 33:357-366.
[44] 刘一戎,一类平面三次系统的焦点量公式、中心条件和中心积分,科学通报,1987,2:85-97.
[45] J. P. Francoise. Successive derivatives of first-return map, application to quatratic vector fields. Ergodic theory dynamical systems.1996,16: 87-96.
[46] 秦元勋,刘尊全,微分方程公式的机器推导(Ⅲ),1981,7:388-391.
[47] 李承治,关于平面二次系统的两个问题,中国科学,1982,12:1087-1096.
[48] 刘一戎,李继彬,论复自治系统的奇点量,中国科学,1989,3:245-255.
[49] Liu Yirong, Li Jibin. Theory of values of singular point in complex autonomous differential system. Science in china (Series A), 1990, 33:10-24.
[50] Yirong Liu and Haibo Chen. Formulas of singurlar point quantities and the first 10 saddle quantities for a class of cubic system. Acta Mathematicae applicatae sinica, 2002, 25(2): 295-302. (in Chinese)
[51] N. G. Loyd, J. M. Pearson. Bifurcation of limit cycles and integrability of Planar dynamical systems in complex form. Computers and mathematics with applicat -ions. 2002, 43:1261-1269.
[52] A. Gasull, A. Guillamon, V. Maqosa. An explicit expression of the first Liapunov and period constants with applications. J. Math. Anal. Appl., 1997, 211:190-212.
[53] Sibirskii K. S. On the number of limit cycles in a neighborhood of singular points. Dif. Eq., 1965,1:36-76.(in Russian)
[54] N. G. Lloyd, T. R. Blows, M. C. Kalenge. Some cubic systems with several limit cycles. Nonlinearity, 1988, 1:65-69.
[55] E. M. James, N. G. Lloyd. A cubic system with eight small amplitude limit cycles. LM. A. J. Applied Math., 1991, 47:163-171.
[56] S. Ma, S. Ning. Derive some new conditions on existence of eight limit cycles for a cubic system. Computers and Mathematics with Appli -cations, 1997,33: 59-64.
[57] S. C. Ning, S. L. Ma, H. K. Keng, Z. M. Zheng. A cubic system with eight small amplitude limit cycles. Appl. Math. lect., 1994,4:23-27.
[58] H. Zoladek. Eleven small limit cycles in a cubic vector field. Nonlinearity, 1995, 8:843-860.
[59] 叶彦谦,多项式微分系统定性理论,上海:上海科学技术出版社,1995,85-106.
[60] M. Han, Y. Lin, P. Yu. A study on the existence of limit cycles of a planar system with 3rd-degree polynomials. Int. J. Bifurcation and Chaos, 2004,14: 65-78.
[61] C. Chicone, M. Jacobs. Bifurcation of critical periods for plane vector fields. Amer. Math. Soc. Trans., 1989, 312:433-486.
[62] J. Guckenheimer, P. Holmes. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Appl. Math. Sci. 42, Second Printing, Springer-Verlag, 1986.
[63] V. I. Arnold, Y. S. Ilyashenko. Dynamical Systems Ⅰ, Ordinary Differential Equations. Encyclopaedia of Mathematical Sciences, Vol. 1, Springer-Verlag, 1986.
[64] H. Giacomini, J. Libre, M. Viano. On the nonexistence, existence And uniqueness of limit cycles. Nonlinearity, 1996,9:501-516.
[65] H. Giacomini, J. Libre, M. Viano. On the shape of limit cycles that bifurcate from Hamiltonian centers, Nonlinear Analysis, Ser. A: Theory Methods, 2000,41:523-537.
[66] C. Li, W. Li, J. Llibre, Z. Zhang. On the limit cycles of Polynomial differential systems with homogeneous nonlinearities. Proc. of the Edinburgh Math. Soc., 2000, 43: 529-543.
[67] J. Llibre, J. S. Perez Delrio, J. A. Roddriguez. Averaging Analysis of A Perturbed Quadratic Center. Nonlinear Anal., 2001, 4:45-51.
[68] F. Dumortier, C. Li, Z.Zhang. Unfolding of a quadratic integrable system with two centers and two unbounded heteroclinic loops. J. Differential Equations, 1997, 139:146-193.
[69] I. D. Iliev. Perturbations of quadratic centers. Bull. Sci. Math. ,1998, 122:107-161.
[70] C. Li, W. Li, J. Llibre, Z.Zhang. Linear estimates for the number of zores of Abelian integrals for quadratic isochronous centers,Nonlinearity, 2000,13:1775-1800.
[71] H. Giacomini, J. Llibre, M. Viano. The shape of limit cycles that Bifurcate from non-Hamiltonian centers. Nonlinear Analysis, Ser. A:Theory Methods, 2001, 43:837-859.
[72] C. Li, W. Li, J. Llibre, Z. Zhang. Linear estimates for the number of zores of Abelian integrals for some cubic isochronous centers. J. Differential Equations, 2002, 180:307-333.
[73] J. Yu, C. Li. Bifuration of a class of planar non-Hamiltonian Integrable systems with one center and one homollinic loop. J. Math. Anal. Appl., 2002, 269(1): 227-243.
[74] A. Gasull, J. Libre. Limit cycles for a class of Abel equations. SIAM J. of Math. Anal., 1990, 21:1235-1244.
[75] F. Dumortier, C, Li. perturbations from an Elliptic Hamiltonian of degree four:Ⅰ. saddle loop and two saddle cycle. J. Differential Equations, 2001,176(1): 114-157.
