论文标题
一个有效的解决方案程序，用于解决高度施加的HOPF和Bogdanov-Takens分叉
An efficient solution procedure for solving higher-codimension Hopf and Bogdanov-Takens bifurcations
论文作者
论文摘要
在解决现实世界中的高度构型分叉问题时，人们通常会面临与广义HOPF分叉相关的正常形式或焦点值的困难，以及与较高的Bogdanov-Takens Bifurcens Bifurcation相关的正常形式。困难不仅来自焦点值的典型符号计算，而且还归因于对系统参数的限制，这通常会导致计算中使用的常规方法的失败，即使是以简单的$ 2 $ 2 $二维非线性动力学系统的失败。在本文中，我们使用了一个简单的二维流行病模型，该模型在分析HOPF分叉产生的极限周期的稳定性中未能为此，以说明如何有效地应用我们的方法来确定HOPF Bifurcation的编辑。此外，我们将最简单的正常形式理论应用于Codimension-3 Bogdanov-Takens分叉，并提出了一种有效的一步转换方法，与经典的六步转换方法相比，我们可以证明我们方法的优势。
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 teadious 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.
