Gaussian Colored Noise Research Articles

Response of Gaussian color noise excited oscillators with inertia nonlinearity based on the radial basis function neural network method

Given the widespread utilization of oscillators featuring inertial nonlinear terms across engineering disciplines and the omnipresence of noise excitations, comprehending their response to noise excitation holds paramount importance. In contrast to Gaussian white noise, colored noise stands out as a more adept mathematical model for simulating ambient noise scenarios. The widely used stochastic averaging method is somewhat ineffective in dealing with the response of oscillators with inertial nonlinearity under Gaussian colored noise excitation, as it cannot display the saddle shaped characteristics of the steady-state probability density function (PDF) caused by inertial nonlinearity. To overcome the shortcomings, this paper adopts a semi-analytical method called a radial basis function neural network method (RBFNN) to address this problem. This method represents the solution of the Fokker-Plank-Kolmogorov (FPK) equation of the system as the sum of a series of weighted radial basis functions, and obtains the optimal weight coefficients through the Lagrange multiplier method. This article investigates two examples, one of which has inertial nonlinearity and the other has multiple potential wells. The effects of inertial nonlinearity coefficients and the delay coefficients of colored noise on response are explored. The mean square errors between Monte Carlo (MC) and RBF are presented, which indicates the RBF results perfectly agree with the MC predictions.

The transport of self-propelled ellipsoidal particles confined in 2D smooth corrugated channel

Directed transport of self-propelled ellipsoidal particles confined in a smooth corrugated channel with asymmetric potential and Gaussian colored noise is investigated. Effects of the channel, potential and colored noise on the system are discussed. Large noise intensity in the x-axis direction inhibits the transport in [Formula: see text] and [Formula: see text] directions. The directed transport speed [Formula: see text] has a maximum with increasing noise intensity in the y-axis direction. Proper size of the bottleneck is good for the directed transport of the ellipsoidal particles, but large or small size of bottleneck inhibits this directed transport. The transport reverse phenomenon appears with increasing load and self-propelled speed. Confined spherical particle is easier to produce directed transport than confined needlelike ellipsoid particle.

Directional transport of coupled self-propelled particles with potential and Gaussian colored noise

The transport phenomenon of coupled self-propelled particles with potential and colored noise is investigated. Large translational motion noise intensity is good for the directional transport in [Formula: see text]-direction, but large self-correlation time of translational motion noise will inhibit this transport. For proper value of the asymmetry parameter, coupled inert particles move always in the [Formula: see text]-direction with increasing angular noise intensity, but in coupled self-propelled particles appears current reversal phenomenon with increasing angular noise intensity. The average velocity has a maximum and a minimum with increasing spring constant k. For inert particles and particles with small self-propelled speed, large number of particles is good for directional movement, but the effect of coupling will become weak when the self-propelled speed is large.

Probabilistic Response Analysis of Nonlinear Tristable Energy Harvester Under Gaussian Colored Noise

Probabilistic Response Analysis of Nonlinear Tristable Energy Harvester Under Gaussian Colored Noise

Homoclinic–Heteroclinic Bifurcations and Chaos in a Coupled SD Oscillator Subjected to Gaussian Colored Noise

The so-called coupled smooth and discontinuous (SD) oscillator whose stiffness term leads to a transcendental function is a simple mass-spring system constrained to a straight line by two parameters, which are the dimensionless distances to the fixed point. This paper studies the homoclinic–heteroclinic chaos in a coupled SD oscillator subjected to Gaussian colored noise. In order to investigate the chaos thresholds analytically, the piecewise linearization approximation is used to fit the transcendental function. Stochastic nonsmooth Melnikov method with homoclinic–heteroclinic orbits is developed to study chaos thresholds of oscillators with tri-stable potential. Based on stochastic Melnikov process, the mean square criterion and the rate of phase-space flux function theory are used to study the chaotic motions of a coupled SD oscillator under weak noise and strong noise, respectively. The obtained results show that it is effective to use the piecewise linear approximation to analyze chaos in the coupled SD oscillator subjected to Gaussian colored noise. It also lays the foundation for chaos research of other nonsmooth mechanical vibration systems under random excitation.

