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

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 to 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 scrutinise 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.自然界与工程应用中，各类控制参数发生变化时会表现出临界转变的复杂系统无处不在。尽管该领域已有海量研究文献，但针对分岔参数（bifurcation parameter）变化速率对临界点（tipping points）影响的探讨仍相对匮乏。本研究针对两种场景下的亚临界随机霍普夫分岔（subcritical stochastic Hopf bifurcation）展开分析：其一为分岔参数以准稳态方式进行变化，其二为分岔参数以有限变化速率实施斜坡调节。在后一场景中，我们观测到了与变化速率相关的分岔延迟现象，并通过燃烧室（combustion chamber）内的热声不稳定性（thermoacoustic instability）实验对该现象进行了实证验证。该延迟随参数变化速率的升高而增大，由此会使得系统产生相较于准稳态参数变化时振幅更大的状态转变。我们还通过实验证实，当分岔参数反向斜坡调节时，分岔延迟会引发动态迟滞（dynamic hysteresis）现象。为预测这类延迟的统计特性并剖析其内在的随机动力学（stochastic dynamics）机制，我们构建了一套替代模型（surrogate model）。本研究凸显了分岔参数有限变化速率对临界点的显著影响，同时明确指出，在探究临界转变问题时，必须考虑这一效应，这具有至关重要的意义。