In solving real world systems for higher-codimension bifurcation problems, one often faces the difficulty in computing the normal form or the focus values associated with generalized Hopf bifurcation, and the normal form with unfolding for higher-codimension Bogdanov–Takens bifurcation. The difficulty is not only coming from the tedious symbolic computation of focus values, but also due to the restriction on the system parameters, which frequently leads to failure of the conventional approach used in the computation even for simple 2-dimensional nonlinear dynamical systems. In this paper, we use a simple 2-dimensional epidemic model, for which the conventional approach fails in analyzing the stability of limit cycles arising from Hopf bifurcation, to illustrate how our method can be efficiently applied to determine the codimension of Hopf bifurcation. Further, we apply the simplest normal form theory to consider codimension-3 Bogdanov–Takens bifurcation and present an efficient one-step transformation approach, compared with the classical six-step transformation approach to demonstrate the advantage of our method.

中文翻译：

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求解高维 Hopf 和 Bogdanov–Takens 分岔的有效求解过程


在解决现实世界系统的高维分岔问题时，人们经常面临计算与广义 Hopf 分岔相关的范式或焦点值以及高维 Bogdanov-Takens 分岔的展开范式的困难。困难不仅来自于焦点值繁琐的符号计算，还来自于系统参数的限制，这常常导致传统的计算方法即使对于简单的二维非线性动力系统也常常失败。在本文中，我们使用一个简单的二维流行病模型来说明如何有效地应用我们的方法来确定 Hopf 分岔的余维数，而传统方法无法分析该模型由 Hopf 分岔引起的极限环的稳定性。此外，我们应用最简单的范式理论来考虑余维 3 Bogdanov-Takens 分岔，并提出一种有效的一步转换方法，与经典的六步转换方法相比，证明了我们方法的优势。