Transverse eigenvalue crossing ($\mathrm{Re}\lambda ({\mu }_0)=0,\frac{d\mathrm{Re}\lambda }{d\mu }({\mu }_0)\ne 0$)

A transverse eigenvalue crossing occurs when the real part of an eigenvalue passes through zero with nonzero parameter derivative. This spectral condition is separate from the nonlinear nondegeneracy conditions in the Hopf theorem.