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These results serve as a guidance for the design of engines and to propose an effective control strategy, which is of great significance to aerospace-related fields. More interestingly and importantly, power-law relationships between the maximum amplitude and the noise parameters are uncovered, and the probability of successfully dodging a thermoacoustic instability is calculated. We find that both a relatively higher rate of change of parameters and change in the correlation time of the noise are beneficial to dodge thermoacoustic instability, while a relatively large noise intensity is a disadvantageous factor. A transient dynamical behavior is identified through a probability density analysis. Thus, in this work, based on a fundamental mathematical model of thermoacoustic systems driven by colored noise, the corresponding Fokker–Planck–Kolmogorov equation of the amplitude is derived by using a stochastic averaging method. It was speculated that such unwanted instability states may be dodged by changing the bifurcation parameters quickly enough, and compared with the white noise discussed in, colored noise with nonzero correlation time is more practical and important to the system. The thermoacoustic instability may lead to the disintegration of rocket engines, gas turbines and aeroengines, so it is necessary to design control measures for its suppression. This phenomenon is manifested in thermoacoustic systems. Tipping in multistable systems occurs usually by varying the input slightly, resulting in the output switching to an often unsatisfactory state.

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This study sheds light on combining stock network analysis and financial time series modeling and highlights that anomalous changes of a stock network can be important criteria for detecting crashes and predicting recoveries of the stock market. It is shown that our proposed method outperforms the benchmark log-periodic power-law model on detecting the 12 major crashes and predicting the subsequent price rebounds by reducing the false alarm rate. Experiments are conducted based on the New York Stock Exchange Composite Index from 4 January 1991 to.

Finally, we proposed a hybrid indicator to predict price rebounds of the stock index by combining the network anomaly metric and the visibility graph-based log-periodic power-law model. To calculate this metric, we design a prediction-guided anomaly detection algorithm based on the extreme value theory. Based on the observation of anomalous changes in stock correlation networks during market crashes, we extend the log-periodic power-law model with a metric that is proposed to measure network anomalies. This study proposes a framework to diagnose stock market crashes and predict the subsequent price rebounds. The out-of-sample tests quantified by error diagrams demonstrate the high significance of the prediction performance. We adapt the LPPL model to these negative bubbles and implement a pattern recognition method to predict the end times of the negative bubbles, which are characterized by rebounds (the mirror images of crashes associated with the standard positive bubbles).

We argue that similar positive feedbacks are at work to fuel these accelerated downward price spirals.

We then present the concept of “negative bubbles”, which are the mirror images of positive bubbles. The power of the LPPL model is illustrated by two recent real-life predictions performed recently by our group: the peak of the Oil price bubble in early July 2008 and the burst of a bubble on the Shanghai stock market in early August 2009. It embodies a positive feedback loop of higher return anticipations competing with negative feedback spirals of crash expectations.

The LPPL model considers the faster-than-exponential (power law with finite-time singularity) increase in asset prices decorated by accelerating oscillations as the main diagnostic of bubbles. By combining (i) the economic theory of rational expectation bubbles, (ii) behavioral finance on imitation and herding of investors and traders and (iii) the mathematical and statistical physics of bifurcations and phase transitions, the log-periodic power law (LPPL) model has been developed as a flexible tool to detect bubbles.

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