Numerical Normalization Techniques for All Codim 2 Bifurcations of Equilibria in ODE's
Numerical Normalization Techniques for All Codim 2 Bifurcations of Equilibria in ODE's
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DOI：
10.1137/s0036142998335005
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发表时间：
1999-04
影响因子：
2.9
通讯作者：
Y. Kuznetsov
中科院分区：
文献类型：
--
作者：
Y. Kuznetsov
导出了自治常微分方程中平衡点的所有codim 2平衡分支的规范形系数的显式计算公式。它们包括Bogdanov-Takens分叉的二阶系数，尖点和折叠Hopf分叉的三阶系数，以及广义Hopf（Bautin）和双Hopf分叉的五阶项的系数。该公式与相空间的维数无关，只涉及临界特征向量的雅可比矩阵的右手边和转置，以及多线性函数的泰勒展开的右手边在临界平衡。利用导出的公式计算了“新”Lorenz模型中fold-Hopf分岔的规范形系数，证明了系统中存在非平凡不变集.
Explicit computational formulas for the coefficients of the normal forms for all codim 2 equilibrium bifurcations of equilibria in autonomous ODEs are derived. They include second-order coefficients for the Bogdanov--Takens bifurcation, third-order coefficients for the cusp and fold-Hopf bifurcations, and coefficients of the fifth-order terms for the generalized Hopf (Bautin) and double Hopf bifurcations. The formulas are independent on the dimension of the phase space and involve only critical eigenvectors of the Jacobian matrix of the right-hand sides and its transpose, as well as multilinear functions from the Taylor expansion of the right-hand sides at the critical equilibrium. The normal form coefficients for the fold-Hopf bifurcation in the "new" Lorenz model are computed using the derived formulas, proving the existence of a nontrivial invariant set in the system.