Dynamical responses of a Gaussian colored noise-driven shape memory alloy oscillator with a periodic force

For shape memory alloy (SMA) oscillator, the stochastic transitions between low-amplitude and undesirable high-amplitude oscillations will affect the safety and normal use of the system. Therefore, the nonlinear dynamical responses of a SMA oscillator with both periodic and Gaussian colored noise excitations are investigated in this paper. Firstly, to reveal the influences of random factors, a stochastic SMA model with a Gaussian colored noise is introduced. Subsequently, the approximate analytical solution of the steady-state amplitude response under the deterministic case is given by the averaging method, and the time-averaging mean-square response of the stochastic SMA system is obtained by the stochastic averaging method. The deduced approximated analytical solutions are verified by Monte Carlo numerical results, and a good consistency is observed. Finally, the effects of the random excitation and temperature on the steady-state responses of the SMA system are explored. We uncover that the temperature influences the frequency of bifurcation points, which can disrupt the stable structure of the SMA oscillator. In addition, noise intensity and correlation time of the colored noise can induce the occurrence of stochastic transitions between low-amplitude and high-amplitude oscillations, which are visible through the steady-state probability density and time history diagrams. All these obtained results may provide theoretical guidance and valuable insights for avoiding catastrophic stochastic transitions in SMA materials.

Noise-induced synchronization stochastic resonance in Hodgkin–Huxley neurons

In this paper, based on the Hodgkin–Huxley (H-H) neuron model, three distinct types of noises are introduced into the input of the neuron. The noises are Gaussian white noise (GWN), Lévy noise and Colored noise, separately. The phenomenon of stochastic resonance under and above threshold in one-dimensional H-H neuron is discussed. First, the responses of H-H neuron to different input signal frequency, amplitude and constant offset stimulation are researched. The neuron membrane potential diagram and spectrum diagram are given. Then, the mutual information rate (MIR) is taken as the evaluation index. The results show that adding a noise of certain intensity can increase the response of neuron to the orignal signal and the MIR between the output and input of neuron has unimodal property, which proves the existence of stochastic resonance under and above the threshold. We also find that neurons have frequency selectivity and different transmission performance for different amplitude of input signals. In addition, the constant offset stimulus should not be too large. Finally, it is discovered that the adaptability of subthreshold stochastic resonance and suprathreshold stochastic resonance (SSR) characteristics characteristics of one-dimensional H-H neurons induced by three distinct types of noises is different. What is more important is that the results of this paper will be instrumental in the future researching on the affect of these three noises in the stochastic resonance of neuron networks.

Stochastic resonance in asymmetric bistable coupled network systems driven by Gaussian colored noise

In this work studied is the synergistic effect of asymmetric bistable coupled network systems under the action of Gaussian colored noise and periodic signal. The system is a network model consisting of a large number of oscillators. The interaction and change between individuals produce complex nonlinear behavior patterns. For further research, firstly, the original N-dimensional system is reduced and approximated by using the mean field theory, the unified colored noise approximation theory and the equivalent nonlinearization method. Secondly, the Langevin equation of simplified model is obtained through the slaving principle by using the two-state model theory to derive the theoretical expression of signal-to-noise ratio. It is found that the system produces the phenomenon of scale stochastic resonance. Finally, the effects of Gaussian color noise parameters, system parameters and periodic signal parameters on the stochastic resonance behavior of asymmetric coupled network systems are discussed. The results show that the increase of Gaussian colored noise correlation time and noise intensity can promote the scale stochastic resonance phenomenon; selecting appropriate coupling coefficient can achieve the optimal stochastic resonance effect. And the stochastic resonance phenomenon of the system driven by the Gaussian colored noise and the Gaussian white noise, respectively, are analyzed and compared with each other. Research result shows that Gaussian colored noise is more conducive to enhancing stochastic resonance phenomenon.

