Communications in Nonlinear Science and Numerical Simulation最新文献

In this paper, a data-driven method is employed to investigate the probability density function (PDF) of nonlinear stochastic ship roll motion. The mathematical model of ship roll motion comprises a linear term with cubic damping and a nonlinear restoring moment represented as an odd-degree polynomial up to the fifth order. The data-driven method integrates maximum entropy, the pseudo-inverse algorithm, and a backpropagation (BP) neural network to obtain the PDF. The process begins with simulating data for the nonlinear stochastic system, followed by dimensional analysis to identify dimensionless parameter clusters. Optimization algorithms are then employed to solve for the coefficients, leading to the development of a BP neural network model trained to predict the PDF across various system characteristics and excitation intensities. The method's effectiveness is validated with Monte Carlo simulations, demonstrating high accuracy and reduced sensitivity to parameter variations.

This paper addresses the input-to-state formation stabilization problem of nonlinear multi-agent systems within a hybrid impulsive framework, considering delay-dependent impulses, strong nonlinearity, and deception attack signals. By leveraging Lyapunov functionals, impulsive comparison theory, average impulsive interval methods, and graph theory, we develop novel criteria for possessing locally input-to-state and integral input-to-state formation stabilization across different impulse sequence classes. These criteria are expressed in terms of continuous/impulsive feedback gains, time delay size, nonlinearity strength, uniform upper bound of impulsive interval, and length of average impulsive interval. Notably, the design of control impulses benefit the destabilizing continuous dynamics in the formation stabilization process. To demonstrate the effectiveness and validity of the analytical results, we provide numerical simulation examples involving various types of external attack signals.

In this paper, we propose inertial Halpern-type algorithms involving a quasi-monotone operator for approximating solutions of variational inequality problems which are fixed points of quasi-nonexpansive mappings in reflexive Banach spaces. We use Bregman distance functions to enhance the efficiency of our algorithms and obtain strong convergence results, even in cases where the Lipschitz constant of the operator involved is unknown a priori. Furthermore, we illustrate the practical applicability of our methods through numerical experiments. Notably, our algorithms excel when compared to recent techniques in the literature. Of particular significance is their successful application in restoring computed tomography medical images that have been affected by motion blur and random noise. Our algorithms consistently outperform established state-of-the-art methods in all conducted experiments, showcasing their competitiveness and potential to advance variational inequality problem-solving, especially in the field of medical image recovery.

This paper delves into the synchronization dynamics of fractional-order memristor Cohen–Grossberg neural network systems with time-varying delays at predefined times (PTS-MFCGNNs). Firstly, leveraging the concept of predefined-time stability, we devise a fractional-order controller, establish sufficient conditions for predefined-time synchronization, and achieve synchronization within the Cohen–Grossberg drive–response system. Secondly, building upon these findings, we scrutinize the synchronization dynamics within the time domain of the PTS-MFCGNNs system. Finally, we validate our theoretical framework through numerical simulations and engage in a comprehensive discussion on predefined-time synchronization within the PTS-MFCGNNs system.

In the present study, we introduce a high-order non-polynomial spline method designed for non-linear time-fractional reaction–diffusion equations with an initial singularity. The method utilizes the L2-1${}_{\sigma }$ scheme on a graded mesh to approximate the Caputo fractional derivative and employs a parametric quintic spline for discretizing the spatial variable. Our approach successfully tackles the impact of the singularity. The obtained non-linear system of equations is solved using an iterative algorithm. We provide the solvability of the novel non-polynomial scheme and prove its stability utilizing the discrete energy method. Moreover, the convergence of the proposed scheme has been established using the discrete energy method in the $L_2$ norm. It is proven that the method is convergent of order $min\{\theta \alpha ,2\}$ in the temporal direction and 4.5 in the spatial direction, where $\alpha \in (0,1)$ denotes the order of the fractional derivative and the parameter $\theta$ is utilized in the construction of the graded mesh. Finally, we conduct numerical experiments to validate our theoretical findings and to illustrate how the mesh grading influences the convergence order when dealing with a non-smooth solution to the problem.

