Narrow Results

This study introduces an innovative examination of the stochastic Kaup–Newell equation within the realm of fiber Bragg grating, signifying a pioneering effort in this area. The study uses two main methods for integration: the $(\frac{G^{\prime }}{G})$-expansion method and the enhanced direct algebraic method to find different one-step solutions to the Kaup–Newell equation. The outcomes of this research are multifaceted, revealing an array of optical soliton solutions such as dark, singular, bell-shaped, kink-shaped, bright, and straddled solitons. These findings contribute a novel perspective on the behavior of optical solitons in fiber Bragg gratings, which are essential for optical communications and signal processing. Additionally, applying numerical schemes to these analytically obtained solutions offers a detailed visualization of their characteristics and behaviors, enhancing our understanding of their potential in practical scenarios. This work extends the range of soliton solution types for application and paves the way for subsequent explorations in the field of nonlinear optical systems.

This study delves into the perturbed stochastic nonlinear Schrödinger equation in an optical coupler for metamaterials, introducing a parabolic nonlocal law for refractive index nonlinearity, eighth-order dispersion, and multiplicative white noise. An enhanced version of Kudryashov’s method, combined with the direct algebraic method, is utilized to examine stochastic effects in nonlinear optical systems. The findings reveal diverse solution types such as dark and singular solitons, Weierstrass and Jacobi doubly periodic solutions, and straddled solitons, showcasing the complex dynamics driven by nonlocal nonlinearity and higher-order dispersion under stochastic conditions. This pioneering work on the nonlinear Schrödinger equation with a parabolic nonlocal nonlinearity and eighth-order dispersion investigates the impact of random fluctuations on nonlinear dynamics, advancing our understanding of light propagation at higher frequencies. The study aims to inspire further research into the behavior of light in unique materials and the development of more efficient light-based devices, leveraging the insights gained to innovate advanced communication tools and other optical devices, highlighting the response of these materials to random disturbances.

This research rigorously investigates a stochastic Susceptible-Infectious-Susceptible (SIS) epidemic model incorporating a generalized nonlinear incidence rate. The study further expands the model scope by introducing white and Lévy noise components. Including jumps and noises enhances the likelihood of disease extinction within the population. Initially, we utilized a comprehensive mathematical analysis to establish the global stability of the equilibrium state devoid of the disease. Our findings demonstrate that the system maintains stability even in the presence of the disease. This serves as a fundamental insight into the dynamics of disease transmission. Moreover, we define a collection of adequate criteria that contribute to the continuous presence of the ailment, thereby guaranteeing its non-complete eradication. By identifying the critical factors that contribute to the sustained presence of the disease, we enhance our understanding of the long-term behavior of the epidemic. To validate our theoretical findings, we conduct numerical simulations. These simulations effectively demonstrate the accuracy and reliability of the presented results. By comparing the simulated data with our analytical predictions, we can confidently confirm the validity of our model and its applicability in practical scenarios.

We present a simple and general result that the sign of the variations or increments of uncorrelated times series are predictable with a remarkably high success probability of 75% for symmetric sign distributions. The origin of this paradoxical result is explained in details. We also present some tests on synthetic, financial and global temperature time series.

We will present the axioms of Bogoliubov’s causal perturbative QFT in which the creation-annihilation operators are interpreted as Hida operators. We will shortly present the results that can be achieved in this theory: (1) removal of UV and IR infinity in the scattering operator, (2) existence of the adiabatic limit for interacting fields in QED, (3) proof that charged particles have nonzero mass, (4) existence of infrared and ultraviolet asymptotics for QED.

The white noise calculus originated by T. Hida is presented as a powerful tool in investigating physical and social systems. Combined with Feynman's sum-over-all histories approach, we parameterize paths with memory of the past, and evaluate the corresponding probability density function. We discuss applications of this approach to problems in complex systems and biophysics. Examples in quantum mechanics with boundaries are also given where Markovian paths are considered.

We analyze nonadiabatic non-Abelian holonomic transformations of spin-qubits confined to a linear time-dependent harmonic trap with time-dependent Rashba interaction. For this system, exact results can be derived for spin-rotation angle which also enables exact treatment of white gate-noise effects. We concentrate, in particular, on the reliability of cyclic transformations quantified by fidelity defined by the probability that the qubit after one full cycle remains in the ground-state energy manifold. The formalism allows exact analysis of spin transformations that optimize final fidelity. Various examples of time-dependent fidelity probability distributions are presented and discussed.