A Four Cumulant-Based Direction Finding Method for Bistatic MIMO Radar with Mutual Coupling

The mutual coupling effects of the transmitter and receiver are known to degrade the performance of direction finding for a bistatic Multiple Input Multiple Output (MIMO) radar system. A four cumulant-combinatorial matrix-based algorithm is proposed to estimate jointly the Direction Of Departure (DOD)and Direction Of Arrival (DOA) of targets under the coexistences of unknown mutual coupling and Gaussian colored noise. Firstly, the multiple groups of four cumulant matrices both on transmitter and receiver are constructed by using the Kronecker product and the banded symmetric Toeplitz characteristics of the mutual coupling matrices. The block four-cumulant matrix is further constructed by combining the transmitter and receiver four cunmulant matrices. Then the new matrix is combined to extract the transmit and receive shift invariance matrices by using the transmitter and receiver four cumulant matrices. The results illustrate that: The proposed method can estimate the DOD and DOA of the targets efficiently in the presence of the strong mutual coupling effect, and parameters are paired automatically without extra pairing operation. The parameter estimation performance of the proposed method is better than those of the existing methods under the strong mutual coupling effect conditions.

Stochastic approach for noise analysis and parameter estimation for RC and RLC electrical circuits

‎The main focus of this paper is to examine the effects of Gaussian white noise and Gaussian colored noise perturbations on the voltage of RC and RLC electrical circuits‎. ‎For this purpose‎, ‎the input voltage is assumed to be corrupted by the white noise and the charge is observed at discrete time points‎. ‎The deterministic models will be transferred to stochastic differential equations and these models will be solved analytically using Ito's lemma‎. ‎Random colored noise excitations‎, ‎more close to real environmental excitations‎, ‎so Gaussian colored noise is considered in these electrical circuits‎. ‎Scince there is not always a closed form analytical solution for stochastic differential equations‎, ‎then these models will be solved numerically based on the Euler‎- ‎maruyama scheme‎. ‎The parameter estimation for these stochastic models is investigated using the least square estimator when the parameters are missing data that it is a concern in electrical engeineering‎. ‎Finally‎, ‎some numerical simulations via Matlab programming are carried out in order to show the efficiency and accuracy of the present work‎.

Response of Cantilever Model with Inertia Nonlinearity under Transverse Basal Gaussian Colored Noise Excitation

Considering the curvature nonlinearity and longitudinal inertia nonlinearity caused by geometrical deformations, a slender inextensible cantilever beam model under transverse pedestal motion in the form of Gaussian colored noise excitation was studied. Present stochastic averaging methods cannot solve the equations of random excited oscillators that included both inertia nonlinearity and curvature nonlinearity. In order to solve this kind of equations, a modified stochastic averaging method was proposed. This method can simplify the equation to an Itô differential equation about amplitude and energy. Based on the Itô differential equation, the stationary probability density function (PDF) of the amplitude and energy and the joint PDF of the displacement and velocity were studied. The effectiveness of the proposed method was verified by numerical simulation.

Heterogeneous diffusion processes and nonergodicity with Gaussian colored noise in layered diffusivity landscapes.

Heterogeneous diffusion processes (HDPs) with space-dependent diffusion coefficients D(x) are found in a number of real-world systems, such as for diffusion of macromolecules or submicron tracers in biological cells. Here, we examine HDPs in quenched-disorder systems with Gaussian colored noise (GCN) characterized by a diffusion coefficient with a power-law dependence on the particle position and with a spatially random scaling exponent. Typically, D(x) is considered to be centerd at the origin and the entire x axis is characterized by a single scaling exponent α. In this work we consider a spatially random scenario: in periodic intervals ("layers") in space D(x) is centerd to the midpoint of each interval. In each interval the scaling exponent α is randomly chosen from a Gaussian distribution. The effects of the variation of the scaling exponents, the periodicity of the domains ("layer thickness") of the diffusion coefficient in this stratified system, and the correlation time of the GCN are analyzed numerically in detail. We discuss the regimes of superdiffusion, subdiffusion, and normal diffusion realisable in this system. We observe and quantify the domains where nonergodic and non-Gaussian behaviors emerge in this system. Our results provide new insights into the understanding of weak ergodicity breaking for HDPs driven by colored noise, with potential applications in quenched layered systems, typical model systems for diffusion in biological cells and tissues, as well as for diffusion in geophysical systems.

