Search

In this paper, we discuss the numerical representation of the linearly implicit Euler method for an age-structured HIV infection model with a general transmission mechanism. We first define the basic reproduction number of the continuous model, and present the stability results of the equilibriums. For the numerical process, we establish the solvability of the system and the non-negativity and convergence of numerical solutions. In the analysis of the long-term dynamical behavior, this paper mainly focus on the existence of the infection equilibrium determined by the numerical reproduction number R 0 Δ t. To overcome the difficulty caused by the complexity of epidemic transmission mechanisms, the 1-order convergence analysis of numerical basic reproduction numbers R 0 Δ t is implemented by using the properties of the fundamental solution matrix. By a comparison principle, we show that the disease-free equilibrium is globally asymptotically stable if R 0 Δ t < 1. Moreover, for R 0 Δ t > 1 , a unique numerical endemic equilibrium exists, which converges to the exact one, is locally asymptotically stable. Hence, numerical processes visually represent the dynamic properties of nonlinear age-structured HIV models. Finally, some numerical experiments demonstrate the verification and the efficiency of our results. • The age-structured HIV model with general transmission is reviewed. • The exact basic reproduction number is recalled. • The linearly implicit Euler method is implemented to the model. • The theoretical and numerical threshold dynamics are investigated. • The convergence of the basic reproduction numbers is proved for general case. [ABSTRACT FROM AUTHOR]

In this paper we investigate a system of coupled inequalities consisting of a variational–hemivariational inequality and a quasi-hemivariational inequality on Banach spaces. The approach is topological, and a wide variety of existence results is established for both bounded and unbounded constraint sets in real reflexive Banach spaces. Applications to Contact Mechanics are provided in the last section of the paper. More precisely, we consider a contact model with (possibly) multivalued constitutive law whose variational formulation leads to a coupled system of inequalities. The weak solvability of the problem is proved via employing the theoretical results obtained in the previous section. The novelty of our approach comes from the fact that we consider two potential contact zones and the variational formulation allows us to determine simultaneously the displacement field and the Cauchy stress tensor. • A system of coupled inequalities is introduced and studied. • The weak solvability of the system of coupled inequalities is proved. • A contact model involving multivalued and nonmonotone constitutive laws is explored. [ABSTRACT FROM AUTHOR]

This paper investigates a novel class of nonlinear dynamical fuzzy systems referred to as fuzzy fractional delay differential hemivariational inequalities (FFDDHVIs) in Banach spaces. These systems integrate fractional fuzzy differential inclusions with delay and hemivariational inequalities. The existence theorem for the HVIs is established based on the KKM theorem. Moreover, specific propositions are proved, covering the superpositional measurability and the upper semicontinuity for the HVIs. Next, by using the fixed points theorem, we establish the existence and compactness of mild solution sets for the FFDDHVIs under certain mild conditions. Finally, As an illustrative application, we investigate a frictional quasistatic contact problem for viscoelastic materials, in which the friction and contact conditions are described by the Clarke generalized gradient of nonconvex and nonsmooth functionals. • We proposed FFDDHVIs are more general than existing results. • The form of HVIs with delay system in this paper is different from previous case. • FFDHVIs address the uncertainty of volume heat source in contact problems. • In contact problems, we have considered heat conduction is affected by delay systems. [ABSTRACT FROM AUTHOR]

For the mixed linear/nonlinear state-space model (ML/NLSSM) with asynchronously dependent noise processes (ADNP), this paper aims at designing Rao-Blackwellized particle smoothing (RBPS) algorithms via the sequential Monte Carlo sampling method to solve its fixed-interval smoothing problem. Asynchronous dependency leads to the current measurement depending not only on the current state, but also on the one-step previous state. This subtle feature makes the use of conditionally linear substructures in the ML/NLSSM complicated and thus brings a technical difficulty to the design of RBPS algorithms. In this paper, we first employ a noise de-correlation technique to covert the ML/NLSSM with ADNP into the one without noise dependency. Then for the converted ML/NLSSM, we propose a particle smoothing algorithm called the basic Rao-Blackwellized backward simulation (RBBSi) for the nonlinear substate. To further alleviate the computational complexity of the basic RBBSi, two improved versions of the basic RBBSi are developed via the Metropolis-Hastings sampling. For the (conditionally) linear substate, two analytical smoothing algorithms are provided by virtue of the forward-backward smoothing formula and the two-filter smoothing formula. By integrating the proposed algorithms, a unified implementation framework enveloping six RBPS algorithms is obtained. Finally, two target tracking examples demonstrate the effectiveness and superiority of the proposed RBPS algorithms. • It solves fixed-interval smoothing for ML/NLSSM with asynchronous noise dependency. • Three particle smoothing (PS) algorithms are proposed for nonlinear substate. • Two analytical smoothing algorithms are designed for conditionally linear substate. • A unified framework including six Rao-Blackwellized PS (RBPS) algorithms is given. • The proposed RBPS algorithms can provide more accurate state smoothing estimates. [ABSTRACT FROM AUTHOR]

The aim of this paper is to develop and investigate a fully-discrete scheme with conforming P 1 element for the nonlinear time-fractional Joule heating problem in which the Caputo derivative is approximated by the classical L 1 method. First, a novel superclose estimate in the H 1 -norm is derived rigorously with some new analysis techniques under low regularity of the solutions u n , ϕ n ∈ L ∞ (0 , T ; H 3 (Ω)) rather than u n ∈ L ∞ (0 , T ; H 4 (Ω)) and ϕ n ∈ L ∞ (0 , T ; H 3 (Ω) ∩ W 2 , ∞ (Ω)) required in the previous studies. Then, the global superconvergence result is deduced by interpolated post-processing approach. Finally, some numerical results are provided to verify the theoretical analysis. It should be mentioned that the analysis and results presented herein are also valid to some other known conforming and nonconforming finite elements. • The aim of this paper is to develop and investigate a fully-discrete scheme with conforming P 1 element for the nonlinear time-fractional Joule heating problem in which the Caputo derivative is approximated by the classical L 1 method. • A novel superclose estimate in the H 1 -norm is derived rigorously with some new analysis techniques under low regularity which improved the results in the previous studies. • The results obtained in our paper are also valid to conforming rectangular Q 1 element, nonconforming Q 1 r o t element on square meshes , and the nonconforming rectangular E Q 1 r o t element and so on. [ABSTRACT FROM AUTHOR]

