Noise Correlation Time Research Articles

Effect of background noise characteristics on early warning indicators of thermoacoustic instability

In this work, we investigate the effects of background noise characteristics — specifically noise color (or correlation time) and intensity — arising from velocity-coupled and additive noise sources, on the early warning indicators (EWIs) of thermoacoustic instability. We employ a nonlinear reduced-order combustion dynamics model for our investigation. In the absence of noise, the nonlinear model undergoes a subcritical Hopf bifurcation, and our focus lies within the linearly stable region of the system (subthreshold region). The studied class of EWIs encompasses those derived from time series (variance), spectral analysis (coherence factor), Hurst exponent, and nonlinear methods (permutation entropy). We find that when the background noise is purely multiplicative, the trends in EWIs are primarily influenced by noise characteristics rather than the control parameter. Further, the EWIs cannot be estimated at low noise levels for large correlation times. In case of purely additive noise driven system, the coherence factor and variance are reliable EWIs across all range of investigated noise characteristics. The Hurst exponent can serve as effective EWI when the system features large noise correlation times, while permutation entropy is effective only when the system features small noise correlation time, i.e., where white noise assumption is acceptable. When the background noise includes contributions from both multiplicative and additive sources, coherence factor and variance emerges as the most reliable EWIs. These results provide insights for selecting appropriate EWIs to be employed in practical systems, considering potential variations in noise characteristics with simultaneous changes in combustor operating conditions.Novelty and significanceThis study conclusively shows that background noise (multiplicative and additive) characteristics – noise correlation time (color) and intensity – can significantly change the trends in early warning indicators (EWIs) for predicting the impending thermoacoustic oscillations. In gas turbine combustors, where thermoacoustic instability poses a critical challenge, inherent noise dynamics undergo variations with changing operating conditions and combustor designs. The development of effective EWIs to foresee the onset of thermoacoustic instability is crucial for preventing potential damage and ensuring the reliable operation of combustion systems. Previous works on stochastic dynamics of combustors simplify noise with the additive white noise assumption. Thus, our results demonstrated on a nonlinear combustion dynamics model advance the state-of-the-art with respect to the early prediction and control of thermoacoustic instability. The results provide valuable insights for selecting appropriate EWIs for engine monitoring in the absence of information on noise properties and their variation with operating conditions. This practical information is of direct relevance and interest to any gas turbine manufacturer/user.

Optimizing power-efficiency dynamics in ambient energy harvesting: Exploring trade-offs, linearity, and synergy

As the demand for low-power electronics and IoT devices grows, ambient energy harvesting appears to be a promising alternative for powering such systems in the long run. However, optimizing power and efficiency concurrently in such systems is challenging, involving balancing a number of variables. This paper investigates the optimization of power and efficiency in ambient energy harvesting systems focusing on nonlinear oscillator electromechanical harvesters subjected to multiplicative time-correlated ambient noise. Through extensive numerical simulations, we reveal distinct relationships between power and efficiency, influenced by various parameters. We observe autonomous stochastic resonance phenomena, elucidating a linear power-efficiency trend for small noise correlation time under fixed noise variance but limiting simultaneous power and efficiency optimization beyond a threshold. Under fixed noise strength, there is a trade-off between power and efficiency. Additionally, damping strength, piezoelectric parameters, and capacitor charging time impact power and efficiency linearly. These insights enhance understanding of power efficiency dynamics in ambient energy harvesting, thereby offering practical recommendations for parameter selection to maximize both power output and efficiency in the next generation of electronics.

Escape rate of a dimer under the influence of additive colored noise: Ornstein–Uhlenbeck process

The escape process of a dimer is studied in a bistable potential in the presence of additive Ornstein-Uhlenbeck (OU) noise. The escape rate of the dimer shows a non-monotonic dependence with respect to the harmonic spring constant, κ. However, an increase in the correlation time, τ of the OU noise causes a delay in the escape mechanism of the dimer. Another important observation is that the optimal spring constant, kopt, at which the escape rate peaks changes with the correlation time. In summary, a complex interplay between the harmonic force and the correlation time of the OU noise is involved in the escape process of the dimer.

