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首页> 外文期刊>Transactions of Tianjin University >Calculation of Coefficients of Simplest Normal Forms of Hopf and Generalized Hopf Bifurcations
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Calculation of Coefficients of Simplest Normal Forms of Hopf and Generalized Hopf Bifurcations

机译：Hopf和广义Hopf分叉的最简单正规形式的系数的计算

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摘要

The coefficients of the simplest normal forms of both high-dimensional generalized Hopf and high-dimensional Hopf bifurcation systems were discussed using the adjoint operator method. A particular nonlinear scaling and an inner product were introduced in the space of homogeneous polynomials. Theorems were established for the explicit expression of the simplest normal forms in terms of the coefficients of both the conventional normal forms of Hopf and generalized Hopf bifurcation systems. A symbolic manipulation was designed to perform the calculation of the coefficients of the simplest normal forms using Mathematica. The original ordinary differential equation was required in the input and the simplest normal form could be obtained as the output. Finally, the simplest normal forms of 6-dimensional generalized Hopf singularity of type 2 and 5-dimensional Hopf bifurcation system were discussed by executing the program. The output showed that the 5th- and 9th-order terms remained in 6-dimensional generalized Hopf singularity of type 2 and the 3rd- and 5th-order terms remained in 5-dimensional Hopf bifurcation system.
机译：使用伴随算子方法讨论了高维广义Hopf分叉系统和高维Hopf分叉系统的最简单范式的系数。在齐次多项式空间中引入了特定的非线性标度和内积。建立了定理，用于根据Hopf的常规范式和广义Hopf分叉系统的系数来明确表达最简单的范式。设计了一种符号操作，以使用Mathematica对最简单的范式的系数进行计算。输入需要原始的常微分方程，而最简单的范式可以作为输出。最后，通过执行该程序，讨论了2型和5维Hopf分叉系统的6维广义Hopf奇异性的最简单正规形式。输出显示，2型的6维广义Hopf奇异性保留了5阶和9阶项，而5维的Hopf分叉系统保留了3阶和5阶项。

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