This work is based on the observation that the first Poincaré–Lyapunov constant is a quadratic function of the coefficients of the two-dimensional vector field at a Hopf bifurcation. From a given parameter point, we find the distance to the “Hopf quadric.” This distance provides a measure of the criticality of the Hopf bifurcation. The viability of the approach is demonstrated through numerical examples.

中文翻译：

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测量Hopf分叉的临界度

这项工作基于以下观察结果：第一个Poincaré-Lyapunov常数是Hopf分支处二维矢量场系数的二次函数。从给定的参数点，我们找到到“ Hopf二次曲面”的距离。该距离提供了霍普夫分支的临界度的量度。通过数值示例证明了该方法的可行性。