Kinetics of escape of ssDNA molecules from α-hemolysin nanopores: a dynamic disorder study

Experimental studies of the escape events of single-stranded DNA (ssDNA) molecules from the membrane embedded α-hemolysin nanopores exhibited the non-exponential kinetics. The data for homopolymers of adenine (dA27) and cytosine (dC27) revealed the profound influence of the sequence dependent DNA–pore interactions and the applied potentials on the escape kinetics. The non-exponential behavior is a signature of the presence of disorder in biochemical reactions. Here, we rationalize the experimental results with a dynamic disorder approach based on Zwanzig’s fluctuating bottleneck (FB) model as a general mechanism where the rate of the reaction is controlled by the passage of a molecule through the cross-sectional area of the vestibule of the α-hemolysin channel (the bottleneck). The stochastic fluctuations of the radius of the latter is represented by the fluctuations of the intersegment distance of a Rouse chain, the dynamics of which is described using a non-Markovian generalized Langevin equation with a memory kernel and Gaussian colored noise. Our analytical results essentially capture the effects of underlying energetics due to the sequence-dependent nature and applied potentials in reproducing the measured survival probabilities for dA27 and dC27 in an excellent way, thereby validating the application of a simple coarse-grained theory to address the thermal escape of biopolymers from bionanopores.

Mechanical Unfolding of Single Polyubiquitin Molecules Reveals Evidence of Dynamic Disorder.

Mechanical unfolding of single polyubiquitin molecules subjected to a constant stretching force showed nonexponentiality in the measured probability density of unfolding (waiting time distribution) and the survival probability of the folded state during the course of the measurements. These observations explored the relevance of disorder present in the system under study with implications for a static disorder approach to rationalize the experimental results. Here, an approach for dynamic disorder is presented based on Zwanzig’s fluctuating bottleneck (FB) model, in which the rate of the reaction is controlled by the passage through the cross-sectional area of the bottleneck. The radius of the latter undergoes stochastic fluctuations that in turn is described in terms of the end-to-end distance fluctuations of the Rouse-like dynamics using a non-Markovian generalized Langevin equation with a memory kernel and Gaussian colored noise. Our results are comprised of analytical expressions for the survival probability and waiting time distribution, which show excellent agreement with the experimental data throughout the range of the applied forces. In addition, by fitting the survival probabilities at different stretching forces, we quantify two system parameters, namely, the average free energy ΔGav and the average distance to the transition state Δxav, both perfectly recovered the experimental estimates. These agreements validate the present model of polymer dynamics, which captures the very essence of dynamic disorder in single-molecule pulling experiments.

Inverse stochastic resonance in Hodgkin–Huxley neural system driven by Gaussian and non-Gaussian colored noises

Inverse stochastic resonance (ISR) is the phenomenon of the response of neuron to noise, which is opposite to the conventional stochastic resonance. In this paper, the ISR phenomena induced by Gaussian and non-Gaussian colored noises are studied in the cases of single Hodgkin–Huxley (HH) neuron and HH neural network, respectively. It is found that the mean firing rate of electrical activities depends on the Gaussian or non-Gaussian colored noises which can induce the phenomenon of ISR. The ISR phenomenon induced by Gaussian colored noise is most obvious under the conditions of low external current, low reciprocal correlation rate and low noise level. The ISR in neural network is more pronounced and lasts longer than the duration of a single neuron. However, the ISR phenomenon induced by non-Gaussian colored noise is apparent under low noise correlation time or low departure from Gaussian noise, and the ISR phenomena show different duration ranges under different parameter values. Furthermore, the transition of mean firing rate is more gradual, the ISR lasts longer, and the ISR phenomenon is more pronounced under the non-Gaussian colored noise. The ISR is a common phenomenon in neurodynamics; our results might provide novel insights into the ISR phenomena observed in biological experiments.