In this paper, the synchronization problem of a class of switching neural networks with leakage time-varying delays is studied. A system solution-based direct analysis method is proposed to derive the sufficient conditions of global exponential synchronization for master–slave systems. Firstly, the state variable expression of the error system is derived by constructing a suitable regulation function, in which the leakage delays are explicitly transformed outside the state variable. Then, on this basis, the corresponding synchronization conditions is obtained by using the transition condition ingeniously. In addition, the obtained sufficient conditions contain only some simple linear scalar inequalities, which is different from most publications and greatly reduces the computational complexity. Finally, the reliability of the theoretical results is verified by numerical simulation. It is worth noting that the synchronization problem of switching neural networks with time-varying leakage delays is studied for the first time in this paper, and the method adopted does not need to construct any Lyapunov–Krasovskii functionals, which simplifies the proof process. • Synchronization problem of switching neural networks with leakage delays is studied. • A system solution-based direct method is proposed to achieve synchronization. • The proposed method simplifies proof process and reduces computational complexity. • The obtained synchronization criteria can be easily solved by using MATLAB. [ABSTRACT FROM AUTHOR]

• Established the model of axially moving beam considering gravity. • The model was validated by comparing with abaqus CAE result. • Discussed the effect of the gravity on the stability of the beam. • Special phenomenon caused by gravity was observed. The transmission device of the astronautic exploration vehicle can be regarded as an inclined beam experiencing axial motion under varying gravitational acceleration and tilt angle. Understanding the instability and vibration characteristics of this structure with axial movement is crucial for the dynamic design of the astronautic exploration vehicle. This paper discusses the stability and non-linear vibration nature of a self-weight inclined beam. The governing equations of the system are established and discretized using the hypothesized mode method and the extended Hamilton principle. The stability of the inclined beam is explored through an analysis of its natural frequency. The amplitude-frequency responses of the first four generalized coordinates of the inclined beam are analyzed by the Matcont toolkit. The influence of the axial velocity, the value of external excitation, the gravitational acceleration, and the tilt angle on the nonlinear vibration characteristics of the beam are discussed. Through the numerical discussion in this paper, a diverse range of nonlinear dynamic phenomena are observed and valuable insights for the stability design of the transmission device are provided. [ABSTRACT FROM AUTHOR]

• Nearly dynamically spherical rigid body motion with viscoelastic part is studied. • We study such dynamical case of rigid body motion about its center of mass. • Viscoelasticity is due moving mass linked by spring-damper to point on main axes. • Numerical integration of asymptotic equations is conducted for the body motion. • Solutions are obtained over infinite time range with asymptotically small error. The paper develops an approximate solution by means of an averaging method to the system of Euler's equations with additional perturbation terms for a nearly dynamically spherical rigid body containing a viscoelastic element. The averaging method is used. The asymptotic approach permits to obtain some qualitative results and to describe evolution of angular motion using simplified averaged equations and numerical solution. The main objective of this paper is to extend the previous results for the problem of motion about a center of mass of a rigid body under the influence of small internal torque (cavity filled with a fluid of high viscosity) or external torques (resistive medium, constant body-fixed torque). This paper can be considered as mainstreaming of previous works. The advantage of this paper is in receiving the original asymptotic and numerical calculations, as well as solutions that describe the evolution of motion a rigid body with a moving mass over an infinite time interval with an asymptotically small error. The paper presents a contribution in several areas, partially in the problems of spacecraft and satellite motion, and the activities of crew members about the vehicles. The importance of the results is in the progress of moving mass control, and in the motion of spinning projectiles. [ABSTRACT FROM AUTHOR]

Anti-disturbance state estimation for reaction–diffusion neural networks (RDNNs) subject to persistent dwell-time (PDT) switching constraints is investigated in this paper. First, PDT switching that can be utilized to characterize both the fast and slow switching processes is used in this paper to accurately model the RDNNs. Moreover, by designing the time-sampling and space-splitting measurement algorithms, the proposed state estimator can significantly reduce the measurement cost while tolerating the frequent asynchrony of the system modes and estimator ones caused by the sensor update lag. Furthermore, a state estimator is constructed to obtain the state of RDNNs affected by matched disturbances. To suppress the impact of the disturbance on the system's state estimation, a disturbance observer and a disturbance-related controller are designed to estimate the disturbance information and ensure state estimation performance. Then, sufficient conditions for the proposed state estimator making the estimation error bounded are derived. Finally, numerical simulations for switched RDNNs with two-dimensional spatial diffusion are presented to illustrate the effectiveness and superiority of the proposed method. • Different from existing 1D spatial switched RDNNs, this paper focuses on switched RDNNs with high-dimensional diffusion and PDT switching rule, which are more in line with the system characteristics. • A combined time-sampling and space-splitting measurement method is proposed to reduce the measurement cost. To address the frequent asynchrony of system modes and estimator ones, stability analysis and estimator design are performed using iterative and recursive methods. • To obtain the system's state and to minimize the effect of disturbances on state estimation, an anti-disturbance state estimation scheme is proposed for PDT-switched RDNNs based on the disturbance observer. [ABSTRACT FROM AUTHOR]

This paper focuses on the p th moment exponentially stability for Markov switching and multi-impulse jumps stochastic time-varying delay system, where the switching behavior among subsystems of the target system is determined by Markov chains, and the occurrence of impulsive jumps is decided according to event-triggered impulsive mechanism when certain well-designed conditions are satisfied. By applying the Itô formula, Gronwall inequality and Razumikhin theorem, some novel sufficient criteria are provided to assure the system stability and get rid of Zeno phenomenon. It is worth pointing out that the multi-impulse jumps are our research aim and the range of delays considered is relatively wide, i.e., the daily bounded delay τ (t) ∈ [ 0 , 1) and the unupper bound delay τ (t) ∈ [ 1 , ∞). Subsequently, two diverse event trigger mechanism about impulsive jumps are proposed for such two types of delays, namely the defined event-triggered impulsive mechanism with delay. Finally, the validity and feasibility of the developed theoretical results are verified by two numerical simulations. • Different from the bounded delay system in Wang et al. (2023), Peng et al. (2021),Li and Zhu (2023),Peng et al. (2010), Zhu (2014), Zhua and Cao (2012),Yang and Zhu (2014), Xu and Zhu (2022), we discuss the daily bounded delay τ (t) ∈ [ 0 , 1) and the unupper bound delay τ (t) ∈ [ 1 , ∞). Two categories of event trigger impulsive method named event-triggered impulsive mechanism with delay (ETIMD) are proposed for two types delay, respectively. And in event-triggered condition, it fully consider the impact of delays. Moreover, the upper bound of the delay τ (t) ∈ [ 0 , 1) is required to avoid Zeno phenomenon. While for τ (t) ∈ [ 1 , ∞) , only the lower bound is required to escape Zeno phenomenon, in which is independent of the delayed upper bound. • Compared with Wang et al. (2022), Li et al. (2020), Peng et al. (2021), Zhu (2014), Xu and Zhu (2022), the Markov switching and multi-impulse jumps stochastic time-varying delay system as a benchmark is considered in this paper, where the switching behavior between subsystems is driven by Markov chains, and the occurrence of impulsive jump is decided according to ETIMD strategy. It should be emphasized that the multi-impulse jumps are the research goal. Thus, the system in [27,28,36,43] could be regarded as a special case of the stochastic delay system when the Markov switching is not considered in this study. • By applying the Itô formula, Gronwall inequality and Razumikhin theorem, some novel sufficient criteria of Lyapunov-Razumikhin type for p-ES are provided for unstable subsystems with stable impulses. [ABSTRACT FROM AUTHOR]

