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Complex systems exhibiting critical transitions when one of their governingparameters varies are ubiquitous in nature and in engineering applications.Despite a vast literature focusing on this topic, there are few studies dealingwith the effect of the rate of change of the bifurcation parameter on thetipping points. In this work, we consider a subcritical stochastic Hopfbifurcation under two scenarios: the bifurcation parameter is first changed ina quasi-steady manner and then, with a finite ramping rate. In the latter case,a rate-dependent bifurcation delay is observed and exemplified experimentallyusing a thermoacoustic instability in a combustion chamber. This delayincreases with the rate of change. This leads to a state transition of largeramplitude compared to the one that would be experienced by the system with aquasi-steady change of the parameter. We also bring experimental evidence of adynamic hysteresis caused by the bifurcation delay when the parameter is rampedback. A surrogate model is derived in order to predict the statistic of thesedelays and to scrutinise the underlying stochastic dynamics. Our studyhighlights the dramatic influence of a finite rate of change of bifurcationparameters upon tipping points and it pinpoints the crucial need of consideringthis effect when investigating critical transitions.

Experiments and modelling of rate-dependent transition delay in a stochastic subcritical bifurcation