The presence of slow–fast Hopf (or singular Hopf) points in slow–fast systems in the plane is often deduced from the shape of a vector field brought into normal form. It can however be quite cumbersome to put a system in normal form. In De Maesschalck et al. (Canards from birth to transition, 2020), Wechselberger (Geometric singular perturbation theory beyond the standard form, Springer, New York, 2020) and Jelbart and Wechselberger (Nonlinearity 33(5):2364–2408, 2020) an intrinsic presentation of slow–fast vector fields is initiated, showing hands-on formulas to check for the presence of such singular contact points. We generalize the results in the sense that the criticality of the Hopf bifurcation can be checked with a single formula. We demonstrate the result on a slow–fast system given in non-standard form where slow and fast variables are not separated from each other. The formula is convenient since it does not require any parameterization of the critical curve.

平面中慢速系统中慢速Hopf（或奇异的Hopf）点的存在通常是根据带成正常形式的矢量场的形状推断出来的。但是，以正常形式放置系统可能非常麻烦。在De Maesschalck等人中。（从出生到过渡期的鸭，2020年），韦氏（超越标准形式的几何奇异摄动理论，施普林格，纽约，2020年）和耶尔伯特和韦氏（非线性（33（5）：2364–2408，2020））将启动–fast向量字段，其中显示了动手公式来检查此类奇异触点的存在。我们可以用一个公式来检查Hopf分叉的临界度，从而概括结果。我们以非标准形式给出的慢速系统演示了结果，其中慢速变量和快速变量没有彼此分开。该公式很方便，因为它不需要对关键曲线进行任何参数化。