A chemostat is an apparatus that sustains a homogeneous environment through continuous inflow and outflow. Presented is a chemostat model that characterizes the dynamics of dormancy-capable microorganisms. This model of coupled systems of nonlinear ordinary differential equations (ODEs) can apply to various types of organisms, such as different species of bacteria, archaea, algae, fungi, viruses, and protozoa. However, these species reside in different environments and rely on different sets of nutrients. Thus, the model adapts to each species' limiting nutrient through nutrient recycling. This paper includes a complete stability analysis, supporting phase plane portraits, and accompanying bifurcation diagrams. The paper also proposes an advanced second-order, positivity-preserving, and elementary stable nonstandard finite difference method for solving the mathematical model. Series of numerical simulations are presented that support the theoretical results and explore different biological scenarios. The stability analysis reveals that (1) if the overall dilution, death, and conversion are less than the overall growth, both the dormant and active populations persist when introduced to a chemostat; and (2) if the overall dilution, death, and conversion are more than the overall growth, the microorganism population in its entirety will die off when introduced to a chemostat. Furthermore, the model study suggests that neither dormancy nor nutrient recycling provides substantial survival advantages in a basic chemostat when no threat to the active microorganism is present. • A chemostat model is developed and analyzed for the dynamics of dormancy-capable microorganisms. • If dilution, death, and conversion are less than growth, both populations will persist. • If dilution, death, and conversion are more than growth, both populations will die out. • When there is no external threat, neither dormancy nor nutrition recycling help to ensure survival. [ABSTRACT FROM AUTHOR]

• A novel physics-informed shallow convolutional neural network is proposed. • The solving of the nonlinear physical operator equation is implemented. • The physical information is reconstructed from some noisy observations. • The effectiveness of current development is illustrated through extensive cases. • The speed acceleration of current development is significantly improved. We introduce a novel methodology for solving nonlinear partial differential equation (PDE) on regular or irregular domains using physics-informed ConvNet, which we call the PICN. The network structure consists of three parts: 1) a convolutional neural network for physical field generation, 2) a pre-trained convolutional layer corresponding to the finite-difference filters to estimate differential fields of the generated physical field, and 3) an interpolation network for loss analysis in irregular geometry domains. From a CNN perspective, the physical field is generated by a deconvolution layer and a convolution layer. Unlike the standard Physics-informed Neural Network (PINN) approach, the convolutions corresponding to the finite-difference filters estimate the spatial gradients forming the physical operator and then construct the PDE residual in a PINN-like loss function. The total loss function involving boundary conditions and the physical constraints in irregular geometry domains can be calculated from an efficient linear interpolation network. The theoretical analysis of PICN convergence is performed on a simplified case for solving a one-dimensional physical field, and several examples of nonlinear PDE of solutions with multifrequency characteristics are executed. The theory and examples confirm the effective learning capability of PICN for the physical field solution with high-frequency components, compared to the standard PINN. A series of numerical cases are performed to validate the current PICN, including the solving (and estimation) of nonlinear physical operator equations and recovering physical information from noisy observations. First, the ability of PICN to solve nonlinear PDE has been verified by executing three nonlinear problems including ODE with sine nonlinearity, PDE involving nonlinear sine-square operators, and Schrödinger equation. The proposed PICN has been assessed by solving some nonlinear PDE on irregular domains such as star-shaped domain, bird-like domain, and starfish domain. Moreover, PICN is applied to identify the thermal diffusivity parameters in an anisotropic heat transfer problem from noisy data, and a denoising display of the temperature field from strong noisy data with standard deviations ranging from 0.1 to 0.4. The numerical results demonstrate the high accuracy approximation and fast convergence performance of PICN. The potential advantage in approximating complex physical field with multi-frequency components indicates that PICN may become an alternative efficient neural network solver in physics-informed machine learning. This paper is adapted from the work originally posted on arXiv.com by the same authors (arXiv:2201.10967, Jan 26, 2022). The data and code accompanying this paper are publicly available at https://github.com/zengzhi2015/PICN. [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate the high-order approximation of a class of SPDEs with cubic nonlinearity driven by multiplicative noise with the help of the amplitude equations. The highlight of our work is the provision of approximate solutions with enhanced accuracy. Precisely, previous researches primarily concentrated on deriving approximate solutions via the first-order amplitude equations. However, this paper constructs approximate solutions by utilizing both first-order and second-order amplitude equations. And, we rigorously prove that such approximate solutions enjoy improved convergence property. To further illustrate our demonstration intuitively, we apply our main theorem to stochastic Allen–Cahn equation and present a numerical analysis. • The high-order amplitude equations of SPDEs with multiplicative noise is obtained. • The provision of approximate solutions with enhanced accuracy is given. • The approximate solutions enjoy improved convergence property is rigorously proved. • Applications and numerical analysis to stochastic Allen–Cahn equation are presented. [ABSTRACT FROM AUTHOR]

This paper presents the lowest-order weak Galerkin finite element method for linear elasticity problems on the convex polygonal meshes. This method uses piecewise constant vector-valued spaces on element interiors and edges. The discrete weak gradient space introduced by this paper is the matrix version of C W 0 space. The discrete weak divergence space is piecewise constant space on each element. This method is simple, efficient, stabilizer-free and symmetric positive-definite. The optimal error estimates in discrete H 1 and L 2 norms are presented. Numerical results are given to demonstrate the efficiency of algorithm and the locking-free property. • The matrix version of C W 0 element for discrete weak gradient is introduced. • The lowest-order weak Galerkin finite element space ( P 0 2 , P 0 2 , C W 0 2 , P 0 ) is adopted. • Our method is suitable for the polygonal and hybrid meshes. [ABSTRACT FROM AUTHOR]

