Bifurcations of Double Homoclinic Flip Orbits With Resonant Eigenvalues
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摘要: 该文研究了具有轨道翻转的双同宿环四维系统,在主特征值共振和沿轨道奇点处切方向共振下的两种分支.我们分别在系统奇点小邻域内利用规范型的解构造一个奇异映射,再在双同宿环的管状邻域内引起局部活动坐标架,利用系统线性变分方程的解定义了一个正则映射,通过复合两个映射而得到分支研究中一类重要的Poincaré映射,经过简单的计算最终得到后继函数的精确表达式.对分支方程细致地研究,我们给出了原双同宿环的保存性条件,并证明了“大” 1-同宿环分支曲面,2-重“大”1-周期轨分支曲面,“大”2-同宿环分支曲面的存在性、存在区域和近似表达式,及其分支出的“大”周期轨和“大”同宿轨的存在性区域和数量.Abstract: Concerns double homoclinic loops with or bitflips and two resonant eigenvalues in a fourdimensional system. We use the solution of a normal form system to construct a singular map in some neighborhood of the equilibrium, and the solution of a linear variational system to construct a regular map in some neighborhood of the double homoclinic loops, then compose them to get the important Poincar map. A simple calculation gives explicitly an expression of the associated successor function. By a delicate analysis of the bifurcation equation, we obtain the condition that the original double homoclinic loops are kept, and prove the existence and the existence regions of the large 1-homo clinic orbit bifurcation surface, 2-fold large 1-periodic or bit bifurcation surface, large 2-homoclinic or bit bifur cation surface and their appro ximate expressions. We also locate the large periodic orbits and large homoclinic orbits and their number.
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Key words:
- double homoclinic orbit /
- orbit flip /
- periodic orbit /
- resonance
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[1] Chow S N,Deng B, Fiedler B. Homoclinic bifurcation at resonant eigenvalues[J].J Dyn Syst Diff Eqs,1990,2(2):177-244. doi: 10.1007/BF01057418 [2] HAN Mao-an, CHEN Jian.On the number of limit cycles in double homoclinic bifurcations[J].Sci China,2000,43(9):914-928. doi: 10.1007/BF02879797 [3] JIN Yin-lai, ZHU De-ming.Bifurcation of rough heteroclinic loop with two saddle points[J].Sci China,2003,46(4):459-468. [4] TIAN Qin-ping,ZHU De-ming.Bifurcation of nontwisted heteroclinic loop[J].Sci China,Ser A,2000,43(8):818-828. doi: 10.1007/BF02884181 [5] ZHU De-ming,XIA Zhi-hong.Bifurcation of heteroclinic loops[J].Sci China,Ser A,1998,41(8),837-848. [6] Sandstede B.Verzweigungstheorie Homokliner Verdopplungen[D].Ph D Thesis.Berlin:Freie Universitat Berlin,1993.Institut fur Angewandte Analysis und Stochastic[R]. Report No.7, Berlin. [7] Sandstede B.Constructing dynamical systems having homoclinic bifurcation points of codimension two[J].J Dyn Diff Equs,1997,9(2):269-288. doi: 10.1007/BF02219223 [8] Homburg A J,Krauskopf B.Resonant homoclinic flip bifurcations[J].J Dyn Diff Eq,2000,12(4):807-850. doi: 10.1023/A:1009046621861 [9] Oldeman B E,Krauskopf B, Champneys A R. Numerical unfoldings of codimension-three resonant homoclinic flip bifurcations[J].Nonlinearity,2001,14(3):597-621. doi: 10.1088/0951-7715/14/3/309 [10] ZHANG Tian-si, ZHU De-ming. Codimension 3 homoclinicbifurcation of orbit flip with resonant eigenvalues corresponding tothe tangent directions[J].Int J Bifurcation Chaos,2004,14(12):4161-4175. doi: 10.1142/S0218127404011880 [11] ZHANG Tian-si, ZHU De-ming.Homoclinic bifurcation of orbit flip with resonant pricipal eigenvalues[J].Acta Math Sin Engl Ser,2006,22(3):855-864. doi: 10.1007/s10114-005-0581-x [12] SHUI Shu-liang, ZHU De-ming.Codimension 3 bifurcations of homoclinic orbits with orbit flips and inclination flips[J].Chin Ann Math,2004,25B(4):555-566. [13] SHUI Shu-liang,ZHU De-ming.Codimension 3 nonresonant bifurcations of homoclinic orbits with two inclination flips[J].Sci China,Ser A,2005,48(2):248-260.
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