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Mathematical models about changes of plankton populations can help study trendence in marine ecosystems. Random noise is common in marine ecosystems, arising from internal dynamics or external environmental factors. To study the dynamic characteristics of oceanic systems more accurately, the state estimation of the phytoplankton–zooplankton model (PZ model’s) Julia sets affected by additive noise and multiplicative noise is performed to mitigate its effects. The study focuses on two state estimation methods: filtering and prediction, drawing inspiration from particle filtering and moving average principles. The final image analysis demonstrates that both methods effectively reduce noise interference.

The photon anticorrelation experiment is simulated with the unquantized Maxwell field by implementing realistic instrument bandwidth and signal to noise ratio. The results strongly violate an inequality believed to be satisfied by all semiclassical theories, which opens up a classical interpretation that rivals quantum optics and casts doubt on potential quantum computer applications that involve optical fields.

The performance of one-way and two-way continuous variable quantum key distribution (CVQKD) protocols can be increased by adding some noise on the reconciliation side. In this paper, we propose to add noise at the reconciliation end to improve the performance of CVQKD with untrusted source. We derive the key rate of this case and analyze the impact of the additive noise. The simulation results show that the optimal additive noise can improve the performance of the system in terms of maximum transmission distance and tolerable excess noise.

This paper presents main paradigms of research for feature extraction methods to further augment the state of art in speaker recognition (SR) which has been recognized extensively in person identification for security and protection applications. Speaker recognition system (SRS) has become a widely researched topic for the last many decades. The basic concept of feature extraction methods is derived from the biological model of human auditory/vocal tract system. This work provides a classification-oriented review of feature extraction methods for SR over the last 55 years that are proven to be successful and have become the new stone to further research. Broadly, the review work is dichotomized into feature extraction methods with and without noise compensation techniques. Feature extraction methods without noise compensation techniques are divided into following categories: On the basis of high/low level of feature extraction; type of transform; speech production/auditory system; type of feature extraction technique; time variability; speech processing techniques. Further, feature extraction methods with noise compensation techniques are classified into noise-screened features, feature normalization methods, feature compensation methods. This classification-oriented review would endow the clear vision of readers to choose among different techniques and will be helpful in future research in this field.

A nonadditive generalization of Klimontovich's S-theorem [Bagci, 2008] has recently been obtained by employing Tsallis entropy. This general version allows one to study physical systems whose stationary distributions are of the inverse power law in contrast to the original S-theorem, which only allows exponential stationary distributions. The nonadditive S-theorem has been applied to the modified Van der Pol oscillator with inverse power law stationary distribution. By using nonadditive S-theorem, it is shown that the entropy decreases as the system is driven out of equilibrium, indicating self-organization in the system. The allowed values of the nonadditivity index q are found to be confined to the regime (0.5,1].

Phantom attractors in nonlinear systems under additive stochastic excitation have been recently discovered. This paper uncovers the existence of phantom attractors in a single-degree-of-freedom smooth nonlinear equation, which characterizes the vibration of an inextensible beam subjected to lateral stochastic excitation. It also elucidates that the stochastic averaging method, in this context, may lead to qualitatively erroneous probability density functions, identified as one of the reasons why these attractors were previously overlooked. The study then proceeds to analyze the formation process of the phantom attractor and the critical noise intensity associated with it. Subsequently, the key nonlinear term related to the emergence of phantom attractors is identified by observing whether the system still exhibits phantom attractors after the corresponding nonlinear terms are removed. It is revealed that in this system, the presence of phantom attractors is closely linked to the inertia nonlinearity of the hardening type. The system investigated in this paper is simpler compared to previously identified systems capable of generating phantom attractors. This simplicity aids in facilitating research focused on unraveling the general principles behind the formation of phantom attractors.