Stochastic noise exists widely in the nervous system, and noise plays an extremely important role in the information processing of the nervous system. Noise can enhance the ability of neurons to process information as well as decrease it. For the dynamic behavior of stochastic resonance and coherent resonance shown by neurons under the action of stochastic noise, this paper uses Fourier coefficient and coherence resonance coefficient to measure the behavior of stochastic resonance and coherence resonance, respectively, and some conclusions are drawn by analyzing the effects of additive noise and multiplicative noise. Appropriate noise can make the nonlinear system exhibit stochastic resonance behavior and enhance the detection ability of external signals. It can also make the coherent resonance behavior of the nonlinear system reach its optimal state, and the system becomes more orderly. By comparing the effects of additive and multiplicative noise on the stochastic resonance behavior and coherent resonance behavior of the system, it is found that additive and multiplicative noise can both make the system appear the phenomenon of stochastic resonance and have almost identical discharge state at the same noise intensity. However, with the increase of noise intensity, the coherent resonance of the system occurs, the multiplicative noise intensity is smaller than that of additive noise, but the coherent resonance coefficient of additive noise is smaller and the coherent resonance effect is better. The system whose system parameters are located near the bifurcation point is more prone to coherent resonance, and the closer the bifurcation point is, the more obvious the coherent resonance phenomenon is, and the more regular the system becomes. When the parameters of the system are far away from the bifurcation point, the coherent resonance will hardly appear. Besides, when additive and multiplicative noise interact together, the stochastic resonance and coherent resonance phenomena are more likely to appear at small noise, and the behavior of stochastic resonance and coherent resonance that the system shown is better in the local range.

In this article, we study the empirical power of Bartlett's Kolmogorov–Smirnov test statistic derived from cumulative periodograms in detecting chaos. In our study, we simulated 10 000 time series of i.i.d. standard normal random variable, logistic map and Gaussian chaotic map of various lengths with and without measurement errors. The empirical power was assessed for the original series and its order transformed series. Although our results showed that the Bartlett's test statistic was not a powerful test statistic in detecting chaos when it is applied directly to the original series generated by the logistic map and the Gaussian chaotic map, the order transformation technique can significantly increase the power of the Bartlett's test statistic.

In this paper, by using Lyapunov functions and exponents, Feller's scale functions, and the Fokker–Planck equations, we investigate the stability and bifurcation of stochastic closed orbit equations with singular diffusion coefficients.

A stochastic model of turbidostat in which two microorganism species compete for an inhibitory growth-limiting nutrient is considered. In the deterministic case, the model has rich dynamics: a coexistence equilibrium and the washout equilibrium can be simultaneously stable, and a stable limit cycle may exist. In the stochastic case, a phenomenon of noise-induced extinction occurs. Namely, the stochastic trajectory near the deterministic coexistence equilibrium will tend to the washout equilibrium. Based on the stochastic sensitivity function technique, in this paper, we construct the confidence ellipse and then estimate the critical value of the intensity for noise generating a transition from coexistence to extinction. We also construct the confidence band to find the configurational arrangement of the stochastic cycle.

In this article, we construct an excitable memristive diffusive neuron model by considering a biophysical slow–fast bursting oscillator and study the effects of electromagnetic induction on the dynamics of the single model as well as the coupled systems. We explore various firing regimes such as tonic spiking, bursting, and mixed-mode oscillations depending on the bifurcation structure with different injected current stimuli, then perform a comparative analysis on the synchronization of the coupled oscillators by setting the model into two different network architectures. First, a diffusively coupled network is considered, and later a global network is constructed. The results suggest that the diffusively connected neurons show complete synchronization at higher couplings for bursting and tonic spiking regimes. Furthermore, we show that the extended spatial system can generate spiral-like patterns in the vicinity of a Hopf bifurcation point and observe the impact of Gaussian white noise to study its effects on pattern formation. These types of patterns are robust in the excitable model. Our results might contribute significantly to the dynamical studies of irregular neural computation.

This paper addresses parameter identification of signals that are smooth and have compact support, but are otherwise arbitrary. We propose a definition of the time reference that results in the time uncertainty estimates that are much smaller than what one might expect with a casual analysis. These estimates are easily calculable given the noise level of the recording device and the signal shape.

The paper deals first with the deterministic analysis of stability and bifurcation of a non-linear model prey-predator system and then with the critical analysis of nonequilibrium fluctuation and stability of the model system under random perturbation including the comparative study of both deterministic and stochastic criteria of stability of the system.

In this paper, a stochastic nutrient–plankton model with seasonal fluctuation was developed to investigate how seasonality and environmental noise affect the dynamics of aquatic ecosystems. First, the survival analysis of plankton was proposed. Then, by using Lyapunov function and Khasminskii’s theory for periodic Markov processes, we derive the sufficient conditions for the existence of positive periodic solution. The numerical simulations were carried out to provide a better understanding of the model, and the results indicate that seasonal fluctuation is beneficial to the coexistence of plankton species.

