一类平面微分系统的广义中心条件与极限环分支
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摘要
本文研究一类平面微分系统,通过作两个适当的变换以及焦点量的仔细计算,得出了该系统的无穷远点与5个初等奇点(-1/2,0),(-1/2,±3~(1/2)/2),(±1/2,-3~(1/2)/6)能够同时成为6个广义中心的条件,进一步得出在一定条件下该系统能够分支出12个极限环的结论,其中2个大振幅极限环来自无穷远点,10个小振幅极限环来自5个初等焦点.我们的工作是有意义的,相似的结论在已经出版的文献中少见。
This paper is concerned with a class of planar differential system of nine degrees.By making two appropriate transformations of system and calculating focal values carefully,we obtain the conditions that the infinity and five elementary foci(-1/2|,0),(-1/2,±2/3~(1/2))(±1/2,-6/3~(1/2)) become six general centers at the same time.Moreover 12 limit cycles including10 small limit cycles from five elementary foci and 2 large hmit cycles from the infinity can occur at the same step of disturbance under a certain condition.What is worth mentioning is that similar conclusions have hardly been seen in published paper up till now and our work is significative.
This paper is concerned with a class of planar differential system of nine degrees.By making two appropriate transformations of system and calculating focal values carefully,we obtain the conditions that the infinity and five elementary foci(-1/2|,0),(-1/2,±2/3~(1/2))(±1/2,-6/3~(1/2)) become six general centers at the same time.Moreover 12 limit cycles including10 small limit cycles from five elementary foci and 2 large hmit cycles from the infinity can occur at the same step of disturbance under a certain condition.What is worth mentioning is that similar conclusions have hardly been seen in published paper up till now and our work is significative.
引文
[1]Hu Haiyan,etc.New Development of Theory and Applications on Nonlinear Dynamics.Beijing:Science Press,2009
[2]Li J B.Hilbert's 16th Problem and Bifurcation of Planar Polynomial Vector Fields.Internal.J.Bifurcation Chaos,2003,13:47-106
[3]Shi S L.A Concrete Example of the Existence of Four Limit Cycles for Quadratic Systems.Science in China,1980,23:16-21
[4]Yu P,Han M.Small Limit Cycles Bifurcation From Fine Focus Points in Cubic Order z_2-equivariant Vector Fields.Chaos,Solitons and Fractals,2005,24:329-348
[5]Liu Y R,Huang W T.A Cubic System with Twelve Small Amplitude Limit Cycles.Bull.Sci.Math.,2005,129:83-98
[6]Wang Q L,Liu Y R,Du C X.Small Limit Cycles Bifurcating From Fine Focus Points in Quartic Order Z3-equivariant Vector Fields.J.Math.Anal.Appl,2008,337:524-536
[7]Li J,Chan H S Y,Chung K W.Bifurcation of Limit Cycles in a Z_6-equivariant Vector Fields of Degree5.Science in China,2002,7:3-12
[8]Blows T R,Rousseau C.Bifurcation at Infinity in Polynomial Vector Fields.J.Diff.Equs,1993,104:215-242
[9]Liu Y R.Theory of Center-focus for a Class of Higher-degree Critical Points and Infinite Points.Science in China(Series A),2001,44:37-48
[10]Liu Y R,Chen H B.Stability and Bifurcations of Limit Cycles of the Equator in a Class of Cubic Polynomial Systems.Computers and Mathematics with Applications,2002,44:997-1005
[11]Liu Y R,Huang W T.Seven Large-amplitude Limit Cyclea in a Cubic Polynomial System.International Journal of Bifurcation,2006,16(2):473-485
[12]Zhang Q,Liu Y R.A Cubic Polynomial System with Seven Limit Cycles at Infinity.Applied Mathematics and Computation,2006,177(1):319-329
[13]Chen H B,Liu Y R.Limit Cycle of the Equator in a Quintic Polynomial System.Chinese Annals of Mathematics,2003,24A:19-224