[76] F. Dumortier, C.Li. Perturbations from an Elliptic Hamiltonian of degree four: Ⅳ. figure eight-loop. J. Differential Equations, 2003,188(2) :473-511.
[77] R. Roussarie. Bifurcation of planar vector fields and Hilbert' s sixteenth problem progr. Math., Vol. 164, Birkh auser, Basel, 1998.
[78] Blows T. R., Rousseau C. Bifurcation at infinity in polynomial Vector fields. Journal of Differential Equations, 1993, 104:215-242.
[79] M. Han. Stability of the equator and bifurcation of limit cycles. World Scientific, 1993.
[80] Liu Yirong. Theory of center-focus for a class of higher-degree critical points and infinite points. Science in china (Series A), 2001, 44:37-48.
[81] Li Jibin , Zhao Xiaohua. Rotation symmetry groups of planar Hamiltonian systems , Ann. Diff. Equs., 1989, 5: 25-33.
[82] Y. Liu, H. Chen. Stability and bifurcations of limit cycles of the equator in a class of cubic polynomial systems. Comuters and Mathematics with applications, 2002, 44:997-1005.
[83] Yirong Liu, Wentao Huang, Seven large-amplitude limit cycles in a cubic polynomial system, International Journal of Bifurcation and Chaos ,2006,16 (2) :473 - 485.
[84] Qi Zhang, Yirong Liu, A cubic polynomial system with seven limit cycles at infinity, Applied Mathematics and Computation ,2006,177 (1) : 319-329.
[85] 刘一戎,赵梅春,一类五次系统赤道环的稳定性与极限环分枝,数学年刊,2002,23A(1):1-4.
[86] 陈海波,刘一戎,一类五次系统赤道环的极限环问题,数学年刊,2003,24A:219-224.
[87] Wentao Huang, Yirong Liu, Bifurcations of limit cycles from infinity for a class of quintic polynomial system, Bulletin des Sciences Mathematiques, 2004, 128:291-301.
[88] Wentao Huang, Li Zhang, Limit cycles of infinity in a quintic Polynomial system, Natural Science Journal of Xiangtan University , 2006, 28 (3): 12 - 16.
[89] Qi Zhang, Yirong Liu, A quintic polynomial system with eleven limit cycles at the infinity, Computer and Mathematics with Applications, 2007,53(10): 1518- 1526.
[90] 陈海波,罗交晚,一类七次系统赤道环的稳定性与极限环分枝,应用数学,2002,15:22-25.
[91] Qi Zhang, Yirong Liu, Haibo Chen, Bifurcation at the equator for a class of quintic polynomial differential system, Applied Mathematics and Computation, 2006, 181(1): 747-755.
[92] W. Huang, Y. Liu, A Polynomial Differential System with Nine Limit Cycles at Infinity, Computers and Mathematics with Applications, 2004, 48:577-588.
[93] Qi Zhang, Weihua Gui, Yirong Liu, Bifurcation of limit cycles at the equator for a class of polynomial differential system, Nonlinear Analysis: Real World Applications, 2009,10(2): 1042-1047.
[94] Chaoxiong Du, Yirong Liu, General center conditions and bifurcation of limit cycles for a quasi-symmetric seventh degree system,Computers and Mathematics with Applications, 2008,56(11): 2957-2969.
[95]Chaoxiong Du, Yirong Liu, The problem of general center-focus and bifurcation of limit cycles for a planar system of nine degrees, Journal of Computational and Applied Mathematics, 2009, 223(2):1043-1057.
[96] H. Poincare. Memoire surles courbes defines par les equations differentielles. J. Math. Pures Appl., 1885,4:167-244.
[97] H. Dulac. Determination et integration dune certaine classe Dequations differentielles ayant pour point singulier un center. Bull. Sci. Nath., 1908,32:230-252.
[98] W. Kapteyn. Over de middeopunten der integral krlnnen van differential vergelijkingen van de eerste orde en den eerstengraad. Kon.Ned. Akad. Wet. Versl., 1911,19:1446-1457.
[99] V. A. Lunkevich, K. S. Sibirskii. Integrals of a general quadratic differential systems in cases of a center. Differential Equations, 1982,18(5):563-568.
[100] F. G(?)bber, K. D. Willamowski. Lyapunov approach to multiple Hopf bifurcation. J. Math. Anal. Appl., 1979, 71:333-350.
[101] D. Schlomiuk. Algebraic particular integrals, integrability and the problem of center, Trans. Amer. Math. Soc., 1993, 338:799-845.
[102] D. Schlomiuk. Elementary first integrals and invariant algebraic curves of differential equations. Expositiones Mathematicae,1993,11:433-454.
[103] K. E. Malkin, Criteria for the center for a differential equation. Volz.Math. Sb., 1964,2:87-91. [104] V. A. Lunkevich, K. S. Sibirskii. Conditions of a center in Homogeneous nonlinearities of third degree. Differential equations,1965,1:1164-1168.
[105] K. S. Sibirsky. Introduction to the algebraic theory of invariants of differential equations. Translated from the Russian. Nonlinear Science:Theory and Applications. Manchester: Manchester University press, 1988.
[106] C. Rousseau, D. Schlomiuk. Cubic systems symmetric with respect to a center. J. Differential Equations, 1995, 123:388-436.
[107] 刘一戎等.二阶非线性系统的奇点量,中心问题与极限环分叉,北京,科学出版社,2008.
[108] I. S. Kukles. Sur quelques cas de distinction entre un foyer et un center. Dokl. Akad. Nauk., 1944,42:653-669.
