关于质量作用网络中通过对偶方程计算折叠和尖点分岔
On the computation of fold and cusp bifurcations in mass action networks via dual equations
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中文总结 AI 辅助
本文利用Gale对偶原理构造替代方程组，以简化质量作用网络中折叠和尖点分岔的判定，避免中心流形或Lyapunov-Schmidt约化，并给出无分岔的推论及实例。
中文摘要 AI 辅助
本文提出了一些有助于识别或排除质量作用网络中折叠和尖点分岔的结果，从而有助于研究这些网络中的多稳态性。主要结果是，质量作用网络中折叠和尖点分岔的发生可以通过使用基于Gale对偶原理得到的替代方程组来确认或排除，而不是使用原始的质量作用方程。对于某些类别的网络，这可以大大简化所涉及的计算，在检查分岔、非退化性和横截性条件时，无需进行显式的中心流形约化或Lyapunov-Schmidt约化。此外，该方法为具有某些组合性质的网络类别中不发生分岔提供了直接推论。我们给出了若干例子来说明这些结果和推论。
英文摘要
This paper presents some results useful in identifying or ruling out fold and cusp bifurcations in mass action networks, and hence in studying multistationarity in these networks. The main result is that the occurrence of fold and cusp bifurcations in mass action networks can be confirmed or ruled out using alternative systems of equations, obtained using principles of Gale duality, rather than the original mass action equations. This can, for certain classes of networks, greatly simplify the calculations involved, avoiding the need for explicit centre manifold reduction or Lyapunov-Schmidt reduction when checking bifurcation, nondegeneracy and transversality conditions. Moreover, the approach leads to immediate corollaries on the non-occurrence of bifurcations in classes of networks with certain combinatorial properties. We present a number of examples illustrating the results and corollaries.