Speech is the primary method of communication among humans; it is the main form of communication to deliver emotions and thoughts. Intuitively, humans would find it convenient to communicate with machines through speech. The automatic speech recognition model’s primary goal is to transcribe or identify the word sequence represented by the acoustic signals or speech. In this advanced era of digital media and e-learning platforms, the effective use of voice recognition technology to deliver a more individualized and effective educational experience in e-learning initiatives. It helps the students to improve their oral pronunciation skills. Even though Arabic is one of the most spoken languages, the research works related to Arabic speech and text are lacking when compared to other languages. Here, we propose a hybrid model of $K$-Nearest Neighbor (KNN) classifier and Dynamic Time Warping (DTW) for implementing a noise-resilient speech recognition system for isolated words in the Arabic language. The model is implemented with Mel-Frequency Cepstral Coefficients (MFCC) and its piecewise first and second derivatives as feature representation models. The proposed model is implemented in Python and simulated using Arabic Speech Corpus for Isolated Words [A. Alalshekmubarak and L. Smith, On improving the classification capability of reservoir computing for Arabic speech recognition, in International Conference on Artificial Neural Networks (Springer, Cham, 2014), pp. 225–232]. The proposed model is implemented with (1) MFCC alone for feature representation, (2) combination of MFCC with Delta coefficients, and (3) combination of MFCC with Delta and Delta–Delta coefficients. The implemented model is evaluated using different test sets of varying sizes of 100, 200, 500, 1000, 1500, and 2000. Evaluation is performed for both noised and noiseless speech using these three feature representation models and performed an evaluation of these three models. For evaluating this model’s performance in noised conditions, both white and babble noises of various signal-to-noise ratio values such as 10 dB, 20 dB, and 30 dB are added into the noiseless speech and estimate the classification accuracy in those situations. The proposed KNN–DTW model with MFCC outperformed all the other models in the literature. MFCC with Delta and Delta–Delta coefficients is effective than other two models using MFCC and MFCC with Delta coefficients. The proposed model could be used to recognize the isolated word recognition of grade 1 textbook vocabulary in classroom noisy environment for providing a better interactive classroom environment to Kuwaiti elementary students to enhance their math learning level. In an interactive computer-based education environment for Mathematics learning, the proposed model could be recognized as the spoken utterances of students.

In this work, a predator–prey model is considered to study its dynamics in a stochastic environment with the addition of predator migration. First, conditions on the existence of positive equilibria, along with their stability, and the emergence of limit cycles in the corresponding deterministic system are derived. Then, two stochastic properties are established: The ultimate boundedness of solutions and the existence of an ergodic stationary distribution for the system under consideration. The theoretical results are supported by numerical simulations, which reveal the significant influence of migration on the dynamical behaviors of both deterministic and stochastic systems.

This work focuses on deriving optical soliton solutions for the stochastic variant of the higher-order Lakshmanan–Porsezian–Daniel (LPD) model, which incorporates cubic-quintic law nonlinearities alongside chromatic and spatio-temporal dispersion terms. The primary approach chosen for this purpose is the new Kudryashov method (nKM). Initially, bright and dark soliton solutions are derived. The stability of these dark and bright solitons is then verified using the Vakhitov–Kolokolov criterion. Subsequently, the impact of the stochastic term on the bright and dark solitons is examined. As an additional aspect, an investigation has been conducted on the effects of the model parameters, namely, group velocity dispersion and spatio-temporal dispersion terms.

The compensated Poisson noise is expressed as a composite sum (splitting) of creation and annihilation operators, whose probabilistic interpretation relies on time changes. We construct an Itô table for this decomposition and obtain continuous and discrete time realizations of Lévy processes on the finite difference algebra and on , e.g. the space–time dual of the Poisson process (compensated gamma process), and the continuous binomial process.

Let H be a real separable Hilbert space and let E⊂H be a nuclear space with the chain {E_m: m=1,2,…} of Hilbert spaces such that E = ∩_m∈ℕE_m. Let E* and E_-m denote the dual spaces of E and E_m, respectively. For γ > 0, let be the collection of complex-valued continuous functions f defined on E* such that is finite for every m. Then is a complete countably normed space equipping with the family {‖·‖_m,γ : m = 1,2,…} of norms. Using a probabilistic approach, it is shown that every continuous linear functional T on can be represented uniquely by a complex Borel measure ν_T satisfying certain exponential integrability condition. The results generalize an infinite dimensional Riesz representation theorem given previously by the first author for the case γ = 2. As applications, we establish a Weierstrass approximation theorem on E* for γ≥1 and show that the space spanned by the class {exp[i(x,ξ)] : ξ ∈ E} is dense in for γ>0. In the course of the proof we give sufficient conditions for a function space on which every positive functional can be represented by a Borel measure on E*.