Normal form coefficients for the nonresonant double Hopf bifurcation
Normal form coefficients for the nonresonant double Hopf bifurcation
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DOI：
10.1016/0375-9601(86)90057-5
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发表时间：
1986-07
影响因子：
2.6
通讯作者：
E. Knobloch
中科院分区：
文献类型：
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作者：
E. Knobloch
动力系统理论的最新进展强调了对多重分岔的研究[1，2]，并集中在低余维分岔的分类上。一个重要的余维二分叉是双Hopf分叉，它发生在系统同时失去稳定的频率为0：1和频率为%的振荡。在这个参数值下，四个特征值+ i：%，_+ io：2位于虚轴上，并且系统可以通过标准方法简化为中心流形上的四维系统[2]。通过坐标的适当线性变化，可以采用这些方程的形式为~ 1 wI 0+ G1（xa，Y1，x2，Y2），（1a）-t2 0 Fz，其中（X1，Y1，x> Y2）是中心流形上的（真实的）坐标，并且函数Fj，Gj（j= 1，2）表示非线性项。因此
Recent advances in dynamical systems theory have emphasized the study of multiple bifurcations [1, 2], and have focused on the classification of low codimension bifurcations. One important codimension-two bifurcation is the double Hopf bifurcation, which occurs when a system loses stability simultaneously to oscillations with frequency 0: 1 and frequency%. At this parameter value four eigenvalues,+ i¢%, _+ io: 2, lie on the imaginary axis, and the system can be reduced by standard methods to a four-dimensional system on the center manifold [2]. By an appropriate linear change of coordinates, one may take the form of these equations to be~ 1 w I 0+ G1 (xa, Yl, x2, Y2),(la)-t2 0 Fz where (Xl, Yl, x> Y2) are (real) coordinates on the center manifold, and the functions Fj, Gj (j= 1, 2) represent nonlinear terms. Thus