Singular Behavior of Observables at Hopf Bifurcations
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 a 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. The time averages of 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 yet non-diverging features emerge without any underlying singular stationary state.
Versions
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BibTeX
@misc{remlein2026singular,
title = {{Singular Behavior of Observables at Hopf Bifurcations}},
author = {Benedikt Remlein and Massimiliano Esposito},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2605.05194},
eprint = {2605.05194},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.05194},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Singular Behavior of Observables at Hopf Bifurcations
AU - Benedikt Remlein
AU - Massimiliano Esposito
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2605.05194
UR - https://arxiv.org/abs/2605.05194
AB - 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 a 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. The time averages of 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 yet non-diverging features emerge without any underlying singular stationary state.
ER -