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In this paper, we investigate the stochastic transitions in a bistable Schlögl chemical reaction subjected to a nonextensive statistical noise (NESN) and a Gaussian white noise. The NESN is a correlated noise, and able to describe heavy-tailed noise distributions with certain deviation degree relative to Gaussian colored noise. With a unified-colored noise approximation method, the theoretical results of the steady state probability density function (PDF) and mean first passage time (MFPT) are obtained to demonstrate the influences of NESN. The validity of theoretical results has been confirmed by numerical simulations of the PDFs. Besides, stochastic P-bifurcation is discovered based on the changing of PDFs. The results show that the NESN will keep the conversion rate of substrate and the generation rate of products in the low state. Finally, from the MFPT, we find that the intensity and deviation degree of NESN, and Gaussian noise intensity can decrease the switching time between two states.

In this paper, we study the effect of Gaussian white noise on escapes in Helmholtz oscillator whose basin of attraction has a integral structure in the absence of noise. Four different ways are adopted to implement the noise perturbation, namely, the external forcing, damping coefficient, strength and phase of the external forcing of oscillator have been subjected to noise, respectively. It is found that in the first, second and third cases of noise perturbation, when the noise intensity D > 0, the escapes are triggered and as D is further increased the escapes become more serious. Whereas, in the last case of noise perturbation, no escape can be observed, even with lager value of noise intensity. These phenomena imply that noise-perturbed external forcing, damping coefficient and forcing strength will make it easier to induce and enhance escapes in dynamical Helmholtz oscillator.

This work examines the role of spatial impurity extension (SIE) on a few nonlinear optical (NLO) properties of impurity-doped GaAs quantum dot (QD). The NLO properties considered are the nonlinear optical rectification (NOR), the second harmonic generation (SHG) and the third harmonic generation (THG). The study also includes 1 (GWN) which has been incorporated into the doped QD through additive and multiplicative pathways. The study uncovers the delicate interplay between GWN and SIE which ultimately fabricates the said NLO properties. NOR, SHG and THG manifest red-shift with enhancement of SIE both with and without GWN. The study also shows possibilities of attaining maximum NLO response by prudently applying GWN in a particular pathway. Depending upon the particular NLO property concerned, the presence of noise and its mode of application can either deplete or amplify it to different extents with respect to the noise-free situation.

A kriging method is presented as a spatial filter for smoothing gray-scale images degraded by Gaussian white noise. The concepts are based on the analysis of semivariances, the linear combination scheme of kriging, and fuzzy sets. Application of fuzzy sets allows a gradual transition between two boundaries of semivariance levels as a criterion for smoothing the pixel values. This fuzzy thresholding also allows some degree of flexibility to suit various desired results for particular problems. Experimental results obtained by the fuzzy kriging filter are smoother and still preserve edges compared with those by the adaptive Wiener filter.

This work investigates the global stability of cellular neural networks with distributive time delays and noise perturbations. Two types of noise perturbations are considered in the system, one is internal noise and another is external noise. We show that the deterministic system preserves the globally exponential stability when it is perturbed by the internal noise with small noise strength. On the other hand, the deterministic system can only preserve weaker stability when it is perturbed by the external noise. We also provide some numerical simulations to support our theoretical analysis.

Detecting chaos in heavy-noise environments is an important issue in many fields of science and engineering. In this paper, first, a new criterion is proposed to recognize chaos from noise based on the distribution of energy. Then, a new method based on stationary wavelet transform (SWT) is presented for chaos detection that is recommended for data that contain more than 60% noise. This method is dependent on the distribution of signal’s energy in different frequency bands based on SWT for chaos detection which is robust to noisy environments. In this method, the effect of white noise and colored noise on the chaotic system is considered. As a case study, the proposed method is applied to detect chaos in two different oscillators based on memristor and memcapacitor. The simulation results are used to display the main points of the paper.

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.