In this paper, we develop strong convergence of the Euler–Maruyama (EM) scheme for approximating the doubly perturbed McKean–Vlasov stochastic differential equations. In contrast to the existing work, a novel feature is that we use more general conditions for parameters α and β. To obtain the desired approximation, this paper also proves the existence and uniqueness of strong solution for this class of McKean–Vlasov SDEs. Combining with the results of propagation of chaos, the overall convergence rate is obtained for the EM scheme. Finally, two numerical examples are provided to demonstrate our results. • We prove the particle system converges (propagation of chaos) with the corresponding rate. • The strong convergence of the EM method in the finite time is proved. Combining this with the propagation of chaos results gives an overall convergence rate. • We use the weaker perturbation coefficient α and β control for conditions. [ABSTRACT FROM AUTHOR]

This paper studies the fixed-time synchronization of a class of quaternion-valued neural networks (QVNNs) with time delays and impulses. In contrast to some existing decomposition methods for studying this problem, this paper presents several strategies to simplify the implicit Lyapunov function method from both controller and implicit function equation perspectives. The benefits of these strategies are, in the first strategy, the controller does not contain the Lyapunov function so that the controller will be more general, and in the second strategy, the implicit function equation is simpler so that the restriction on the bounded Lyapunov function can be dispensed with, which also makes the technique of handling time delays simpler. Furthermore, three sufficient conditions for fixed-time synchronization of the above QVNN are presented, despite the impulsive influence. Finally, we verify the feasibility of our methods with three examples. • Using a flexible implicit function approach to study fixed-time synchronization of QVNNs. • Fixed-time synchronization is studied without sign functions in the controller. • Non-decomposition method is utilized. [ABSTRACT FROM AUTHOR]

In this paper, we propose a fully implicit nonconforming virtual element method for solving the Sobolev equations with Burger's type nonlinearity by utilizing the backward Euler scheme. The boundedness, existence and uniqueness of the fully discrete numerical solution are proven strictly. Furthermore, by virtue of the careful estimation of the consistency error caused by the nonlinear term, we obtain the optimal order error estimate of the numerical algorithm in H 1 -norm. Finally, four numerical examples are presented to verify the theoretical findings, showing that the constructed scheme in this paper is effective under various grids and maintains the energy dissipation property. • The boundedness, existence and uniqueness of the fully discrete numerical solution are proven strictly. • The optimal order error estimate of the numerical algorithm in H 1 -norm is obtained. • Four numerical examples are presented to verify the theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, a novel and fast two-grid finite element method (FEM) is proposed for efficiently solving semilinear time-fractional diffusion equations with variable coefficients. To handle the initial singularity, the nonuniform L1 scheme is employed in the temporal domain. A novel two-grid FEM technique is used in the spatial domain to reduce computational cost. The two-grid algorithm solves the original nonlinear fractional equation on a much coarser grid with size H , obtaining the coarse-grid solution u H n based on the fine-grid solutions at previous time levels. This approach avoids repetitive discrete convolutional summation when approximating the Caputo derivative on the coarse grid, resulting in reduced computational cost. Furthermore, a fast nonuniform L1 scheme with two-grid technique is developed to accelerate the evaluation of the Caputo derivative. The paper also proposes a novel, fast and highly accurate algorithm for computing the coefficients involved in the discretized Caputo fractional derivative. The α -robust stability and optimal L 2 - and H 1 -norm error estimates for fully discrete scheme are derived, where the error bound remains valid as α → 1 − . Numerical results validate the theoretical findings and demonstrate the superior efficiency of the proposed two-grid algorithms compared to the standard FEM. • A novel, fast two-grid FEM is proposed for time-fractional diffusion equations. • A innovative method for computing the discrete convolution kernels is proposed. • The α -robust stability and optimal L 2 - and H 1 -norm error estimates are derived. • Numerical results show the efficiency and accuracy of the proposed algorithms. [ABSTRACT FROM AUTHOR]

For the stability analysis and stabilization synthesis problems, this paper considers networked control systems (NCSs) with the transmission delay and the deception attack under aperiodic samplings, where the deception attack and its activation function are represented as a sector bound function and a random variable with Bernoulli distribution, respectively. This paper proposes the stability and stabilization criteria for the NCSs by constructing Lyapunov–Krasovskii (L–K) functionals with continuous functionals and looped-functionals. Compared with the literature, the proposed continuous functionals take into account the mixed delay including the transmission delay and the maximum allowable sampling interval, as well as the augmented vector and integral terms. Also, the proposed looped-functionals construct two augmented vectors with integral vectors that are zeros at t = t k and t = t k + 1 , respectively. By utilizing these two augmented vectors, the looped-functionals fully utilize the sampling patterns compared with the literature. Based on the proposed L–K functionals, this paper derives not only the stability criterion for the NCSs with the transmission delay, but also the stabilization criterion for the NCSs with the transmission delay and the deception attack in terms of linear matrix inequalities (LMIs), respectively. Numerical example demonstrates the validity of the proposed approach. [ABSTRACT FROM AUTHOR]

This paper introduces a conservative higher-order finite difference scheme for solving the coupled nonlinear Schrödinger equations. The Crank–Nicolson method is employed to discretize time derivatives and the sixth-order difference operator is used to discretize space derivatives, correspondingly, the resulting difference scheme has second-order accuracy in time and sixth-order accuracy in space. By utilizing the discrete energy method, the conservation of discrete mass and energy, the boundedness, existence and uniqueness of solution, unconditional stability and the convergence of the new scheme are proved. Then making use of Richardson extrapolation, the time accuracy is increased to the fourth order. Finally, the numerical experiments are conducted to validate the theoretical results presented in the paper. • A novel conservative higher-order finite difference method with sixth-order accuracy in space and fourth-order accuracy in time is proposed for solving the coupled nonlinear Schrödinger equations. • The conservation of discrete mass and energy, the boundedness, existence and uniqueness of solution, unconditional stability and the convergence of the new scheme are proved. • Numerical results demonstrate that the presented scheme is superior to the methods in the existing literature in terms of accuracy. [ABSTRACT FROM AUTHOR]

For a class of nonlinear cyber physical systems (CPSs) under intermittent denial-of-service attacks (DoS), this paper investigates a novel adaptive security control scheme. A switched fuzzy observer is designed to estimate unmeasurable states. In addition, the relative threshold-based event-triggered approach is introduced to reduce the released data, and the dynamic surface control technique is used to remove the 'explosion of terms' problem. Meanwhile, an observer-based adaptive event-triggered prescribed performance controller is constructed, which ensures that all closed-loop signals remain bounded and system output approaches a designed performance bound within a predefined finite time. Simultaneously, the Zeno behavior can be thoroughly eliminated. Finally, a numerical simulation is used to demonstrate the validity of the results. • This paper proposes a novel security control scheme for nonlinear CPSs. • A new performance function is introduced to ensure system performances. • The adopted event-triggered method effectively reduces the communication burden. [ABSTRACT FROM AUTHOR]

