Time-averaged observables exhibit kink or higher-order derivative singularities at supercritical Hopf bifurcations because phase averaging eliminates odd powers of the limit-cycle amplitude while the squared amplitude varies smoothly.
Wiggins,Introduction to Applied Nonlinear Dynamical Sys- tems and Chaos, 2nd ed., Texts in Applied Mathematics, Vol
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Singular Behavior of Observables at Hopf Bifurcations
Time-averaged observables exhibit kink or higher-order derivative singularities at supercritical Hopf bifurcations because phase averaging eliminates odd powers of the limit-cycle amplitude while the squared amplitude varies smoothly.