A numerical algorithm coupling a bifurcating indicator and a direct method for the computation of Hopf bifurcation points in fluid mechanics

International audienceThis paper deals with the computation of Hopf bifurcation points in fluid mechanics. This computation is done by coupling a bifurcation indicator proposed recently and a direct method which consists in solving an augmented system whose solutions are Hopf bifurcation points. The bifurcation indicator gives initial critical values (Reynolds number, Strouhal frequency) for the direct method iterations. Some classical numerical examples from fluid mechanics, in two dimensions, are studied to demonstrate the efficiency and the reliability of such an algorithm

Predicting climate tipping as a noisy bifurcation: A review

Copyright © 2011 World Scientific PublishingElectronic version of an article published in International Journal of Bifurcation and Chaos, Vol. 21 (2), pp. 399 – 423. DOI: 10.1142/S0218127411028519. Copyright © World Scientific Publishing Company. www.worldscientific.com/worldscinet/ijbcThere is currently much interest in examining climatic tipping points, to see if it is feasible to predict them in advance. Using techniques from bifurcation theory, recent work looks for a slowing down of the intrinsic transient responses, which is predicted to occur before an instability is encountered. This is done, for example, by determining the short-term autocorrelation coefficient ARC(1) in a sliding window of the time-series: this stability coefficient should increase to unity at tipping. Such studies have been made both on climatic computer models and on real paleoclimate data preceding ancient tipping events. The latter employ reconstituted time-series provided by ice cores, sediments, etc., and seek to establish whether the actual tipping could have been accurately predicted in advance. One such example is the end of the Younger Dryas event, about 11 500 years ago, when the Arctic warmed by 7°C in 50 yrs. A second gives an excellent prediction for the end of "greenhouse" Earth about 34 million years ago when the climate tipped from a tropical state into an icehouse state, using data from tropical Pacific sediment cores. This prediction science is very young, but some encouraging results are already being obtained. Future analyses will clearly need to embrace both real data from improved monitoring instruments, and simulation data generated from increasingly sophisticated predictive models

Correlation lags give early warning signals of approaching bifurcations

Identifying approaching bifurcations and regime transitions from observations is an important challenge in time series analysis with practical applications in many fields of science. Well-known indicators are the increase in spatial and temporal correlations. However, the performance of these indicators depends on the system under study and on the type of approaching bifurcation, and no indicator provides a reliable warning for any system and bifurcation. Here we propose an indicator that simultaneously takes into account information about spatial and temporal correlations. By performing a bivariate correlation analysis of signals recorded in pairs of adjacent spatial points, and analyzing the distribution of lag times that maximize the cross-correlation, we find that the variance of the lag distribution displays an extreme value that is a consistent early warning indicator of the approaching bifurcation. We demonstrate the reliability of this indicator using different types of models that present different types of bifurcations, including local bifurcations (transcritical, saddle-node, supercritical and subcritical Hopf), and global bifurcations.This work was funded by the Spanish Ministerio de Ciencia, Innovación y Universidades (PGC2018-099443-B-I00 ) and the ICREA ACADEMIA program of Generalitat de Catalunya.Peer ReviewedPostprint (published version

Modeling and Optimal Operation of Hydraulic, Wind and Photovoltaic Power Generation Systems

The transition to 100% renewable energy in the future is one of the most important ways of achieving "carbon peaking and carbon neutrality" and of reducing the adverse effects of climate change. In this process, the safe, stable and economical operation of renewable energy generation systems, represented by hydro-, wind and solar power, is particularly important, and has naturally become a key concern for researchers and engineers. Therefore, this book focuses on the fundamental and applied research on the modeling, control, monitoring and diagnosis of renewable energy generation systems, especially hydropower energy systems, and aims to provide some theoretical reference for researchers, power generation departments or government agencies

