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.
FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophysical Journal1, 445 (1961)
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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.