In this paper, we establish some sufficient conditions for the existence of random limit cycle generated by stochastic Liénard equation. Our technique involve Lyapunov functions and truncation arguments. Furthermore, using polar coordinate transformation and rigid rotation, we further established existence (non-existence) of a possible minimal period of random periodic solution of stochastic van der Pol oscillator. • In section 1 and 2, we introduced the work and reviewed the notion of random periodicity. • In section 3, we proved the existence and uniqueness of stable random periodic solutions for stochastic Liénard equations. • In section 4, we proved the existence of possible minimal period for stochastic van der Pol oscillator. • In section 5, we concluded the paper and pointed out a follow-up of this paper in our future publications. [ABSTRACT FROM AUTHOR]

This paper discusses the problem of tracking control for nonlinear CPSs in the presence of resource constraints and multiple cyber-attacks. Firstly, a fuzzy model is built to solve the nonlinear characteristics of CPSs. Secondly, to reduce the burden on communication-resources, two dynamic event-triggered schemes are put forward from plant and reference. Considering the existence of cyber-attacks in dual channel, the deception-attacks model and DoS-attacks model are constructed by stochastic variables. This paper aims to obtain a dynamic event-based tracking controller under cyber-attacks. A sufficient condition of nonlinear CPSs is received to ensure the stochastic stability and desired tracking-performance. Finally, examples are simulated to testify the advantage of the proposed method. • A T-S fuzzy model is introduced to describe the nonlinear CPSs. • The dynamic event-triggered schemes are applied to overcome resource-constraints. • Two novel models with cyber-attacks are founded to analyze the stability of system. • An output tracking model is established to obtain the desired tracking performance. • The LKF and LMI techniques are used to derive the sufficient conditions. [ABSTRACT FROM AUTHOR]

Based on the thin plate theory and D'Alembert's principle to establish the equilibrium equation for viscoelastic axially moving plates, this paper establishes the ternary governing equation of variable fractional order for viscoelastic axially moving plates by using a variable fractional order constitutive relationship for viscoelastic materials. Shifted Chebyshev wavelets are introduced for approximating the deflection function, and numerical solutions for the governing equations are given. The effectiveness and accuracy of the algorithm in this paper is illustrated by convergence analysis, error correction and numerical examples. Finally, the algorithm is used to simulate the vibration of axially moving plates with different moving speeds and different boundary conditions. Meanwhile, Gaussian white noise was introduced to investigate the vibration of the viscoelastic axially moving plate under pure noise environment, simple harmonic load-same direction noise environment and simple harmonic load-opposite direction noise environment, respectively. And the vibration comparison of PP material plate and LLDPE material plate is also carried out. The above research conclusions are consistent with the existing literature, indicating that algorithm proposed by this paper is applicable to numerical simulation and research on viscoelastic axially moving plates. • This algorithm solves the fractional axially moving plate equation in time domain. • The simulated plate vibration matches reality under Gaussian white noise. • Research shows PP boards are stiffer than LLDPE boards. • Research shows tighter clamping reduces board vibration. • Algorithm's accuracy and effectiveness shown via analysis and error correction. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the convergence of an adaptive two-grid weak Galerkin (ATGWG) finite element method for second order semilinear elliptic partial differential equations (PDEs). First, we propose an ATGWG method and then prove that the sum of the energy error and the error estimator of ATGWG method between two consecutive adaptive loops is a contraction. The weak Galerkin (WG) elements (P j (T) , P ℓ (∂ T) , R T j (T)) (Wang and Ye, 2013) are studied in this paper and numerical experiments based on the lowest order case with j = l = 0 are provided to support the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, the problem of resilient distributed secure consensus control for uncertain networked agent systems under hybrid denial-of-service (DoS) attacks is investigated. The current situation is that there exists a cyber layer connecting the network control units and a physical layer with specific physical links in the networked agent systems. Both the communication networks connecting the two layers and within the cyber layer may be attacked by DoS maliciously. These two attack scenarios have different impacts on the networked system. The former focuses on updating the control inputs timely, while the latter influences the connection weight of the cyber communication topology. Firstly, a distributed security consensus framework is proposed for the case that updates of signals are destroyed by DoS attacks between two layers, in which an acknowledgment (ACK)-based attack detection and a recovery mechanism are introduced, and a self-triggered based distributed control protocol is designed. On the premise of avoiding Zeno behavior, the relationship between trigger intervals and DoS attack characteristics is revealed. Secondly, corresponding asynchronous switching topology method is developed for secure consensus of the networked systems when DoS attacks are launched within the cyber layer. In addition, we found that large signals jump during switching and triggering will generate pulses and affect system stability. Therefore, a saturation function is introduced to constrain the fluctuation range of the signal. Finally, the effectiveness of the design scheme is verified by the simulation results of multi-robot systems. • In existing security control schemes against DoS attacks, most of the considered DoS attacks occur in the communication channels between the physical layer and the cyber layer [15,16,28,29,30]. However, in practice, the channels between these two layers are relatively fixed. The communication channels connected by the corresponding network control unit of each agent in the cyber layer are more vulnerable to DoS attacks. This paper analyzes the impacts of DoS attacks on the system under the above two scenarios in detail. • Instead of supposing that the occurrence of cyber attacks follows some specific probabilistic characterizations [31,32], we only introduce two basic features of DoS attacks, which make the designed control protocol able to resist more general attack forms. Meanwhile, unlike the existing event-triggered anti-attack control protocols [28,29,30], a control protocol based on self-triggered is proposed to improve the resilience of the control system against DoS attacks and reasonably regulate the communication resources of the system. • For the problem of actuator input saturation in the system, this paper constructs a bounded function with auxiliary variables to reduce its influence. Under the limit of actuator input saturation, the designed control scheme based on self-triggered reduces the pulse phenomenon caused by signal triggering and switching, which is ignored in many existing works. [ABSTRACT FROM AUTHOR]

