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.

The problem of output-only parameter identification for nonlinear oscillators forced by colored noise is considered. In this context, it is often assumed that the forcing noise is white, since its actual spectral content is unknown. The impact of this white-noise forcing assumption upon parameter identification is quantitatively analyzed. First, a Van-der-Pol oscillator forced by an Ornstein-Uhlenbeck process is considered. Second, the practical case of thermoacoustic limit cycles in combustion chambers with turbulence-induced forcing is investigated. It is shown that in both cases, the system parameters are accurately identified if time signals are appropriately band-pass-filtered around the oscillator eigenfrequency.

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This paper shows the importance of considering the thermal state of a combustor to investigate or predict its thermoacoustic stability. This aspect is often neglected or regarded as less important than the effect of the operating parameters, such as thermal power or equivalence ratio, but under certain circumstances it can have a dramatic influence on the development of the instabilities. The paper presents experimental results collected from a combustor featuring a lean swirl-stabilized flame exhibiting thermoacoustic instability at some operating conditions. It is shown that this instability is caused by a change of the flame topology that is induced by the progressive increase of the wall temperature with the thermal power. This dependence of the instability on wall temperature leads to inertial effects and hysteresis when the operating condition is changed dynamically. A low-order model of the system reproducing this remarkable dynamics is proposed and validated against the experimental data.

Varying one of the governing parameters of a dynamical system may lead to a critical transition, where the new stable state is undesirable. In some cases, there is only a limited range of the bifurcation parameter that corresponds to that unwanted attractor, while the system runs problem-less otherwise. In this study, we present experimental results regarding a thermoacoustic system subject to two consecutive and mirrored supercritical Hopf bifurcations: the system exhibits high amplitude thermoacoustic limit cycles for intermediate values of the bifurcation parameter. Changing quickly enough the bifurcation parameter, it was possible to dodge the unwanted limit cycles. A low-order model of the complex thermoacoustic system was developed, in order to describe this interesting transient dynamics. It was afterward used to assess the risk of exceeding an oscillation amplitude threshold as a function of the rate of change of the bifurcation parameter.

We report experimental evidence of thermoacoustic bistability in a lab-scale turbulent combustor over a well-defined range of fuel–air equivalence ratios. Pressure oscillations are characterized by an intermittent behavior with “bursts,” i.e., sudden jumps between low and high amplitudes occurring at random time instants. The corresponding probability density functions (PDFs) of the acoustic pressure signal show clearly separated maxima when the burner is operated in the bistable region. The gain and phase between acoustic pressure and heat release rate fluctuations are evaluated at the modal frequency from simultaneously recorded flame chemiluminescence and acoustic pressure. The representation of the corresponding statistics is new and particularly informative. It shows that the system is characterized, in average, by a nearly constant gain and by a drift of the phase as function of the oscillation amplitude. This finding may suggest that the bistability does not result from an amplitude-dependent balance between flame gain and acoustic damping, but rather from the nonconstant phase difference between the acoustic pressure and the coherent fluctuations of heat release rate.

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Sequential combustion constitutes a major technological step-change for gas turbines applications. This design provides higher operational flexibility, lower emissions and higher efficiency compared to today’s conventional architectures. Like any constant pressure combustion system, sequential combustors can undergo thermoacoustic instabilities. These instabilities potentially lead to high-amplitude acoustic limit cycles, which shorten the engine components’ lifetime and therefore reduce their reliability and availability. In case of a sequential system, the two flames are mutually coupled via acoustic and entropy waves. This additional inter-stages interaction markedly complicates the already challenging problem of thermoacoustic instabilities. As a result, new and unexplored system dynamics are possible. In this work, experimental data from our generic sequential combustor are presented. The system exhibits many different distinctive dynamics, as function of the operation parameters and of the combustor arrangement. This paper investigates a particular bifurcation, where two thermoacoustic modes synchronize their self-sustained oscillations over a range of operating conditions. A low-order model of this thermoacoustic bifurcation is proposed. This consists of two coupled stochastically driven non-linear oscillators, and is able to reproduce the peculiar dynamics associated with this synchronization phenomenon. The model aids in understanding what the physical mechanisms that play a key role in the unsteady combustor physics are. In particular, it highlights the role of entropy waves, which are a significant driver of thermoacoustic instabilities in this sequential setup. This research helps to lay the foundations for understanding the thermoacoustic instabilities in sequential combustion systems.