Bifurcation Regulations Induced by Joint Noise in a Tri-Rhythmic Van Der Pol System

Tri-rhythmical nature has attracted extensive attention from scholars in describing the dynamical behaviors of self-sustained systems. In this paper, we consider a tri-rhythmic van der Pol system and give a bifurcation analysis of a stochastic tri-rhythmic self-sustained system under joint noise perturbation. Based on an approximate approach, we give the stationary probability density function of amplitudes, and we find that the noise and time-delay feedback, regulating the velocity time-delay feedback strength parameter, may not cause transitions among unimodal, bimodal and trimodal in the tri-rhythmic system. More stochastic bifurcations appear by regulating the time delay in this system. The system, surprisingly, undergoes five times of stochastic bifurcations when the time delay is monotonically increased. It is shown that the time delay is more sensitive to the tri-rhythmic system and a much stronger dependence on the system. From a biological point of view, the reaction rate of biological molecules can be enhanced or diminished by the change of the noise intensity or correlation time of Gaussian colored noise. More surprisingly, an increase in displacement feedback will delay the reaction rate; however, the effect of an increase in velocity feedback on the reaction rate depends on the time delay. A detailed research on the parameter space indicates that time delay and colored noise parameters can effectively control the multirhythmicity of the system. Finally, this paper verifies the effectiveness of the theoretical solution through the numerical results of Monte Carlo simulation of the original system. These results may help to further explore forking bifurcations in real-world applications. This research provides new insight into the understanding of nontrivial effects of joint noise on the tri-rhythmic system.

Effect of correlation time of combustion noise on early warning indicators of thermoacoustic instability.

In this paper, we analyze the effects of finite correlation time (noise color) of combustion noise on noise-induced coherence and early warning indicators (EWIs) via numerical and experimental studies. We consider the Rijke tube as a prototypical combustion system and model combustion noise as an additive Ornstein-Uhlenbeck process while varying noise intensity and correlation time. We numerically investigate corresponding effects on coherence resonance and multi-fractal properties of pressure fluctuations. Subsequently, we experimentally validate results and elucidate the influence of noise color and intensity on trends in coherence resonance and multi-fractal measures that can be expected in a practical scenario using an electroacoustic simulator. We find that the coherence factor, which quantifies the relative contribution of coherent oscillations in a noisy signal, increases as the system approaches the thermoacoustic instability-irrespective of the correlation time. It works at most levels of combustion noise (except for too low and too high noise levels). The Hurst exponent reduces as the system approaches thermoacoustic instability only when the correlation time is small. These results have implications on the prediction and monitoring of thermoacoustic instability in practical combustors.

Detection of colored noise correlation time with Josephson junctions

It is shown that the effect of an exponentially correlated noisy source on the escape times of a Josephson junction depends upon the noise correlation time. A careful analysis reveals that two effects are preeminent: the increase of the stability (as measured by the average escape time from the metastable state) and the increase of the extreme events in the tails of the escape time distribution. The changes occur if the noise correlation exceeds the characteristic time of the Josephson junction. As the superconducting device is very fast, with a characteristic time easily below the nanosecond, the device is responsive even to very short noise correlation. We speculate that it is thus possible to estimate in situ noise correlation with a superconducting Josephson junction through the measurements of the escapes, without a high-frequency sampling of the noise. Moreover, as the superconducting circuit can be cooled as much as necessary, the device offers the further advantage to be intrinsically very quiet.

An underdamped and delayed tri-stable model-based stochastic resonance

Stochastic resonance (SR) is investigated in an underdamped tri-stable potential system driven by Gaussian colored noise and a periodic excitation, where both displacement and velocity time-delayed states feedback are considered. It is challenging to study SR in a second-order delayed multi-stable system analytically. In this paper, the improved energy envelope stochastic average method is developed to derive the analytical expressions of stationary probability density (SPD) and spectral amplification. The effects of noise intensity, damping coefficient, and time delay on SR are analyzed. The results show that the shapes of joint SPD can be adjusted to the desired structure by choosing the time delay and feedback gains. For fixed time delay, the SR peak is increased for negative displacement or velocity feedback gain. Meanwhile, the SR peak is decreased while the optimal noise intensity increases with increasing correlation time of noise. The Monte Carlo simulations (MCS) confirm the effectiveness of the theoretical results.

Impact of Non-Gaussian Noise and Time Delay on Stability and Stochastic Resonance for a FitzHugh-Nagumo Neural System Subjected to a Multiplicative Periodic Signal

In this paper, we focus on the investigations on the stochastic stability and the stochastic resonance (SR) phenomena for a FitzHugh-Nagumo system with time delay induced by a multiplicative non-Gaussian colored noise and an additive Gaussian colored noise. By use of the fast descent method, the unified colored noise approximation and the two-state theory for the SR, the stationary probability density function (SPDF) and the signal-to-noise ratio (SNR) caused by different noise terms and time delay are explored. The investigation results indicate that the two noise intensities, time delay and the departure parameter from the Gaussian noise can all reduce the probability density around the two stable states and destroy the stability of the neural system; while the two noise correlation times [Formula: see text] and [Formula: see text] can both improve the probability density around both stable states and reinforce the biological stability of the neural system. As regards the SNR, it is found that the two noise intensities and the departure coefficient can all weaken the SR effect, while time delay [Formula: see text] and the correlation time [Formula: see text] of the multiplicative noise will always magnify the SR phenomenon. It is worth to mention that the correlation time [Formula: see text] of the additive noise can stimulate the SR effect, but not alter the maximum of the SNR.