[109] X. Jin, D. Wang. On the conditions of Kukles for the existence of a center. Bull. London Math. Soc.,1990,22:1-4.
[110] C. J. Christopher, N. G. Lloyd. Invariant algebraic curves and conditions for a center. proc. Roy. Soc. Edinburgh, 1994, 124A:1209-1229.
[111] N. G. Lloyd, J. M. Pearson. Conditions for a center and the bifurcation of limit cycles in a class of cubic systems. Berlin: Springer, 1990, LNM 1455, 230-242.
[112] Qi Zhang, Weihua Gui, Yirong Liu. The generalied center problem Of gedenerate resonant singular point, Bulletin des Math(?)matiques, in press, Available online 23 December 2007.
[113] Qinlong Wang, Yirong Liu. Linearizability of the polynomial differential systems with a resonant singular point, Bulletin des Sciences Mathematiques, 2008,132(2):97-111.
[114] D. J. Needham. A center theorem for two-dimensional complex holomorphic systems and its generalizations. London: Proc. Roy. Soc., 1995, A450:225-232.
[115] G. R. Fowles, G. L. Cassidy. Analytical Mechanics. Orlando: Saunders College Publishing, Philadelphia, 1997.
[116] W. S. Loud. Behavior of the period of solutions of certain plane autonomous systems near centers. Contributions to Differential Equations, 1964, 3:21-36.
[117] I. Pleshkan. A new method of investigating the isochronicity of a system of two differential equations. Differential equations, 1969,5:796-802.
[118] J. Chavarriga, J. Gin(?), I. Garc(?)a. Isochronous centers of a linear center perturbed by fifth degree homogeneous polynomial. J.Comput.Appl. Math., 2000,126:351-368.
[119] J. Chavarriga, J. Gin(?), I. Garc(?)a. Isochronous centers of a linear center perturbed by fourth degree homogeneous polynomial. Bull. des Sci. Math., 1999, 123:77-96.
[120] J. Chavarriga, I. Garcia, J. Gine. Isochromicity into a family of time-reversible cubic vector fields. Applied mathematics and computation, 2001,121:129-145.
[121] J. Chavrriga L. Cairo, J. Gine, J. Llibre. A class of reversible cubic systems with an isochronous center. Computers and Mathematics with Applications, 1999,38:39-43.
[122] W. Zhang, X. Hou, Z. Zeng.Weak centers and bifurcation of critical periods in reversible cubic systems. Computers and Mathematics with Applications, 2000,40:77-82.
[123] M. Sabatini. A connection between isochronous Hamiltonian centers and the jacobian conjecture. Nonlinear Analysis, 1998, 34:29-38.
[124] J. Chavarriga, J. Gine, I. Garcia. Isochronous centers of cubic systems with degenerate infinity. Differential Equations Dynamical Systems, 1999, 7:221-238.
[125] N. G. Lloyd, J. Christopher, J. Devlin, J. M. Pearson, N. Uasmin. Quadratic like cubic systems. Differential Equations Dynamical Systems, 1997, 5(3-4): 329-345.
[126] 林怡平,李继彬,平面自治系统的规范型与闭轨周期的临界点,数学学报,1991,34:490-501.
[127] Y. Liu, W. Huang. A new method to determine isochronous center conditions for polynomial differential systems. Bull. des Sci. Math., 2003,127:133-148.
[128] Y.Liu, J.Li, Periodic constants and time-angle difference of Isochronous centers for complex analytic system, Int. J. Bifurcation Chaos, 2006, 12(16) :3747-3757.
[129] 王勤龙,平面多项式微分系统的中心问题与极限环分支:[博士学位论文].长沙:中南大学,2006.
[130] 张齐,多项式微分自治系统的极限环与广义等时中心:[博士学位论文].长沙:中南大学,2006.
[131] Wentao Huang, Yirong Liu, Weinian Zhang. Conditions of infinity to be an isochronous centre for a rational differential system, Mathematical and Computer Modelling, 2007, 46(5-6): 583-594.
[132] Yirong Liu, Wentao Huang, Center and isochronous center at infinity for differential systems, Bulletin des Sciences Mathematiques, 2004, 128(2): 77-89.
[133] Qinlong Wang, Yirong Liu, Linearizability of the polynomial differential systems with a resonant singular point, Bulletin des Sciences Mathematiques, 2008, 123(2): 97-111.
[134] Chaoxiong Du, Yirong Liu, Heilong Mi, A class of ninth degree system with four isochronous centers, Computers & Mathematics with Applications, 2008,56(10): 2609-2620.
[135] Lloyd N. G, Pearson J.M, Sabz E, Szant I. A Cubic Kolmogorov System with six limit cycles. Computers and Mathematics with Applications 2002,44:445-455.
[136] Lu Zhengyi, He Bi, Multiple Stable Limit cycles for a cubic Kolmogorov prey-predator system, Chinese Journal of Engineering Mathematics, 2001, 4:115-117.
[137] Saez E and Szanto I. Limit cycles of a cubic kolmogorov system , Appl Math. Lett, 1996;9:15-18.
[138] 杜超雄,刘一戎,米黑龙,一类三次Kolmogorov系统的极限环分枝,工程数学学报,2007,4:746-752.
[139] 崔景泰,陈兰荪.稀疏效应下功能性反应系统的数学研究,工程数学学报,1995,13(1):27-35.
[140] 吴兴歧,刘学生,赵振海.一类稀疏效应下捕食者—被捕食者系统极限环的存在性和唯一性,数学研究与评论,1997,17(1):69-73.
