Singular Behavior of Observables at Hopf Bifurcations

TL;DR

This paper reveals that at Hopf bifurcations, time-averaged observables typically display universal singularities due to geometric phase averaging, even without singular stationary states.

Contribution

It uncovers a universal mechanism for nonanalytic behavior of observables at Hopf bifurcations, independent of stationary state singularities.

Findings

Observables exhibit finite discontinuities in derivatives at bifurcation points.

Phase averaging eliminates odd powers of oscillation amplitude, leading to singularities.

The mechanism is demonstrated in chemical, electronic, and climate oscillators.

Abstract

Hopf bifurcations are a universal route to self-sustained oscillations in driven systems. Despite the absence of any singular stationary state, we show that time-averaged observables generically exhibit singularities at the onset of oscillations. The origin of this behavior is geometric: phase averaging over the emergent periodic attractor eliminates odd powers of the oscillation amplitude, while the squared amplitude varies smoothly with the distance from the bifurcation. Consequently, the excess of any smooth time-averaged observable admits an integer-power expansion; observables remain finite but display discontinuities in finite-order derivatives. This yields an Ehrenfest-like hierarchy of Hopf singularities, in which the first nonanalytic derivative is determined by the lowest-order coupling between the observable and the limit-cycle waveform that survives phase averaging. Generic observables therefore exhibit kink singularities, while symmetry or geometric cancellations can suppress lower-order couplings and shift nonanalyticity to higher derivatives. We demonstrate this mechanism in chemical, electronic, and climate oscillators. Our results identify supercritical Hopf bifurcations as a universal mechanism for nonanalytic observable behavior, where singular features arise without any underlying singular stationary state. They thus provide a generic setting for singular behavior without divergence.