This paper is devoted to introduce a new double phase elliptic inclusion problem (DPEI) involving nonlinear and nonhomogeneous partial differential operator which has unbalanced growth and logarithmic perturbation terms, and two multivalued functions which are defined in the domain and its boundary. The main goal of this paper is to establish the existence and extremality results to the elliptic inclusion problem under consideration. More exactly, we give the definitions of weak solutions, subsolutions and supersolutions to (DPEI). Then, under the coercive setting, an existence theorem of weak solutions to (DPEI) is obtained by employing a surjectivity theorem for pseudomonotone operators. Moreover, in the noncoercive framework, we apply the method of sub-supersolution combined with the nonsmooth calculus analysis and truncation techniques to prove that (DPEI) has at least a weak solution within an ordered interval of sub-supersolution. Finally, when the constraint set K satisfies a lattice condition, the existence of smallest and greatest elements of solution set to (DPEI) is established. • A nonlinear double phase elliptic inclusion is studied. • Two existence theorem are established. • The existence of smallest and greatest elements of solution set is proved. [ABSTRACT FROM AUTHOR]

A special kind of Poisson stable trajectory, which is called unpredictable and leads to sensitivity in the quasi-minimal set, was proposed by Akhmet and Fen (2016) for semiflows. In the present paper we carry this finding one step further by defining a new kind of trajectory, called asymptotically unpredictable. We prove that such motions also lead to sensitivity in the dynamics. This feature is now achieved under a weaker hypothesis. Benefiting from the Bebutov dynamical system, continuous asymptotically unpredictable functions on the real axis are defined, and it is shown that the set of these functions properly includes the set of unpredictable ones. Moreover, results on producing new asymptotically unpredictable functions from a given one are obtained. The existence and uniqueness of bounded asymptotically unpredictable solutions of quasi-linear systems are also investigated, and an application to Hopfield neural networks is provided. • A new type of motion, called asymptotically unpredictable, is introduced on semiflows. • It is proved that asymptotically unpredictable trajectories lead to sensitivity in the dynamics. • Sensitivity is achieved under a weaker hypothesis compared to unpredictable motions. • Continuous asymptotically unpredictable functions are defined via the Bebutov dynamical system. • Asymptotically unpredictable solutions of quasi-linear systems are additionally discussed. [ABSTRACT FROM AUTHOR]

In this paper, the unconditional stability and optimal error estimate of the velocity, pressure and angular velocity for the modified characteristics FEMs of the unsteady Micropolar Naiver–Stokes Equations (MNSE) are presented. In this method, the nonlinear equation is linearized by the characteristic finite element method for dealing with the time derivative term and the convection term. Basing on the characteristic time-discrete system, the error function is split into a temporal error and a spatial error. With a rigorous analysis to the characteristic time-discrete system, we prove that the error between the numerical solution and the solution of the time-discrete system is τ -independent, where τ denotes the time stepsize. The stability results and optimal error estimates in L 2 norm and H 1 norm will be given. Finally, some numerical results will be provided to confirm our theoretical analysis. • The unconditional stability and optimal error estimate for the modified characteristics FEMs are presented. • The nonlinear equation is linearized by the characteristic method. • Basing on the characteristic time-discrete system, the error function is split into a temporal error and a spatial error. • The stability results and optimal error estimates in $L̂2$ norm and $Ĥ1$ norm will be given. [ABSTRACT FROM AUTHOR]

We study two-dimensional, two-piece, piecewise-linear maps having two saddle fixed points. Such maps reduce to a four-parameter family and are well known to have a chaotic attractor throughout open regions of parameter space. The purpose of this paper is to determine where and how this attractor undergoes bifurcations. We explore the bifurcation structure numerically by using Eckstein's greatest common divisor algorithm to estimate from sample orbits the number of connected components in the attractor. Where the map is orientation-preserving the numerical results agree with formal results obtained previously through renormalisation. Where the map is orientation-reversing or non-invertible the same renormalisation scheme appears to generate the bifurcation boundaries, but here we need to account for the possibility of some stable low-period solutions. Also the attractor can be destroyed in novel heteroclinic bifurcations (boundary crises) that do not correspond to simple algebraic constraints on the parameters. Overall the results reveal a broadly similar component-doubling bifurcation structure in the orientation-reversing and non-invertible settings, but with some additional complexities. [ABSTRACT FROM AUTHOR]

This paper introduces, analyses, and implements efficient time parallel methods for solving the Cahn–Hilliard (CH) equation. Efficient numerical methods for the CH equation are crucial due to its wide range of applications. In particular, simulating the CH equation often requires long computational times to obtain the solution during the phase coarsening stage. Therefore, there is a need to accelerate the computations using parallel methods in the time dimension. We propose linear and nonlinear Parareal methods for the CH equation, depending on the choice of the fine approximation. The effectiveness of our approach is demonstrated through numerical experiments. • Formulation of linear and non-linear Parallel-in-time methods for the Cahn–Hilliard equation. • Convergence analysis of proposed time parallel methods. • Numerical illustration of convergence behaviour of proposed time parallel methods. • Possibility of parallel computing. [ABSTRACT FROM AUTHOR]

In this paper, based on the scalar auxiliary variable (SAV) approach, the superconvergence error analysis is investigated for the time-dependent Navier–Stokes equations. In which, an equivalent system of the Navier–Stokes equations with three variables and a fully-discrete scheme is developed with semi-implicit Euler discretization for the temporal direction and low-order bilinear-constant finite element discretization for the spatial direction, respectively. With the help of the high-precision estimations of the bilinear-constant finite element pair on the rectangular meshes, the superclose error estimates for velocity in H 1 -norm and pressure in L 2 -norm are obtained by treating the trilinear term carefully and skillfully. The global superconvergence results are also derived in terms of a simple and efficient interpolation post-processing technique. Finally, some numerical results are provided to demonstrate the correctness of the theoretical analysis. • A semi-implicit SAV Galerkin FEM is investigated for the time-dependent N–S equations. • The high-precision estimations are employed in the error analysis. • The trilinear term is treated skillfully in the error analysis. • The superclose and superconvergence results are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we propose a second-order decoupled finite element method based on lagging a part of the interfacial coupling terms for the time dependent Stokes–Darcy problem, which only need to solve two sub-physical problems sequentially. Under a modest time step restriction Δ t ≤ C (physical parameters), the optimal long time error estimates are obtained both in the L 2 norm and in the H 1 norm. Numerical results are provided to illustrate the convergence rate O (Δ t 2) of the temporal approximation. • A second-order decoupled finite element method based on lagging a part of the interfacial coupling terms is proposed. • The optimal long time error estimates are obtained in the L 2 norm. • The optimal long time error estimates are obtained in the H 1 norm. [ABSTRACT FROM AUTHOR]