[141] 阳平化,徐瑞,张士军.一类具有稀疏效应的生态系统的极限环,生物数学学报,1997,12(4):335-338.
[142] 沈伯骞,沈聪.食饵具有常投放的一类稀疏效应捕食系统,数学杂志,2000,20(2):173-179.
[143] 郑清波,余昭旭,孙继涛.一类稀疏效应下食饵—捕食者系统极限环的存在唯一性,生物数学学报,200l,16(2):156-161.
[144] 沈聪,沈伯骞.一类稀疏效应下的捕食系统存在唯一极限环的充要条件,生物数学学报,2003,18(2):207-210.
[145]Yonghui Xia, Jinde Cao, Suisun Cheng. Multiple periodic solutions of a delayed stage-structureed predator-prey model with non-monotone functional responses, Applied Mathematical Modelling, 2007, 31:1947-1959.
[146] Yonghui Xia, Maoan Han. Multiple periodic solutions of a ratio-dependent predator-prey model, Chaos, Solitons and Fractals, In Press.
[147] Tapan Kumar Kar, Swarnakamal Misra. Influence of prey reserve in a prey - predator fishery, Nonlinear Analysis, 2006, 65(9): 1725-1735.
[148] J. Bai, Y.Liu. A class of of planar degree n (even number) Polynomial systems with a fine focus of order n_2·n, Chinese Sci.Bull.,1992, 12:1063-1065.
[149] C.J.Christopher, N.G.Lloyd. Polynomial systems:A lower bound for the Hilbert numbers, Proc. R. Soc. Lond. Ser. A., 1995, 450(1938):219-224.
[150] M.Han, Y.Lin, P. Yu. A study on the existence of limit cycles of a planar system With 3~(rd)-degree polynomials, International Journal of Bifurcation and Chaos, 2004,14:41-60.
[151] Zhang Zhifen. On the uniqueness of limit cycles of some non-linear oscillation equations, Dokl. Akad. Nauk. S S S R, 1958, 119: 659 - 662.
[2] Smale S, Dynamics retrospective: great problems attempts that failed,Physica D, 1991, 51: 261-273.
[3] Smale S, Mathematical problem for the next century, Mathematical Intelligencer, 1998, 20(2): 7-15.
[4] Yu. Ilyashemko, Centennial history of Hilbert's 16th problem, Bull. Amer. Math Soc. (N.S.) 2002, 39 (3) : 301-354.
[5] J. Li, Hilbert's 16th problem and bifurcation of Planar polynomial vector fields, Internat. J. Bifur. Chaos. 2003, 13: 47-106.
[6] J. Gine, On some open problems in planar differential systems and Hilbert's 16th problem, Chaos, Solitons Fractals . 2007, 31 (5): 1118 -1134.
[7] Dulac H, Surles cycles limites, Bull. Soc. Math. France. 1923, 51: 45-188.
[8] Petrovskii I. G.., Landis E. M., On the number of limit cycles of the equation dy/dx=P(x, y)/Q(x, y) where P and Q are polynomials of the second degree (Russian), Mat. Sb. N. S. 1955, 37(79): 209-250.
[9] Petrovskii I. G.., Landis E. M., On the number of limit cycles of the Equation dy/dx=P(x, y)/Q(x, y) where P and Q are polynomials of the second degree (Russian), Mat. Sb. N. S. 1957, 85: 209-250.
[10] Chen L., Wang M., The relative position and number of limit cycles of quadratic differential system, Acta Math. Sinica, 1979,22: 751-758.
[11] Shi S., A concrete example of the existence of four limit cycles for plane quadratic system, Sci. Sinica(A), 1980,23: 153-158.
[12]Ilyashenko Yu. S., Singular points and limit cycles of different -ial equations in the real and the complex planes, NIVTS AN SSSR, Pushchino, 1982, 38.
[13] Ilyashenko Yu. S., Dulac' s memoir" on limit cycles" and related questions of the local theory of differential equations (Russian), Uspekhi Mat. Nauk, 1985, 40(6):41-78.
[14] Ilyashenko Yu.S.,Finiteness for limit cycles, Transl. Math. Monographs, 1991, 94.
[15] Ecalle J., Introduction aux Functions Analysables et Preuve Constructive de la Conjecture de Dulac, Hermann Pulb, Paris, 1992.
[16] J. Li, Q. Huang. Bifurcation set and limit cycles forming compound eyes in the cubic system. Chin. Ann. of Math. 1987, 8B:391-403.
[17] Li J., Liu Z. Bifurcation set and limit cycles forming compound eyes in a perturbed Hamiltonian system. Publ. Math. 1991, 35:487-506.
[18] Viktor Levandovskyy, Valery G. Romanovski, Douglas S. Shafer. The cyclicity of a cubic system with nonradical Bautin ideal, Journal of ifferential Equations, 2009, 246(3): 1274-1287.
[19] T. Zhang, M. Han, H. X. Meng. Bifurcations of limit cycles for a cubic Hamiltonian system under quartic perturbations. Chaos Solitons and Fractals, 2004, 22:1127-1138.
[20] P. Yu, M. Han. Twelve limit cycles in cubic case of the Hilbert problem.Int.J. Bifurcation & Chaos, 2005,15(7):2191-2205.
[21] P. Yu, M. Han. Small limit cycles bifurcating from fine focus points in cubic order Z2-equivariant vector fields. Chaos, Solitons & Fractals, 2005, 24: 329-348.
[22] Y. Liu, W. Huang. A cubic system with twelve small amplitude Limit cycles.Bull, des Sci. Math., 2005, 129:83-98.