ON THE STABILITY OF VARIABLE HELIX MILLING TOOLS

One of the main aims of the manufacturing industry has been to maximise the material removal rate of machining processes. However, this goal can be restricted by the appearance of regenerative chatter vibrations. In milling, one approach for regenerative chatter suppression is the implementation of variable-helix cutters. However, these tools can lead to isolated unstable regions in the stability diagram. Currently, variable-helix unstable islands have not been extensively researched in the literature. Therefore, the current thesis focuses on studying and experimentally validating these islands. For the validation, an experimental setup that scaled not only the structural dynamics but also the cutting force coefficients was proposed. Therefore, it was possible to attain larger axial depths of cut while assuming linear dynamics. The variable-helix process stability was modelled using the semi-discretization method and the multi-frequency approach. It was found that the variable helix tools can further stabilise a larger width of cut due to the distributed time delays that are a product of the tool geometry. Subsequently, a numerical study about the impact of structural damping on the variable-helix stability diagram revealed a strong relationship between the damping level and instability islands. The findings were validated by performing trials on the experimental setup, modified with constrained layer damping to recreate the simulated conditions. Additionally, a convergence analysis using the semi-discretization method (SDM) and the multi-frequency approach (MFA) revealed that these islands are sensitive to model convergence aspects. The analysis shows that the MFA provided converged solutions with a steep convergence rate, while the SDM struggled to converge. In this work, it is demonstrated that variable-helix instability islands only emerge at relatively high levels of structural damping and that they are particularly susceptible to model convergence effects. Meanwhile, the model predictions are compared to and validated against detailed experimental data that uses a specially designed configuration to minimise experimental error. To the authors' knowledge, this provides the first experimentally validated study of unstable islands in variable helix milling, while also demonstrating the importance of accurate damping estimates and convergence studies within the stability predictions

Abrupt transitions in turbulent thermoacoustic systems

Abrupt transitions to the state of thermoacoustic instability (TAI) in gas turbine combustors are a significant challenge plaguing the development of next-generation low-emission aircraft and power generation engines. In this paper, we present the observation of abrupt transition in three disparate turbulent thermoacoustic systems: an annular combustor, a swirl-stabilized combustor, and a preheated bluff-body stabilized combustor. Using a low-order stochastic thermoacoustic model, we show that the reported abrupt transitions occur when an initially stable, supercritical limit cycle becomes unstable, leading to a secondary bifurcation to a large amplitude limit cycle solution. The states of combustion noise and intermittency observed in these turbulent combustors are well captured by the additive stochastic noise in the model. Through amplitude reduction, we analyze the underlying potential functions affecting the stability of the observed dynamical states. Finally, we make use of the Fokker-Planck equation, educing the effect of stochastic fluctuations on subcritical and secondary bifurcation. We conclude that a high enough intensity of stochastic fluctuations which transforms a subcritical bifurcation into an intermittency-facilitated continuous transition may have little effect on the abrupt nature of secondary bifurcation. Our findings imply the high likelihood of abrupt transitions in turbulent combustors possessing higher-order nonlinearities where turbulence intensities are disproportionate to the large amplitude limit cycle solution

Tracking the distance to criticality in systems with unknown noise

Many real-world systems undergo abrupt changes in dynamics as they move across critical points, often with dramatic and irreversible consequences. Much of the existing theory on identifying the time-series signatures of nearby critical points -- such as increased signal variance and slower timescales -- is derived from analytically tractable systems, typically considering the case of fixed, low-amplitude noise. However, real-world systems are often corrupted by unknown levels of noise which can obscure these temporal signatures. Here we aimed to develop noise-robust indicators of the distance to criticality (DTC) for systems affected by dynamical noise in two cases: when the noise amplitude is either fixed, or is unknown and variable across recordings. We present a highly comparative approach to tackling this problem that compares the ability of over 7000 candidate time-series features to track the DTC in the vicinity of a supercritical Hopf bifurcation. Our method recapitulates existing theory in the fixed-noise case, highlighting conventional time-series features that accurately track the DTC. But in the variable-noise setting, where these conventional indicators perform poorly, we highlight new types of high-performing time-series features and show that their success is underpinned by an ability to capture the shape of the invariant density (which depends on both the DTC and the noise amplitude) relative to the spread of fast fluctuations (which depends on the noise amplitude). We introduce a new high-performing time-series statistic, termed the Rescaled Auto-Density (RAD), that distils these two algorithmic components. Our results demonstrate that large-scale algorithmic comparison can yield theoretical insights and motivate new algorithms for solving important practical problems.Comment: The main paper comprises 18 pages, with 5 figures (.pdf). The supplemental material comprises a single 4-page document with 1 figure (.pdf), as well as 3 spreadsheet files (.xls