Impact of Double Time Delays on Regime Shift and Stochastic Resonance for a Species Population System Driven by Colored Correlated Multiplicative and Additive Noises

In this paper, the characteristics for the state transition between the boom state and the extinction one, varius stochastic resonance (SR) phenomena for a species population system induced by double time delays and colored cross-correlated Gaussian noises are investigated. The control of the species population system has an important effect on ecological balance and the development of human living environment. By means of the numerical calculation of the mean extinction time and the stationary probability density function (SPDF), we can figure out some fact: both external and internal noises, the noise correlation strength and its correlation time, one time delay can all reduce the ecological stability and shorten the survival time of the system; while the other time delay can enhance significantly the biological stability and postpone the extinction of the population. For the mean extinction time (MET) of the species population, it is found that noise correlation strength, time delay, the noise correlation time can all reduce the peak value of the MET, which leads to the reduction of the life cycle. Inversely, the other time delay will not change the maximum of the MET, and induce the peak value to shift to the left. With respect to the SNR, it is discovered that in the noise correlation strength and its correlation time, one-time delay mainly plays a prominent role in inhibiting the SR effect. On the contrary, the multiplicative noise can result in the enhancement of the resonant peak. In particular, the other time delay and the additive noise can stimulate the SR effect for the SNR during the initial period. However, the further increase of them will bring about the reduction of the resonant phenomenon.

Tumor state transitions driven by Gaussian and non-Gaussian noises

Tumor state transitions between the excited (high-concentration) and nonexcited (low-concentration) basins under the Gaussian white noise and non-Gaussian colored noise are investigated via the most probable steady states (MPSS) and the first escape probability (FEP)-based stochastic basin of attraction (SBA), respectively. Reducing the non-Gaussian colored noise and then utilizing the unified colored noise approximation (UCNA), the Markov system is derived. The extremal controlling equation of stationary probability density function (SPDF) is derived to analyze the impacts of noise on transitions in terms of MPSS. The existence of the ‘color’ of the non-Gaussian colored noise induces the reappearance of the uncorrelated additive white noise parameter that had vanished from the extremal controlling equation, completely reversing the inability of the uncorrelated additive Gaussian white noise to operate on transitions. The FEP-dependent SBA characterizing the excited basin stability is performed to further analyze the role of noise on the likelihood of escaping to the nonexcited state. Results show that the cross-correlated noises play a dual role in regulating SBA. The increased SBA indicating more difficulty to escape to the nonexcited state reflects a worse therapeutic effect. Therefore, enhancing the negatively correlated noise intensities and augmenting the non-Gaussian noise correlation time is essential for destabilizing the excited basin and achieving optimal therapeutic efficacy.

Influence of colored cross-correlated noise on stochastic resonance for an underdamped bistable system subjected to multiplicative and additive noises

The stochastic resonance (SR) behavior for an underdamped bistable system with colored cross-correlated noise between multiplicative and additive noise is investigated. The stationary probability density is obtained under the condition of the detailed balance. The expressions for the signal-to-noise ratios (SNRs) for two initial states is deduced by applying two-state theory under the adiabatic condition. The analysis result indicates that the SR phenomenon takes place when the SNRs vary with the coupling strength and the correlation time of the cross-correlated noise. Double SR phenomenon occurs on SNRs’ curves with the increase of the strength of the additive noise. One resonance peak exists when the SNRs change with the damping coefficient and with the intensity of the multiplicative noise.

Finite-time correlations boost large voltage angle fluctuations in electric power grids

Decarbonization in the energy sector has been accompanied by an increased penetration of new renewable energy sources in electric power systems. Such sources differ from traditional productions in that, first, they induce larger, undispatchable fluctuations in power generation and second, they lack inertia. Recent measurements have indeed reported long, non-Gaussian tails in the distribution of local voltage frequency data. Large frequency deviations may induce grid instabilities, leading in worst-case scenarios to cascading failures and large-scale blackouts. In this article, we investigate how correlated noise disturbances, characterized by the cumulants of their distribution, propagate through meshed, high-voltage power grids. For a single source of fluctuations, we show that long noise correlation times boost non-Gaussian voltage angle fluctuations so that they propagate similarly to Gaussian fluctuations over the entire network. However, they vanish faster, over short distances if the noise fluctuates rapidly. We furthermore demonstrate that a Berry–Esseen theorem leads to the vanishing of non-Gaussianities as the number of uncorrelated noise sources increases. Our predictions are corroborated by numerical simulations on realistic models of power grids.