[14]Huang W T,Liu Y R.Bifurcations of Limit Cycles From Infinity for a Class of Quintic Polynomial System.Bull.Sci.Math.,2004,128:291-301
[15]Huang W T,Zhang Li.Limit Cycles of Infinity in a Quintic Polynomial System.Natural Science Journal of Xiangtan University,2006,28(3):12-16
[16]Huang W T,Liu Y R.A Polynomial Differential System with Nine Limit Cycles at Infinity.Computers and Mathematics with Applications,2004,48:577-588
[17]Zhang L,Huang W T.Polynomial Differential System with Ten Limit Cycles at Infinity.Journal of Chongqing University(Natural Science Edition),2006,29(8):146-149
[18]Liu Y R,Zhao M C.Stability And Bifurcation of Limit Cycles of the Equator in a Class of Fifth Polynomial Systems.Chinese Journal of Contemporary Mathematics,2002,23:67-72
[19]Liu Y R,Li J B.Theory of Values of Singular Point in Complex Autonomous Differential System.Science in China(Series A),1990,33:10-24
[2]Li J B.Hilbert's 16th Problem and Bifurcation of Planar Polynomial Vector Fields.Internal.J.Bifurcation Chaos,2003,13:47-106
[3]Shi S L.A Concrete Example of the Existence of Four Limit Cycles for Quadratic Systems.Science in China,1980,23:16-21
[4]Yu P,Han M.Small Limit Cycles Bifurcation From Fine Focus Points in Cubic Order z_2-equivariant Vector Fields.Chaos,Solitons and Fractals,2005,24:329-348
[5]Liu Y R,Huang W T.A Cubic System with Twelve Small Amplitude Limit Cycles.Bull.Sci.Math.,2005,129:83-98
[6]Wang Q L,Liu Y R,Du C X.Small Limit Cycles Bifurcating From Fine Focus Points in Quartic Order Z3-equivariant Vector Fields.J.Math.Anal.Appl,2008,337:524-536
[7]Li J,Chan H S Y,Chung K W.Bifurcation of Limit Cycles in a Z_6-equivariant Vector Fields of Degree5.Science in China,2002,7:3-12
[8]Blows T R,Rousseau C.Bifurcation at Infinity in Polynomial Vector Fields.J.Diff.Equs,1993,104:215-242
[9]Liu Y R.Theory of Center-focus for a Class of Higher-degree Critical Points and Infinite Points.Science in China(Series A),2001,44:37-48
[10]Liu Y R,Chen H B.Stability and Bifurcations of Limit Cycles of the Equator in a Class of Cubic Polynomial Systems.Computers and Mathematics with Applications,2002,44:997-1005
[11]Liu Y R,Huang W T.Seven Large-amplitude Limit Cyclea in a Cubic Polynomial System.International Journal of Bifurcation,2006,16(2):473-485
[12]Zhang Q,Liu Y R.A Cubic Polynomial System with Seven Limit Cycles at Infinity.Applied Mathematics and Computation,2006,177(1):319-329
[13]Chen H B,Liu Y R.Limit Cycle of the Equator in a Quintic Polynomial System.Chinese Annals of Mathematics,2003,24A:19-224
[14]Huang W T,Liu Y R.Bifurcations of Limit Cycles From Infinity for a Class of Quintic Polynomial System.Bull.Sci.Math.,2004,128:291-301
[15]Huang W T,Zhang Li.Limit Cycles of Infinity in a Quintic Polynomial System.Natural Science Journal of Xiangtan University,2006,28(3):12-16
[16]Huang W T,Liu Y R.A Polynomial Differential System with Nine Limit Cycles at Infinity.Computers and Mathematics with Applications,2004,48:577-588
[17]Zhang L,Huang W T.Polynomial Differential System with Ten Limit Cycles at Infinity.Journal of Chongqing University(Natural Science Edition),2006,29(8):146-149
[18]Liu Y R,Zhao M C.Stability And Bifurcation of Limit Cycles of the Equator in a Class of Fifth Polynomial Systems.Chinese Journal of Contemporary Mathematics,2002,23:67-72
[19]Liu Y R,Li J B.Theory of Values of Singular Point in Complex Autonomous Differential System.Science in China(Series A),1990,33:10-24