In the framework of perturbed Keplerian systems we deal with the Delaunay normalisation of a wide class of perturbations such that the radial distance is raised to an arbitrary real number $\gamma$. The averaged function is expressed in terms of the Gauss hypergeometric function ${}_2{F}_1$ whereas the associated generating function is the so called Appell hypergeometric function $F_1$. The Gauss hypergeometric function related to the average depends on the eccentricity, $e$, whereas the Appell function depends additionally on the eccentric anomaly, $E$, and both special functions are properly defined and evaluated for all $e\in [0,1)$ and $E\in [-\pi ,\pi ]$. We analyse when the functions we determine can be extended to $e=1$. When the exponent of the radial distance is an integer, the usual values of the averaged and generating functions are recovered.

Delayed resonator (DR), which enables complete vibration suppression through loop delay tuning, has been extensively investigated as a linear active dynamic vibration absorber since its invention. Besides, the nonlinear high-static-low-dynamic stiffness (HSLDS) has been widely used in vibration isolators for broadband (yet incomplete) vibration reduction. This work combines the benefits of DR and the nonlinear HSLDS characteristics, thus creating a nonlinear DR (NDR). Three unexplored topics are considered: (1). To address the nonlinear dynamics that are made complicated by the introduction of delay and the nonlinearity coupled between the NDR and the primary structure; (2). To tune the control parameters to seek possible complete vibration suppression in the nonlinear case, and accordingly, to determine the operable frequency band; (3). To evaluate how the HSLDS characteristics can enhance the performance of the linear DR (LDR) and how to design an NDR to maximize its benefits. Without loss of generality, a classic nonlinear HSLDS structure with three springs and two links is considered, and mathematical tools and computational algorithms are introduced or proposed to efficiently address theoretical analysis. Using the parameters of an experimental setup, we show that a properly tuned NDR suppresses the vibrations on the primary structure to a sufficiently low level. Besides, the HSLDS characteristics extend the operable frequency band compared with the LDR, while strong nonlinearity limits such extension. The proposed analysis procedures for delay-coupled nonlinear dynamics, alongside the control parameter tuning, determination of operable frequency bands, and structural design rules, establish a basic theoretical framework for the NDR design.

In this study, we investigate a control problem involving a reaction–diffusion partial differential equation (PDE). Specifically, the focus is on optimizing the chemotherapy scheduling for brain tumor treatment to minimize the remaining tumor cells post-chemotherapy. Our findings establish that a bang-bang increasing function is the unique solution, affirming the MTD scheduling as the optimal chemotherapy profile. Several numerical experiments on a real brain image with parameters from clinics are conducted for tumors located in the frontal lobe, temporal lobe, or occipital lobe. They confirm our theoretical results and suggest a correlation between the proliferation rate of the tumor and the effectiveness of the optimal treatment.

Methods that distinguish dynamical regimes in networks of active elements make it possible to design the dynamics of models of realistic networks. A particularly salient example of such dynamics is partial synchronization, which may play a pivotal role in emergent behaviors of biological neural networks. Such emergent partial synchronization in structurally homogeneous networks is commonly denoted as chimera states. While several methods for detecting chimeras in networks of spiking neurons have been proposed, these are less effective when applied to networks of bursting neurons. In this study, we propose the correlation dimension as a novel approach that can be employed to identify dynamic network states. To assess the viability of this new method, we study networks of intrinsically bursting Hindmarsh–Rose neurons with non-local connections. In comparison to other measures of chimera states, the correlation dimension effectively characterizes chimeras in burst neurons, whether the incoherence arises in spikes or bursts. The generality of dimensionality measures inherent in the correlation dimension renders this approach applicable to a wide range of dynamic systems, thereby facilitating the comparison of simulated and experimental data. This methodology enhances our ability to tune and simulate intricate network processes, ultimately contributing to a deeper understanding of neural dynamics.

To avoid solving a saddle-point system, in this paper, we study two-level Arrow–Hurwicz finite element methods for the steady bio-convection flows problem which is coupled by the steady Navier–Stokes equations and the steady advection–diffusion equation. Using the mini element to approximate the velocity, pressure, and the piecewise linear element to approximate the concentration, we use the linearized Arrow–Hurwicz iteration scheme to obtain the coarse mesh solution and use three different one-step Stokes/Oseen/Newton linearized scheme to obtain the fine mesh solution. The optimal error estimate $O(h+H^2+{\chi }^{m/2})$ of the velocity and concentration in the $H^1$-norm and the pressure in the $L^2$-norm are derived, where $h$ and $H$ are fine and coarse mesh sizes, respectively, and ${\chi }^{m/2}$ denotes the iteration error with $0<\chi <1$. Numerical results are given to support the theoretical analysis and confirm the efficiency of the proposed two-level methods.