This manuscript focuses on investigating a class of Caputo fractional fuzzy differential system with time delay. Firstly, we understand the granular form of fuzzy numbers from a novel perspective, which contains more information than the usual membership function. Subsequently, using a successive approximation approach under the granular arithmetic context, we demonstrate the existence of the solution to this system, and the uniqueness is obtained by the completeness of fuzzy space. Furthermore, we establish a criterion for determining the finite-time stability of the system, and which is an evaluation function containing optional parameter ζ ∈ (0 , ν) by employing a Gronwall inequality with delay form, where ν is the order of the Caputo derivative. Finally, we present a numerical example to verify our primary findings and discuss the appropriate selection of parameter ζ to optimize the evaluation function. • From a novel point of view, it is found that the horizontal membership function contains more distribution information than the membership function. • The existence and uniqueness of the solution of system (1) is obtained in the sense of granular fuzzy environment, and this paper improved the proof method of the reference Luo et al. (2023). • The finite-time stability result for the system (1) has been obtained and optimizing parameter to obtain more suitable result. [ABSTRACT FROM AUTHOR]

This paper presents a Crank–Nicolson leap-frog (CNLF) scheme for the unsteady incompressible magnetohydrodynamics (MHD) equations. The spatial discretization adopts the Galerkin finite element method (FEM), and the temporal discretization employs the CNLF method for linear terms and the semi-implicit method for nonlinear terms. The first step uses Stokes style's scheme, the second step employs the Crank–Nicolson extrapolation scheme, and others apply the CNLF scheme. We establish that the fully discrete scheme is stable and convergent when the time step is less than or equal to a positive constant. Firstly, we show the stability of the scheme by means of the mathematical induction method. Next, we focus on analyzing error estimates of the CNLF method, where the convergence order of the velocity and magnetic field reach second-order accuracy, and the pressure is the first-order convergence accuracy. Finally, the numerical examples demonstrate the optimal error estimates of the proposed algorithm. • Crank–Nicolson leap-frog (CNLF) scheme for the unsteady incompressible magnetohydrodynamics (MHD) equations. • The CNLF method for linear terms and the semi-implicit method for nonlinear terms. • Fully discrete scheme is stable and convergent. • The convergence order of the velocity and magnetic field reach second-order accuracy. [ABSTRACT FROM AUTHOR]

This paper considers the finite element method to solve a generalized constant delay diffusion equation. The regularity of the solution of the considered model is investigated, which is the first time to discover that the solution has non-uniform multi-singularity in time compared with Tan et al. (2022). To overcome the multi-singularity, a symmetrical graded mesh is used to devise the fully discrete finite element scheme for the considered problem based on L1 formula of the Caputo fractional derivative and fractional trapezoidal formula of the Riemann–Liouville fractional integral. Then we investigate the unconditional stability of this scheme. Next, the local truncation errors of the L1 formula and the fractional trapezoidal formula are analyzed in detail, especially the later one is discussed at the first time, under the multi-singularity of the solution and the symmetrical graded mesh. Using these error results, we obtain the convergence of the proposed numerical scheme. Finally, some numerical tests are provided to verify the obtained theoretical results. • The regularity of the solution is investigated and shows multi-singularity in time. • A symmetrical graded mesh is used to overcome the multi-singularity. • The unconditional stability and optimal temporal convergence rate are obtained. • The truncation errors of L1 formula and fractional trapezoidal formula are discussed. • Some numerical tests are provided to verify the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper it is exhibited the bifurcation scenario concerning a typical singularity of planar piecewise smooth vector fields in two zones. This singularity is characterized by a quadratic contact of one vector field and a quartic contact of another vector field at the same point of the switching manifold. By means of a three parameter perturbation, we observe the presence of bifurcations like: Hopf Bifurcation, Sliding Hopf Bifurcation, the birth and annihilation of chaotic scenarios, among others. The two and three dimensional bifurcation diagrams are included. • quartic–quadratic fold singularity of planar piecewise smooth vector fields. • normal form of the quartic–quadratic fold singularity. • 3-parameter bifurcation of a planar piecewise smooth vector field. • Planar bifurcation diagrams. • tridimensional bifurcation diagram. [ABSTRACT FROM AUTHOR]

In this paper a second order dynamical system model is proposed for computing a zero of a maximally comonotone operator in a Hilbert space. Under mild conditions, we prove existence and uniqueness of a strong global solution of the proposed dynamical system. A proper tuning of the parameters can allow us to establish fast convergence properties of the trajectories generated by the dynamical system. The weak convergence of the trajectory to a zero of the maximally comonotone operator is also proved. Furthermore, a discrete version of the dynamical system is considered and convergence properties matching to that of the dynamical system are established under a same framework. Finally, the validity of the proposed dynamical system and its discrete version is demonstrated by two numerical examples. • Propose a second order dynamical system method for a maximally comonotone inclusion problem. • Analyze convergence rates of the proposed system and prove weak convergence of the trajectory to a zero. • Propose a rate-matching momentum-based algorithm with a relaxation factor and a correction term. • Demonstrate the validity of the system and its resulting algorithm by numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, a neurodynamic optimization approach based on a p -power transformation Lagrangian function is developed for distributed nonconvex optimization. A new Lagrangian function is proposed to eliminate dual gaps of nonconvex problems, and a distributed average tracking approach is developed for estimating global objective function value. Based on the Lagrangian function and the distributed average tracking approach, a neurodynamic model is developed for distributed nonconvex optimization, and its convergence to a local minimum is proven. Two numerical examples are provided to demonstrate the validity and effectiveness of the proposed approach. • A Lagrangian function with a p -power transformation is designed. • A neurodynamic model for distributed nonconvex optimization is developed. • The convergence of the neurodynamic model to a local minimum is proven. • Two numerical examples are provided to demonstrate the validity of the approach. [ABSTRACT FROM AUTHOR]

• A novel piezoelectric inertial stick-slip driven pipeline robot is proposed to meet the detection requirements. • The main body is hollowed out to reduce mass and install actuators. • The nonlinear dynamics model of the pipeline robot is established, and analysis results for the chaos are obtained. • The occurrence conditions of chaotic behavior are determined, which provides theoretical reference for the design of the robot. Pipeline robots using the conventional driving mode have encountered a bottleneck in miniaturization. To address this problem, a micro piezoelectric pipeline robot based on the inertia stick-slip driving principle is proposed in this paper. The robot is well suited to the inspection needs of micro pipes. However, undesirable design parameters found during the structural optimization phase can lead to unstable operation and reduced load capacity of the robot, and can even cause the drive system to stop with chaotic behavior. Therefore, the nonlinear dynamics model of the four degree of freedom discrete system of the pipeline robot is established. The Runge-Kutta method is used to solve the dynamic response of each nonlinear subsystem. The nonlinear dynamical behavior of the system is analyzed through the bifurcation diagram, time domain diagram, phase diagram, Poincaré map and power spectrum diagram of each subsystem response. The stable operation intervals of the machine and electrical parameters in the drive system are given, and the conditions for the occurrence of chaotic behavior are determined. This research can be applied to most of the drive systems consisting of piezoelectric stacks and flexure hinges. It helps to determine and optimize the structural parameters of the piezoelectric actuator, thus avoiding chaotic behavior and ensuring that the actuator maintains good operational performance. [ABSTRACT FROM AUTHOR]