This paper studies the effect of Gaussian white noise on homoclinic bifurcations and chaotic dynamics of a bistable, vibro-impact Smooth-and-Discontinuous (SD) oscillator. First, the SD oscillator is reproduced and generalized by installing a slider on a fixed rod, so the slider is connected by a pair of linear springs initially pre-compressed in the vertical direction to achieve bistable vibration characteristics, and two screw nuts are installed on the rod as two adjustable bilateral rigid constraints to generate the vibro-impact. A discontinuous dynamical equation with a map defined on switching boundaries to represent velocity loss during each collision is derived to describe the vibration pattern of the bistable, vibro-impact SD oscillator through studying the persistence of the unique, unperturbed, nonsmooth, homoclinic structure. Second, the general framework of random Melnikov process for a class of bistable, vibro-impact systems contaminated with Gaussian white noise is derived and employed through the corresponding Melnikov function to obtain the necessary parameter thresholds for homoclinic tangency and possible chaos of the bistable, vibro-impact SD oscillator. Third, the effectiveness of a semi-analytical prediction by the Melnikov function is verified using the largest Lyapunov exponent, bifurcation series, and 0–1 test. Finally, the sensitivity to the initial values of chaos is verified by the fractal attractor basins, and the influence of the Gaussian white noise on periodic and chaotic structures is studied through Poincaré mapping to show the rich dynamical geometric structures.

This paper explores the role of spatial extension of impurity (SEI) on two nonlinear optical (NLO) properties and two other related properties of impurity doped GaAs quantum dot (QD). The NLO properties include the electro-optic effect (EOE) and the third-order nonlinear optical susceptibility (TONOS) whereas the other two related properties being total optical dielectric function (TODF) and the intraband transition lifetime (ITL). The study also involves Gaussian white noise (GWN) which has been introduced to the doped QD through additive and multiplicative pathways. The investigation unveils the subtle interplay between GWN and SEI which ultimately fine-tunes the said NLO properties. TODF, EOE and TONOS manifest red-shift with enhancement of SEI both with and without GWN. The ITL, on the other hand, exhibits monotonic growth with increase in SEI under all conditions. Moreover, the application of GWN causes noticeable quenching of the NLO properties. And the quenching becomes more stringent in presence of multiplicative noise than its additive counterpart.

We investigate the properties of the Wick square of Gaussian white noises through a new method to perform nonlinear operations on Hida distributions. This method lays in between the Wick product interpretation and the usual definition of nonlinear functions. We prove an Itô-type formula and solve stochastic differential equations driven by the renormalized square of the Gaussian white noise. Our approach works with standard assumptions on the coefficients of the equations, global Lipschitz continuity, and produces existence and uniqueness results in the space where the noise lives. The linear case is studied in details and positivity of the solution is proved.

The suspension system of the maglev train will inevitably be disturbed by random factors such as track irregularities, which will cause random vibration of the train and even affect the safety of the train. Therefore, the research on the response and reliability of suspension system under random disturbance is crucial to its safe operation. In this paper, the response and the reliability of a suspension system are investigated using the theory and methods of stochastic dynamics. First, the magnetic gap and vertical velocity of the suspension system are random due to the random disturbance. Thus, the stochastic response is investigated through the probability density function (PDF), which is governed by the Fokker–Planck–Kolmogorov (FPK) equation corresponding to suspension system. And the response statistics of the suspension system under different system parameters and disturbance intensities are analyzed by solving the corresponding FPK equation using the finite difference (FD) method. Second, random disturbance may lead to the vibration amplitude of the suspension system exceeding the safety domain and causing safety incident, which is a reliability problem in stochastic dynamical systems. The probability that response is still in the safety domain at a given time is the reliability function of the suspension system, which is governed by the backward Kolmogorov equation. The time that the response first passes through the safety domain is the first-passage time, and its n-order moment satisfies the generalized Pontryagin equation. Reliability of the suspension system is analyzed by solving these governing equations using the FD method. In addition, the results of the FD method in this paper are verified with those of Monte Carlo (MC) simulation, which shows the correctness of FD method.

In this paper, an improved stochastic averaging method is introduced to solve a monostable piezoelectric vibration energy harvester (VEH) model subject to basal Gaussian white and colored noises. First, the electric voltage is expressed as perturbations to the linear stiffness term and the damping term of the vibrational equation. Second, the original model is simplified into an Itô differential equation by the improved stochastic averaging method. Third, stationary solution of the Fokker–Plank–Kolmogorov function driven from the Itô equation is obtained, so that the stationary probability density functions (PDFs) of the displacement, velocity, and electrical voltage are obtained. With the help of the Monte Carlo, the effectiveness of these theoretical predictions is verified. Finally, the impacts of the nonlinear stiffness coefficient and nonlinear inertia coefficient on the responses are compared. It is found that the stiffness nonlinearity has stronger effect on the outputn electric voltage than the inertia nonlinearity.