[23] Q. Wang, Y.Liu, C. Du, Small limit cycles bifurcating from fine focuspoints in quartic order Z3-equivariant vector fields, Journal of Mathematical Analysis and Applications, 2008,337(1): 524-536.
[24] Yuhai Wu , Yongxi Gao , Maoan Han, Bifurcations of the limit cycles in a Z3-equivariant quartic planar vector field, Chaos, Solitons & Fractals, 2008, 38(4): 1177-1186.
[25] J. Li, HSY Chan, KW Chung. Bifurcations of limit cycles in a Z6-equivariant planar vector field of degree 5. Sci. China Ser. A, 2002,45: 817-26.
[26] Yuhai Wu, Maoan Han, New configurations of 24 limit cycles in a quintic system, Computers & Mathematics with Applications, 2008, 55(9): 2064-2075.
[27] Wang S., Yu P., Bifurcation of limit cycles in a quintic Hamiltonian system under a sixth-order perturbation, Chaos, Solitons & Fractals, 2005, 26: 1317-1335.
[28] Li J., Zhang M., Bifurcation of limit cycles in a Z8-equivariant vector field, J.Diff. Eqs. Dyn. Syst., 2004, 16:1123-1139.
[29] Wang S., Yu P., Li J., Bifurcation of limit cycles in a Z10-equivariant vector field of degree 9, Int. J. Bifurcation and chaos, 2006, 16(8): 2309-2324.
[30] Wang S., Yu P., Existence of 121 limit cycles in a perturbed planar polynom -ial Hamiltonian vector field of degree 11, Chaos, Solitons & Fractals, 2006, 30(3):606-621.
[31] C. Christopher, N. G. Lloyd, Polynomial systems: A lower bound for the Hilbert numbers, Proc. Royal Soc. London Ser., 1995 450(A): 219-224.
[32] J. Li, HSY Chan, KW Chung, Some lower bounds for H(n) in Hilbert' s 16 th Problem, Qualitative Theory of Dynamical, 2003, 3(44): 345-360.
[33] Itenberg I., Shustin E., Singular points and limit cycles of planar polynomial vector fields Duke Math. J., 2000, 102(1):1-37.
[34] 秦元勋,蔡隧林,史松林,关于平面二次系统的极限环,中国科学,1981,8:929-938.
[35] N. Bautin. On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center type. Amer. Math. Soc. Trans. 1954,100:397-413.
[36] W. W. Farr, C. Li, I. S. Labouriau, W. F. Langford. Degenerate hopf bifurcation formulas and Hilbert' 16th problem. SIAM. J. Math. Anal. 1989,20:13-30.
[37] F. Gobber, K. Willamowski. Liapunov approach to multiple Hopf bifurcation. J. Math. Anal. Appl. 1979, 71(98):330-350.
[38] S. Shi. A Method of constructing cycles without contact around a weak focus. J. Differential Equations, 1980, 41:301-312.
[39] S. Shi. On the structure of Poincare-Liapunov constants for the focus of polynomial vector fields. J. Differential Equations, 1984, 52:52-57.
[40] J. Chavarriga. Integrable systems in the plane with center type linear part. Application(?)s Math(?)matica(?), 1994, 22:285-309.
[41] 杜乃林,曾宪武,计算焦点量的一类递推式,科学通报,1994,39(19):1742-1745.
[42] J. Gine. Conditions for the existence of a center for the Kukles homogeneous systems. Computers and mathematics with applications. 2002, 43:1261-1269.
[43] N. G. Lloyd, J. M. Pearson. Symmetry in planar dynamical systems. J. Symbolic Computation, 2002 33:357-366.
[44] 刘一戎,一类平面三次系统的焦点量公式、中心条件和中心积分,科学通报,1987,2:85-97.
[45] J. P. Francoise. Successive derivatives of first-return map, application to quatratic vector fields. Ergodic theory dynamical systems.1996,16: 87-96.
[46] 秦元勋,刘尊全,微分方程公式的机器推导(Ⅲ),1981,7:388-391.
[47] 李承治,关于平面二次系统的两个问题,中国科学,1982,12:1087-1096.
[48] 刘一戎,李继彬,论复自治系统的奇点量,中国科学,1989,3:245-255.
[49] Liu Yirong, Li Jibin. Theory of values of singular point in complex autonomous differential system. Science in china (Series A), 1990, 33:10-24.
[50] Yirong Liu and Haibo Chen. Formulas of singurlar point quantities and the first 10 saddle quantities for a class of cubic system. Acta Mathematicae applicatae sinica, 2002, 25(2): 295-302. (in Chinese)
[51] N. G. Loyd, J. M. Pearson. Bifurcation of limit cycles and integrability of Planar dynamical systems in complex form. Computers and mathematics with applicat -ions. 2002, 43:1261-1269.
[52] A. Gasull, A. Guillamon, V. Maqosa. An explicit expression of the first Liapunov and period constants with applications. J. Math. Anal. Appl., 1997, 211:190-212.
[53] Sibirskii K. S. On the number of limit cycles in a neighborhood of singular points. Dif. Eq., 1965,1:36-76.(in Russian)
[54] N. G. Lloyd, T. R. Blows, M. C. Kalenge. Some cubic systems with several limit cycles. Nonlinearity, 1988, 1:65-69.
[55] E. M. James, N. G. Lloyd. A cubic system with eight small amplitude limit cycles. LM. A. J. Applied Math., 1991, 47:163-171.
[56] S. Ma, S. Ning. Derive some new conditions on existence of eight limit cycles for a cubic system. Computers and Mathematics with Appli -cations, 1997,33: 59-64.