Effect of color cross-correlated noise on the growth characteristics of tumor cells under immune surveillance.

Based on the Michaelis-Menten reaction model with catalytic effects, a more comprehensive one-dimensional stochastic Langevin equation with immune surveillance for a tumor cell growth system is obtained by considering the fluctuations in growth rate and mortality rate. To explore the impact of environmental fluctuations on the growth of tumor cells, the analytical solution of the steady-state probability distribution function of the system is derived using the Liouville equation and Novikov theory, and the influence of noise intensity and correlation intensity on the steady-state probability distributional function are discussed. The results show that the three extreme values of the steady-state probability distribution function exhibit a structure of two peaks and one valley. Variations of the noise intensity, cross-correlation intensity and correlation time can modulate the probability distribution of the number of tumor cells, which provides theoretical guidance for determining treatment plans in clinical treatment. Furthermore, the increase of noise intensity will inhibit the growth of tumor cells when the number of tumor cells is relatively small, while the increase in noise intensity will further promote the growth of tumor cells when the number of tumor cells is relatively large. The color cross-correlated strength and cross-correlated time between noise also have a certain impact on tumor cell proliferation. The results help people understand the growth kinetics of tumor cells, which can a provide theoretical basis for clinical research on tumor cell growth.

Tipping points in spatial ecosystems driven by short-range correlated noise.

Complex spatial systems can experience critical transitions or tippings on crossing a threshold value in their response to stochastic perturbations. While previous studies have well characterized the impact of white noise on tipping, the effect of correlated noise in spatial ecosystems remains largely unexplored. Here, we investigate the effect of both multiplicative and additive Ornstein-Uhlenbeck (OU) correlated noise on the occurrence of critical transitions in spatial ecosystems. We find that decreasing the noise correlation time of OU (exponentially correlated) noise aggravates the chance of critical transitions in spatiotemporal ecological systems. Our results hold good and are supported by the analysis of three well-studied spatial ecological models of varying nonlinearity. We also compute spatial early warning indicators (e.g., spatial variance, spatial skewness, and spatial correlation) to determine their reliability in anticipating tipping points with variations in noise correlation. The indicators of critical transitions exhibit mixed success in forewarning the occurrence of a tipping point, as indicated by the distribution of Kendall's rank correlation.

Phase rearrangement shell mapping in single-span nonlinear optical fiber communication system

The importance of single-span systems is emphasized by the fact that they currently constitute the fastest growing section of the fiber communication markets. In such systems, the correlation time of the nonlinear phase noise is too short to allow the nonlinear phase-noise mitigation such as adaptive phase-recovery or equalization, while it is sufficient for the shaping approaches. The multi-dimensional (MD) coded-modulation based standard shell mapping (SM) algorithm can realize signal shaping and mitigate the nonlinear phase noise in single-span optical fiber transmissions. However, the standard SM algorithm has poor performance resulted from the decoding penalty in two-dimensional (2D) decoding with a low signal-to-noise (SNR) ratio. In this paper, we propose a phase rearrangement shell mapping (PRSM) algorithm to improve the 2D decoding performance by rearranging the constellation sequence in order of signal phase for each shell. This proposed technique is verified through the simulation of a 5 × 70-GBaud 4D/8D modulated signal in a 5-channel wavelength division multiplexing (WDM) system after a single span transmission of 100-km standard single mode fiber (SSMF). We also conduct an experiment to verify the performance of 3 × 45-GBaud 4D/8D-16QAM with PRSM algorithm after unrepeated transmission of 100-km SSMF. The simulation and experimental results show that the Q2 factor gain of PRSM-4D/8D-16QAM is about 0.2∼0.3 dB compared with standard SM-4D/8D-16QAM using 2D decoding. There is about 1.6-dB or 1.8-dB launch optical power (LOP) dynamic range improvement by PRSM compared with standard SM for 4D/8D-16QAM, respectively.