In this paper, a study was conducted regarding the stability of a binary airfoil under stochastic disturbances with feedback control. A new stochastic controller has been developed and investigated via Monte Carlo simulation. In order to design the new controller, the original system is transformed to the second-order moment one. Then the optimal control parameters can be obtained by searching for the effectual control parameters in the second-order moment system. Finally, the numerical simulations are included to visualize the good performances in flutter suppression.

We investigate the Gross–Pitaevskii equation with the tilted bichromatical optical lattice potential for finding the dynamics of a Bose–Einstein condensate system under the Gaussian white noise. We construct the Poincare sections of system based on the relations between the system parameters and solution behaviors to understand how its shock wave like dynamic could be affected by the noise. Also the hierarchical cluster analysis method investigation of the system is presented.

A detailed examination of the effect of harvesting on a population has been carried out by extending the standard cubic deterministic model by considering a population under Allee effect with a quadratic function representing harvesting. Weak and strong Allee effect transitions, carrying capacity, and Allee threshold change according to harvesting are first discussed in the deterministic model. A Fokker–Planck equation has been obtained starting from a Langevin equation subject to correlated Gaussian white noise with zero mean, and an Approximate Fokker–Planck Equation has been obtained from a Langevin equation subject to correlated Gaussian colored noise with zero mean. This allowed to calculate the stationary probability distributions of populations, and thus to discuss the effects of linear and nonlinear (Holling type-II) harvesting for populations under Allee effect and subject to white and colored noises, respectively.

Ramanujan Sums (RS) have been found to be very successful in signal processing recently. However, as far as we know, the RS have not been applied to image analysis. In this paper, we propose two novel algorithms for image analysis, including moment invariants and pattern recognition. Our algorithms are invariant to the translation, rotation and scaling of the 2D shapes. The RS are robust to Gaussian white noise and occlusion as well. Our algorithms compare favourably to the dual-tree complex wavelet (DTCWT) moments and the Zernike's moments in terms of correct classification rates for three well-known shape datasets.

This paper concerns the evolutionary transitional probability density functions (PDFs) and the mean upcrossing rates (MCRs) of responses of the non-smoothly damped stochastic oscillator under both modulated random and harmonic excitations. The damping expressed by a non-smooth function of velocity is considered in the stochastic dynamical systems. A procedure is presented by making the time-related weighted residue of the formulated Fokker–Planck–Kolmogorov (FPK) equation vanish in the weak sense. Thus, a group of stochastic ordinary differential equations regarding the undetermined parameters in the assumed solution can be formulated and solved numerically. Three cases are studied under different conditions. The achieved evolutionary transitional PDF and MCR solutions attest to the outstanding effectiveness of the presented procedure even in the tail ends of the PDF solutions. Moreover, the achieved solutions of the stochastic dynamical systems are found to have significant non-Gaussian behaviors due to the interaction of the random and harmonic excitations and the system nonlinearity. Notably, the presented procedure greatly improves the running efficiency when compared with Monte Carlo simulation.

The generation and propagation of tsunami caused by a curvilinear stochastic seismic faulting driven by two Gaussian white noise processes in the $x$- and $y$-directions are investigated. This model is used to study the tsunami build up and propagation during and after a realistic curvilinear source model represented by a random spreading slip-fault model. The amplification of tsunami amplitudes builds up progressively as time increases during the generation process due to wave focusing while the maximum wave amplitude decreases with time during the propagation process due to geometric spreading and dispersion. The increase of the normalized noise intensities on the random bottom leads to an increase in oscillations and amplitude of the free surface elevation. Tsunami waveforms using linearized shallow water theory for constant water depth are analyzed analytically by transform methods. The mean and variance of the random tsunami waves are derived and analyzed as a function of the noise intensities, propagated uplift length and the average depth of the ocean along the generation and propagation path.

This work presents a discussion on the probability density function of Intrinsic Mode Functions (IMFs) provided by the Empirical Mode Decomposition of Gaussian white noise, based on experimental simulations. The influence on the probability density functions of the data length and of the maximum allowed number of iterations is analyzed by means of kernel smoothing density estimations. The obtained results are confirmed by statistical normality tests indicating that the IMFs have non-Gaussian distributions. Our study also indicates that large data length and high number of iterations produce multimodal distributions in all modes.

We review some recent developments in white noise analysis and quantum probability. We pay a special attention to spaces of test and generalized functionals of some Lévy white noises, as well as to the structure of quantum white noise on these spaces.