In general, dynamic systems of higher dimensions or with more complex nonlinearities exhibit more intricate behaviors. Conversely, it is worthwhile to discuss whether a complex phenomenon persists in simpler systems. This paper investigates a single-degree-of-freedom vibration system with a velocity-dependent stiffness affected by additive noise. Although the underlying deterministic system possesses only one stable equilibrium point, under noise actions, it has the potential for a stochastic P-bifurcation to occur. This bifurcation causes the central peak of the joint probability density function to split into two symmetric peaks. At this stage, the behavior of the system resembles the development of two phantom attractors that deviate from the equilibrium point, causing the system’s random states to linger around them for extended periods. The effects of the damping ratio and noise intensity on the phantom attractors are discussed, together with the critical parameter curve associated with the onset of phantom attractors. Moreover, the generation mechanism of phantom attractors is disclosed by investigating the phase trajectories of the underlying conservative system. The distribution law of those critical parameter values is also proven by the stochastic averaging method, which is associated with the most probable amplitude. This study highlights that phantom attractors can manifest in dynamic systems even in the absence of Hopf bifurcation.

In this paper, we present a class of stochastic system of fractional space diffusion equations forced by additive noise. Our goal here is to approximate the solutions of this system via a system of ordinary differential equations. Moreover, we study the influence of the same degenerate additive noise on the stability of the solutions of the stochastic system of fractional diffusion equations. We are interested in the systems that have nonlinear polynomial and give applications as Lotka–Volterra system from biology and the Brusselator model for the Belousov–Zhabotinsky chemical reaction from chemistry to illustrate our results.

In this article, we obtain the existence and uniqueness of strong solutions to 3D viscous stochastic primitive equations (PEs) and the random attractor for 3D viscous PEs with additive white noise.

An asymptotic stability analysis of numerical methods used for simulating stochastic differential equations with additive noise is presented. The initial part of the paper is intended to provide a clear definition and discussion of stability concepts for additive noise equation derived from the principles of stability analysis based on the theory of random dynamical systems. The numerical stability analysis presented in the second part of the paper is based on the semi-linear test equation dX(t) = (AX(t) + f(X(t))) dt + σ dW(t), the drift of which satisfies a contractive one-sided Lipschitz condition, such that the test equation allows for a pathwise stable stationary solution. The θ-Maruyama method as well as linear implicit and two exponential Euler schemes are analysed for this class of test equations in terms of the existence of a pathwise stable stationary solution. The latter methods are specifically developed for semi-linear problems as they arise from spatial approximations of stochastic partial differential equations.

Using compactly supported wavelets, Giné and Nickl [Uniform limit theorems for wavelet density estimators, Ann. Probab.37(4) (2009) 1605–1646] obtain the optimal strong $L^{\infty }(ℝ)$ convergence rates of wavelet estimators for a fixed noise-free density function. They also study the same problem by spline wavelets [Adaptive estimation of a distribution function and its density in sup-norm loss by wavelet and spline projections, Bernoulli16(4) (2010) 1137–1163]. This paper considers the strong $L^p(ℝ)(1\le p\le \infty )$ convergence of wavelet deconvolution density estimators. We first show the strong $L^p$ consistency of our wavelet estimator, when the Fourier transform of the noise density has no zeros. Then strong $L^p$ convergence rates are provided, when the noises are severely and moderately ill-posed. In particular, for moderately ill-posed noises and $p=\infty$, our convergence rate is close to Giné and Nickl’s.

This paper deals with the asymptotic behavior of solutions to non-autonomous, fractional, stochastic $p$-Laplacian equations driven by additive white noise and random terms defined on the unbounded domain ${ℝ}^N$. We first prove the existence and uniqueness of solutions for polynomial drift terms of arbitrary order. We then establish the existence and uniqueness of pullback random attractors for the system in $L^2({ℝ}^N)$. This attractor is further proved to be a bi-spatial $(L^2({ℝ}^N),L^r({ℝ}^N))$-attractor for any $r\in [2,\infty )$, which is compact, measurable in $L^r({ℝ}^N)$ and attracts all random subsets of $L^2({ℝ}^N)$ with respect to the norm of $L^r({ℝ}^N)$. Finally, we show the robustness of these attractors as the intensity of noise and the random coefficients approach zero. The idea of uniform tail-estimates as well as the method of higher-order estimates on difference of solutions are employed to derive the pullback asymptotic compactness of solutions in $L^r({ℝ}^N)$ for $r\in [2,\infty )$ in order to overcome the non-compactness of Sobolev embeddings on ${ℝ}^N$ and the nonlinearity of the fractional $p$-Laplace operator.