Abstract
Complex systems exhibiting critical transitions when one of their governing parameters varies are ubiquitous in nature and in engineering applications. Despite a vast literature focusing on this topic, there are few studies dealing with the effect of the rate of change of the bifurcation parameter on the tipping points. In this work, we consider a subcritical stochastic Hopf bifurcation under two scenarios: the bifurcation parameter is first changed in a quasi-steady manner and then, with a finite ramping rate. In the latter case, a rate-dependent bifurcation delay is observed and exemplified experimentally using a thermoacoustic instability in a combustion chamber. This delay increases with the rate of change. This leads to a state transition of larger amplitude compared with the one that would be experienced by the system with a quasi-steady change of the parameter. We also bring experimental evidence of a dynamic hysteresis caused by the bifurcation delay when the parameter is ramped back. A surrogate model is derived in order to predict the statistic of these delays and to scrutinize the underlying stochastic dynamics. Our study highlights the dramatic influence of a finite rate of change of bifurcation parameters upon tipping points, and it pinpoints the crucial need of considering this effect when investigating critical transitions.
Highlights
Complex systems exhibiting critical transitions when one of their governing parameters varies are ubiquitous in nature and in engineering applications
We focus on the transient thermoacoustic dynamics associated with the passage through the Hopf bifurcation when one of the critical operating parameters—the equivalence ratio—is ramped
Nonlinear coupling among different thermoacoustic modes can occur in some practical configurations, we focus on situations where, like in the present case, one mode is dominant in the thermoacoustic dynamics
Summary
Thermoacoustic coupling is a phenomenon that has fascinated scientists for over two centuries. Thermoacoustic instabilities became a recurrent issue in the development phase of heavy-duty gas turbines for power generation and turbofans for air transportation. This is because the resulting intense acoustic fields induce high-cycle fatigue of the combustion chambers [47]. The flame exhibits a periodic motion at the frequency of the first acoustic mode (150 Hz), with sound intensity at the anti-nodes exceeding 150 dB, which is considerable for a burner operated at atmospheric pressure This dynamic state would not be acceptable in a commercial aeronautical engine or in a heavy-duty gas turbine, because the acoustic loading, which scales with the engine operating pressure, would be highly detrimental for the mechanical components. This is relevant for the development of new aeronautical and land-based gas turbines, which require fast loading or deloading, and which may be at risk due to such rate-delayed tipping points
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