[57] S. C. Ning, S. L. Ma, H. K. Keng, Z. M. Zheng. A cubic system with eight small amplitude limit cycles. Appl. Math. lect., 1994,4:23-27.
[58] H. Zoladek. Eleven small limit cycles in a cubic vector field. Nonlinearity, 1995, 8:843-860.
[59] 叶彦谦,多项式微分系统定性理论,上海:上海科学技术出版社,1995,85-106.
[60] M. Han, Y. Lin, P. Yu. A study on the existence of limit cycles of a planar system with 3rd-degree polynomials. Int. J. Bifurcation and Chaos, 2004,14: 65-78.
[61] C. Chicone, M. Jacobs. Bifurcation of critical periods for plane vector fields. Amer. Math. Soc. Trans., 1989, 312:433-486.
[62] J. Guckenheimer, P. Holmes. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Appl. Math. Sci. 42, Second Printing, Springer-Verlag, 1986.
[63] V. I. Arnold, Y. S. Ilyashenko. Dynamical Systems Ⅰ, Ordinary Differential Equations. Encyclopaedia of Mathematical Sciences, Vol. 1, Springer-Verlag, 1986.
[64] H. Giacomini, J. Libre, M. Viano. On the nonexistence, existence And uniqueness of limit cycles. Nonlinearity, 1996,9:501-516.
[65] H. Giacomini, J. Libre, M. Viano. On the shape of limit cycles that bifurcate from Hamiltonian centers, Nonlinear Analysis, Ser. A: Theory Methods, 2000,41:523-537.
[66] C. Li, W. Li, J. Llibre, Z. Zhang. On the limit cycles of Polynomial differential systems with homogeneous nonlinearities. Proc. of the Edinburgh Math. Soc., 2000, 43: 529-543.
[67] J. Llibre, J. S. Perez Delrio, J. A. Roddriguez. Averaging Analysis of A Perturbed Quadratic Center. Nonlinear Anal., 2001, 4:45-51.
[68] F. Dumortier, C. Li, Z.Zhang. Unfolding of a quadratic integrable system with two centers and two unbounded heteroclinic loops. J. Differential Equations, 1997, 139:146-193.
[69] I. D. Iliev. Perturbations of quadratic centers. Bull. Sci. Math. ,1998, 122:107-161.
[70] C. Li, W. Li, J. Llibre, Z.Zhang. Linear estimates for the number of zores of Abelian integrals for quadratic isochronous centers,Nonlinearity, 2000,13:1775-1800.
[71] H. Giacomini, J. Llibre, M. Viano. The shape of limit cycles that Bifurcate from non-Hamiltonian centers. Nonlinear Analysis, Ser. A:Theory Methods, 2001, 43:837-859.
[72] C. Li, W. Li, J. Llibre, Z. Zhang. Linear estimates for the number of zores of Abelian integrals for some cubic isochronous centers. J. Differential Equations, 2002, 180:307-333.
[73] J. Yu, C. Li. Bifuration of a class of planar non-Hamiltonian Integrable systems with one center and one homollinic loop. J. Math. Anal. Appl., 2002, 269(1): 227-243.
[74] A. Gasull, J. Libre. Limit cycles for a class of Abel equations. SIAM J. of Math. Anal., 1990, 21:1235-1244.
[75] F. Dumortier, C, Li. perturbations from an Elliptic Hamiltonian of degree four:Ⅰ. saddle loop and two saddle cycle. J. Differential Equations, 2001,176(1): 114-157.
[76] F. Dumortier, C.Li. Perturbations from an Elliptic Hamiltonian of degree four: Ⅳ. figure eight-loop. J. Differential Equations, 2003,188(2) :473-511.
[77] R. Roussarie. Bifurcation of planar vector fields and Hilbert' s sixteenth problem progr. Math., Vol. 164, Birkh auser, Basel, 1998.
[78] Blows T. R., Rousseau C. Bifurcation at infinity in polynomial Vector fields. Journal of Differential Equations, 1993, 104:215-242.
[79] M. Han. Stability of the equator and bifurcation of limit cycles. World Scientific, 1993.
[80] Liu Yirong. Theory of center-focus for a class of higher-degree critical points and infinite points. Science in china (Series A), 2001, 44:37-48.
[81] Li Jibin , Zhao Xiaohua. Rotation symmetry groups of planar Hamiltonian systems , Ann. Diff. Equs., 1989, 5: 25-33.
[82] Y. Liu, H. Chen. Stability and bifurcations of limit cycles of the equator in a class of cubic polynomial systems. Comuters and Mathematics with applications, 2002, 44:997-1005.
[83] Yirong Liu, Wentao Huang, Seven large-amplitude limit cycles in a cubic polynomial system, International Journal of Bifurcation and Chaos ,2006,16 (2) :473 - 485.
[84] Qi Zhang, Yirong Liu, A cubic polynomial system with seven limit cycles at infinity, Applied Mathematics and Computation ,2006,177 (1) : 319-329.
[85] 刘一戎,赵梅春,一类五次系统赤道环的稳定性与极限环分枝,数学年刊,2002,23A(1):1-4.
[86] 陈海波,刘一戎,一类五次系统赤道环的极限环问题,数学年刊,2003,24A:219-224.
[87] Wentao Huang, Yirong Liu, Bifurcations of limit cycles from infinity for a class of quintic polynomial system, Bulletin des Sciences Mathematiques, 2004, 128:291-301.