Stochastic behaviors of an improved Gompertz tumor growth model with coupled two types noise

In this work, an improved Gompertz tumor growth model has been introduced. The expressions of steady probability distributions (SPD) of stochastic Gompertz tumor growth models are studied by using the technique of Fokker-Planck equation (FPE), and their dynamic behaviors are also further investigated. Moreover, the expressions for mean, variance, skewness, as well as the mean first-passage time (MFPT) also have been derived. And the influence of noise intensity, correlation coefficient, and noise correlation time of SPD are further analyzed. It is worthy noting that the colored noise intensity has an important impact on SPD. Furthermore, adjusting birth and death parameters also significantly impact SPD, MFPT, mean, variance as well as skewness.

Ambient energy harvesting from colored multiplicative noise

The present study examines the effect of multiplicative Gaussian noise on the conversion of vibration energy into electrical energy. Nonlinear oscillators are considered in the present model. The results demonstrate that similar to additive noise case, the harvesting of the vibration energy to electricity could be good if the voltage can closely track the advancement of the amplitude of the oscillator. Another observation is that the power of the micro-electromechanical systems (MEMS) goes through a maximum as with inverse of noise correlation time. Further, the position of the maxima in power generation occurs for higher noise correlation time due to the space dependency associated with the multiplicative Gaussian noise. Along with these, we have investigated how the power generation is dependent on the environment, properties of the oscillating systems and the piezoelectric dynamics.

System Parameter Identification of a Colored-Noise-Driven Rijke Tube Simulator

Abstract In this paper, we report an experimental investigation on the influence of colored noise (generated by the Ornstein–Uhlenbeck (OU) process) on thermoacoustic coupling in an electro-acoustic Rijke tube simulator. In the absence of noise, the simulator exhibits subcritical Hopf bifurcation. Although noise in a practical system has a finite correlation time, yet the system identification methods are based on the assumption of white noise. In this study, we investigate the effects of correlation time, and intensity of colored noise on the estimation of the growth rates of acoustic oscillations determined using Fokker–Planck equation in stable, bistable, and linearly unstable regions. Subsequently, we compare the findings against results obtained considering white noise approximations. We report the observed deviation of the estimated growth rates from the actual values as a function of noise intensity and correlation time. We find that with the colored noise model, the deviation in the estimated growth rates lies within the range of 0–10% compared to the deviation of 5–25% observed considering the white noise approximation. We also report that increasing noise amplitudes leads up to a deviation of approximately 30% in the estimated growth rates from the actual values.

Stochastic Stability and Bifurcation of Centrifugal Governor System Subject to Color Noise

This paper presents a detailed investigation of stochastic stability and complex dynamics of a centrifugal governor system with approximately uniform color noise. The centrifugal governor system excited by noise is transformed into Itö equation using polar coordinate transformation and stochastic average method. According to the boundary conditions of attraction and repulsion, the stochastic stability is ensured. In addition, analyses concerning the influence of parameter variation and validity are carried out by employing numerical method. The results manifest that the effects of noise intensity and correlation time on stationary probability density are opposite. The amplitudes of probability density finally tend to a limit value, and the only limit cycle appears, which shows that when the bifurcation occurs, the trivial solution of the system converges to a limit cycle with a higher probability. Finally, the two-dimensional parameter bifurcation analysis of the centrifugal governor system subject to color noise excitation is studied. An interesting distribution characteristic is found that the periodic region is organized according to the sequence of Stern–Brocot trees, and this typical characteristic is a universal characteristic of the system on the two parameter planes. Furthermore, it is concluded that based on the largest Lyapunov exponent diagram and bifurcation diagram in two-dimensional parameter plane, the effects of noise intensity and correlation time on the periodic oscillation state are opposite, but both of them can transform the quasi-periodic oscillation into periodic oscillation. It should be emphasized that with the increase of noise intensity, the coexisting oscillation behavior of the centrifugal governor system will change, which is manifested by the destruction of coexisting attractors and the generation of chaotic attractors.

Propagation of non-Gaussian voltage angle fluctuations in high-voltage power grids

Recent measurements have reported non-Gaussian tails in the distribution of frequency data in electric power grids. Large frequency deviations may induce grid instabilities and it is therefore crucial to understand how noise disturbances with long, non-Gaussian tails propagate. Here, we investigate how fluctuations in power feed-in, characterized by non-zero cumulants of their distribution, propagate through high-voltage power grids. Unlike previous investigations which focused on the white-noise limit, we consider the limit of long noise correlation time, where power feed-in fluctuates over times longer than the inherent dynamical time scales of the grid - the relevant regime for large-scale, high-voltage distribution grids. We show that in this limit, the skewness and kurtosis of the power feed-in distribution propagate similarly as its variance, independently of the distribution of inertia. Non-Gaussianities from individual sources of noise therefore persist throughout the entire network. This finding is corroborated by numerical results on a realistic model of the synchronous grid of continental Europe.