The Stochastic Stability of Internal HIV Models with Gaussian White Noise and Gaussian Colored Noise

In this paper, the stochastic stability of internal HIV models driven by Gaussian white noise and Gaussian colored noise is analyzed. First, the stability of deterministic models is investigated. By analyzing the characteristic values of endemic equilibrium, we could obtain that internal HIV models reach a steady state under the influence of RTI and PI drugs. Then we discuss the stochastic stability of internal HIV models driven by Gaussian white noise and Gaussian colored noise, based on probability density functions. The functional methods are carried out to derive the approximate Fokker-Planck equation of stochastic internal HIV systems and further obtain the marginal probability density functions. Finally, numerical results show that the noise intensities have a great influence on uninfected cell, infected cell, and virus particles, for predicting the stability of stochastic dynamic systems subjected to Gaussian white noise and Gaussian colored noise.

The transport phenomenon of inertia Brownian particles in a periodic potential with non-Gaussian noise

The transport phenomenon (movement and diffusion) of inertia Brownian particles in a periodic potential with non-Gaussian noise is investigated. It is found that proper noise intensity Q will promote particles directional movement (or diffusion), but large Q will inhibit this phenomenon. For large value of Q, the average velocity [Formula: see text] (or the diffusion coefficient D) has a maximum with increasing correlation time [Formula: see text]. But for small value of Q, [Formula: see text] (or D) decreases with increasing [Formula: see text]. In some cases, for the same value of Q and the same value of [Formula: see text], non-Gaussian noise can induce particles directional movement (or diffusion), but Gaussian colored noise cannot.

Population system with coupling between non-Gaussian and Gaussian colored noise under Allee effect

We investigate a stochastic model for single species population growth with strong and weak Allee effects subjected to coupling between non-Gaussian and Gaussian colored noise as well as nonzero cross-correlation in between. Stationary probability distribution of population model is obtained depending on the Fokker–Planck equation. The mean first-passage time is also calculated in order to quantify the time of transition between survival state and extinction state with Allee effect in population. The intensity of non-Gaussian colored noise can induce phase transition, and population may be vulnerable to extinction due to the increase in the intensity of non-Gaussian colored noise. Whether Allee effect is strong or weak, the increase in Allee threshold will not contribute to the survival and stability of the population. Further, the phenomenon of resonant activation is firstly discovered in the study of population dynamics with Allee effect. These behaviors can be interpreted on the basis of a biological model of population evolution.

Stochastic heat equation with fractional Laplacian and fractional noise: existence of the solution and analysis of its density

In this paper we study a fractional stochastic heat equation on Rd(d ≥ 1) with additive noise ∂∂tu(t,x)=dδ_α_u(t,x)+b(u(t,x))+W˙H(t,x) where dδ_α_ is a nonlocal fractional differential operator and W˙H is a Gaussian-colored noise. We show the existence and the uniqueness of the mild solution for this equation. In addition, in the case of space dimension d=1, we prove the existence of the density for this solution and we establish lower and upper Gaussian bounds for the density by Malliavin calculus.

Pressure in an exactly solvable model of active fluid.

We consider the pressure in the steady-state regime of three stochastic models characterized by self-propulsion and persistent motion and widely employed to describe the behavior of active particles, namely, the Active Brownian particle (ABP) model, the Gaussian colored noise (GCN) model, and the unified colored noise approximation (UCNA) model. Whereas in the limit of short but finite persistence time, the pressure in the UCNA model can be obtained by different methods which have an analog in equilibrium systems, in the remaining two models only the virial route is, in general, possible. According to this method, notwithstanding each model obeys its own specific microscopic law of evolution, the pressure displays a certain universal behavior. For generic interparticle and confining potentials, we derive a formula which establishes a correspondence between the GCN and the UCNA pressures. In order to provide explicit formulas and examples, we specialize the discussion to the case of an assembly of elastic dumbbells confined to a parabolic well. By employing the UCNA we find that, for this model, the pressure determined by the thermodynamic method coincides with the pressures obtained by the virial and mechanical methods. The three methods when applied to the GCN give a pressure identical to that obtained via the UCNA. Finally, we find that the ABP virial pressure exactly agrees with the UCNA and GCN results.