[88] Wentao Huang, Li Zhang, Limit cycles of infinity in a quintic Polynomial system, Natural Science Journal of Xiangtan University , 2006, 28 (3): 12 - 16.
[89] Qi Zhang, Yirong Liu, A quintic polynomial system with eleven limit cycles at the infinity, Computer and Mathematics with Applications, 2007,53(10): 1518- 1526.
[90] 陈海波,罗交晚,一类七次系统赤道环的稳定性与极限环分枝,应用数学,2002,15:22-25.
[91] Qi Zhang, Yirong Liu, Haibo Chen, Bifurcation at the equator for a class of quintic polynomial differential system, Applied Mathematics and Computation, 2006, 181(1): 747-755.
[92] W. Huang, Y. Liu, A Polynomial Differential System with Nine Limit Cycles at Infinity, Computers and Mathematics with Applications, 2004, 48:577-588.
[93] Qi Zhang, Weihua Gui, Yirong Liu, Bifurcation of limit cycles at the equator for a class of polynomial differential system, Nonlinear Analysis: Real World Applications, 2009,10(2): 1042-1047.
[94] Chaoxiong Du, Yirong Liu, General center conditions and bifurcation of limit cycles for a quasi-symmetric seventh degree system,Computers and Mathematics with Applications, 2008,56(11): 2957-2969.
[95]Chaoxiong Du, Yirong Liu, The problem of general center-focus and bifurcation of limit cycles for a planar system of nine degrees, Journal of Computational and Applied Mathematics, 2009, 223(2):1043-1057.
[96] H. Poincare. Memoire surles courbes defines par les equations differentielles. J. Math. Pures Appl., 1885,4:167-244.
[97] H. Dulac. Determination et integration dune certaine classe Dequations differentielles ayant pour point singulier un center. Bull. Sci. Nath., 1908,32:230-252.
[98] W. Kapteyn. Over de middeopunten der integral krlnnen van differential vergelijkingen van de eerste orde en den eerstengraad. Kon.Ned. Akad. Wet. Versl., 1911,19:1446-1457.
[99] V. A. Lunkevich, K. S. Sibirskii. Integrals of a general quadratic differential systems in cases of a center. Differential Equations, 1982,18(5):563-568.
[100] F. G(?)bber, K. D. Willamowski. Lyapunov approach to multiple Hopf bifurcation. J. Math. Anal. Appl., 1979, 71:333-350.
[101] D. Schlomiuk. Algebraic particular integrals, integrability and the problem of center, Trans. Amer. Math. Soc., 1993, 338:799-845.
[102] D. Schlomiuk. Elementary first integrals and invariant algebraic curves of differential equations. Expositiones Mathematicae,1993,11:433-454.
[103] K. E. Malkin, Criteria for the center for a differential equation. Volz.Math. Sb., 1964,2:87-91. [104] V. A. Lunkevich, K. S. Sibirskii. Conditions of a center in Homogeneous nonlinearities of third degree. Differential equations,1965,1:1164-1168.
[105] K. S. Sibirsky. Introduction to the algebraic theory of invariants of differential equations. Translated from the Russian. Nonlinear Science:Theory and Applications. Manchester: Manchester University press, 1988.
[106] C. Rousseau, D. Schlomiuk. Cubic systems symmetric with respect to a center. J. Differential Equations, 1995, 123:388-436.
[107] 刘一戎等.二阶非线性系统的奇点量,中心问题与极限环分叉,北京,科学出版社,2008.
[108] I. S. Kukles. Sur quelques cas de distinction entre un foyer et un center. Dokl. Akad. Nauk., 1944,42:653-669.
[109] X. Jin, D. Wang. On the conditions of Kukles for the existence of a center. Bull. London Math. Soc.,1990,22:1-4.
[110] C. J. Christopher, N. G. Lloyd. Invariant algebraic curves and conditions for a center. proc. Roy. Soc. Edinburgh, 1994, 124A:1209-1229.
[111] N. G. Lloyd, J. M. Pearson. Conditions for a center and the bifurcation of limit cycles in a class of cubic systems. Berlin: Springer, 1990, LNM 1455, 230-242.
[112] Qi Zhang, Weihua Gui, Yirong Liu. The generalied center problem Of gedenerate resonant singular point, Bulletin des Math(?)matiques, in press, Available online 23 December 2007.
[113] Qinlong Wang, Yirong Liu. Linearizability of the polynomial differential systems with a resonant singular point, Bulletin des Sciences Mathematiques, 2008,132(2):97-111.
[114] D. J. Needham. A center theorem for two-dimensional complex holomorphic systems and its generalizations. London: Proc. Roy. Soc., 1995, A450:225-232.
[115] G. R. Fowles, G. L. Cassidy. Analytical Mechanics. Orlando: Saunders College Publishing, Philadelphia, 1997.
[116] W. S. Loud. Behavior of the period of solutions of certain plane autonomous systems near centers. Contributions to Differential Equations, 1964, 3:21-36.
[117] I. Pleshkan. A new method of investigating the isochronicity of a system of two differential equations. Differential equations, 1969,5:796-802.
[118] J. Chavarriga, J. Gin(?), I. Garc(?)a. Isochronous centers of a linear center perturbed by fifth degree homogeneous polynomial. J.Comput.Appl. Math., 2000,126:351-368.
[119] J. Chavarriga, J. Gin(?), I. Garc(?)a. Isochronous centers of a linear center perturbed by fourth degree homogeneous polynomial. Bull. des Sci. Math., 1999, 123:77-96.
[120] J. Chavarriga, I. Garcia, J. Gine. Isochromicity into a family of time-reversible cubic vector fields. Applied mathematics and computation, 2001,121:129-145.
[121] J. Chavrriga L. Cairo, J. Gine, J. Llibre. A class of reversible cubic systems with an isochronous center. Computers and Mathematics with Applications, 1999,38:39-43.
[122] W. Zhang, X. Hou, Z. Zeng.Weak centers and bifurcation of critical periods in reversible cubic systems. Computers and Mathematics with Applications, 2000,40:77-82.
[123] M. Sabatini. A connection between isochronous Hamiltonian centers and the jacobian conjecture. Nonlinear Analysis, 1998, 34:29-38.
[124] J. Chavarriga, J. Gine, I. Garcia. Isochronous centers of cubic systems with degenerate infinity. Differential Equations Dynamical Systems, 1999, 7:221-238.
[125] N. G. Lloyd, J. Christopher, J. Devlin, J. M. Pearson, N. Uasmin. Quadratic like cubic systems. Differential Equations Dynamical Systems, 1997, 5(3-4): 329-345.
[126] 林怡平,李继彬,平面自治系统的规范型与闭轨周期的临界点,数学学报,1991,34:490-501.
[127] Y. Liu, W. Huang. A new method to determine isochronous center conditions for polynomial differential systems. Bull. des Sci. Math., 2003,127:133-148.
[128] Y.Liu, J.Li, Periodic constants and time-angle difference of Isochronous centers for complex analytic system, Int. J. Bifurcation Chaos, 2006, 12(16) :3747-3757.
[129] 王勤龙,平面多项式微分系统的中心问题与极限环分支:[博士学位论文].长沙:中南大学,2006.
[130] 张齐,多项式微分自治系统的极限环与广义等时中心:[博士学位论文].长沙:中南大学,2006.
[131] Wentao Huang, Yirong Liu, Weinian Zhang. Conditions of infinity to be an isochronous centre for a rational differential system, Mathematical and Computer Modelling, 2007, 46(5-6): 583-594.
[132] Yirong Liu, Wentao Huang, Center and isochronous center at infinity for differential systems, Bulletin des Sciences Mathematiques, 2004, 128(2): 77-89.
[133] Qinlong Wang, Yirong Liu, Linearizability of the polynomial differential systems with a resonant singular point, Bulletin des Sciences Mathematiques, 2008, 123(2): 97-111.
[134] Chaoxiong Du, Yirong Liu, Heilong Mi, A class of ninth degree system with four isochronous centers, Computers & Mathematics with Applications, 2008,56(10): 2609-2620.
[135] Lloyd N. G, Pearson J.M, Sabz E, Szant I. A Cubic Kolmogorov System with six limit cycles. Computers and Mathematics with Applications 2002,44:445-455.
[136] Lu Zhengyi, He Bi, Multiple Stable Limit cycles for a cubic Kolmogorov prey-predator system, Chinese Journal of Engineering Mathematics, 2001, 4:115-117.
[137] Saez E and Szanto I. Limit cycles of a cubic kolmogorov system , Appl Math. Lett, 1996;9:15-18.
[138] 杜超雄,刘一戎,米黑龙,一类三次Kolmogorov系统的极限环分枝,工程数学学报,2007,4:746-752.
[139] 崔景泰,陈兰荪.稀疏效应下功能性反应系统的数学研究,工程数学学报,1995,13(1):27-35.
[140] 吴兴歧,刘学生,赵振海.一类稀疏效应下捕食者—被捕食者系统极限环的存在性和唯一性,数学研究与评论,1997,17(1):69-73.
[141] 阳平化,徐瑞,张士军.一类具有稀疏效应的生态系统的极限环,生物数学学报,1997,12(4):335-338.
[142] 沈伯骞,沈聪.食饵具有常投放的一类稀疏效应捕食系统,数学杂志,2000,20(2):173-179.
[143] 郑清波,余昭旭,孙继涛.一类稀疏效应下食饵—捕食者系统极限环的存在唯一性,生物数学学报,200l,16(2):156-161.
[144] 沈聪,沈伯骞.一类稀疏效应下的捕食系统存在唯一极限环的充要条件,生物数学学报,2003,18(2):207-210.
[145]Yonghui Xia, Jinde Cao, Suisun Cheng. Multiple periodic solutions of a delayed stage-structureed predator-prey model with non-monotone functional responses, Applied Mathematical Modelling, 2007, 31:1947-1959.
[146] Yonghui Xia, Maoan Han. Multiple periodic solutions of a ratio-dependent predator-prey model, Chaos, Solitons and Fractals, In Press.
[147] Tapan Kumar Kar, Swarnakamal Misra. Influence of prey reserve in a prey - predator fishery, Nonlinear Analysis, 2006, 65(9): 1725-1735.
[148] J. Bai, Y.Liu. A class of of planar degree n (even number) Polynomial systems with a fine focus of order n_2·n, Chinese Sci.Bull.,1992, 12:1063-1065.
[149] C.J.Christopher, N.G.Lloyd. Polynomial systems:A lower bound for the Hilbert numbers, Proc. R. Soc. Lond. Ser. A., 1995, 450(1938):219-224.
[150] M.Han, Y.Lin, P. Yu. A study on the existence of limit cycles of a planar system With 3~(rd)-degree polynomials, International Journal of Bifurcation and Chaos, 2004,14:41-60.
[151] Zhang Zhifen. On the uniqueness of limit cycles of some non-linear oscillation equations, Dokl. Akad. Nauk. S S S R, 1958, 119: 659 - 662.