Active subspace analysis is a useful computational tool to identify and exploit the most important linear combinations in the space of a model's input parameters. These directions depend inherently on a quantity of interest, which can be represented as a function from input parameters to model outputs. As the dynamics of many plasma models are driven by potentially uncertain parameter values, the utilization of active subspaces to perform global sensitivity analysis represents an important step in understanding how certain physical phenomena depend upon fluctuations in the values of these parameters. In the current paper, we construct and implement new computational methods to quantify the induced uncertainty within the growth rate generated by perturbations in a collisionless plasma modeled by the one-dimensional Vlasov–Poisson system near an unstable, spatially-homogeneous steady state in the linear regime. [ABSTRACT FROM AUTHOR]

In this paper, we propose a predator–prey fishery model with prey harvesting, variable price and marine protected area. We assume the price changes faster than other processes such as population growth and predation, and get a slow fast Ordinary Differential Equation (ODE) system. A simplified three-dimensional model is obtained by using approximate aggregation methods. The results show that there are two main scenarios, one is the depletion of fish stocks due to overfishing by fishermen, which is called a "catastrophic" equilibrium; and the other is a stable and sustainable fishery equilibrium. In order to avoid the extinction of fish, we consider establishing marine protected area (MPA) near fishing areas where fishermen only catch the prey. The results of the study provide the conditions for the establishment of MPA, allowing us to avoid the extinction of prey populations and establish sustainable fisheries. As another possibility, we consider increasing taxes to discourage overfishing by fishermen. In addition, when the tax revenue increases, the optimal harvest strategy is obtained. The optimal policy ensures the sustainable development of fisheries and maximizes the interests of fishermen. • A prey-predator fishery model with variable price and marine protected area is described. • The establishment of marine protected area can avoid large amplitude variations of fishing effort and the extinction of prey. • Optimal taxation prevents the extinction of the prey species caused by overfishing. [ABSTRACT FROM AUTHOR]

In this paper, a fuzzy-dependent adaptive event-triggered mechanism (FAETM) for synchronizing Takagi–Sugeno (T–S) fuzzy reaction–diffusion neural networks (RDNNs) is developed while considering deception attacks. Firstly, a general neural network model considering both fuzzy logic rules and reaction–diffusion terms is established. Secondly, a FAETM based on an aperiodic sampling period is presented under deception attacks to alleviate the communication burden, wherein the adaptive threshold function is dependent on membership functions. Moreover, a membership-function-dependent asymmetric Lyapunov functional (LF) is constructed, and some positive-definite and symmetric constraints of matrices are removed as a result. Based on the LF method and integral inequality techniques, the H ∞ synchronization criteria for the T–S fuzzy RDNNs are presented by linear matrix inequalities (LMIs). Finally, one simulation example is exploited to demonstrate the feasibility and validity of the established method, and the outcome is applied to image secure communication. • A T–S fuzzy reaction diffusion neural networks is developed. • A fuzzy adaptive event-triggered mechanism under deception attacks is designed. • A membership-functions-dependent asymmetric Lyapunov functional is constructed. • A less conservative stability criterion is obtained. • An application to image encryption is proposed. [ABSTRACT FROM AUTHOR]

This paper investigates the bifurcations of homoclinic orbits to hyperbolic saddle points in a simplified railway wheelset model with cubic and quintuple nonlinear terms. Using Melnikov's method, the sufficient conditions for the existence of the supercritical and the subcritical pitchfork bifurcations of homoclinic orbits are proven. To determine the integrability of the variational equations around homoclinic orbits in the meaning of differential Galois theory, the corresponding Fuchsian second-order differential equation for the normal variational equation and the Riemann P function are obtained. It is shown that the coefficients of the linear terms and the cubic coupling terms play a very significant role on influencing the existence of homoclinic orbits. While, the cubic coupling terms have little effect on the size of the left-hand and right-hand potential wells of homoclinic orbits. These results are beneficial to explore the key mechanism of hunting stability of a simplified railway wheelset model. • The sufficient conditions for pitchfork bifurcations of homoclinic orbits are proven. • The integrability of variational equations around homoclinic orbits is determined. • Linear and cubic coupling term coefficients affect the existence of homoclinic orbits. [ABSTRACT FROM AUTHOR]

In this paper, a space–time spectral method for solving the Korteweg–de Vries equation is considered. The discrete schemes of the method are based on the Legendre–Petrov–Galerkin method in spatial direction and the Legendre-tau method in temporal direction with nonperiodic boundary conditions. Stability analysis results and error estimates are obtained in L 2 -norm by introducing a cut-off function without Lipschitz condition. The method is also applicable to solve some (2 m + 1) th-order differential equations. Comparison of our numerical results with those of other spectral methods exhibits the accuracy of our methods for the Korteweg–de Vries equation. • An efficient space–time spectral method with pseudospectral treatment of nonlinear term is presented. • Legendre–Petrov–Galerkin method is used in space and Legendre-tau method used in time. • The method allows high-order accuracy and is easy to implement in solving the KdV equation. • Stability and error estimates are obtained without the constraints of Lipschitz conditions for nonlinear terms. [ABSTRACT FROM AUTHOR]

This paper proposes an asymmetric functional-coefficient autoregressive conditional heteroscedasticity in mean (ARCH-M) model, which allows for asymmetry in the volatility. The profile likelihood approach is applied to estimate the parametric and nonparametric components. Under some regularity assumptions, we derive asymptotic behavior of the proposed estimator. To avoid model misspecification, the Wald, quasi-likelihood ratio test statistic and generalized likelihood ratio test statistic are put forward to detect ARCH effect, asymmetric effect and goodness-of-fit, respectively. Moreover, their asymptotic distributions are established under both null and alternative hypotheses. Some Monte Carlo simulations are conducted to evaluate the finite sample performance of the proposed estimation methodology and testing procedure. Also, real data sets are analyzed to demonstrate the applications of the proposed model. [ABSTRACT FROM AUTHOR]

The present study delves into the projective synchronization for distinct fractional-order neural networks consist of inconsistent orders, employing the principles of sliding mode control and incorporating fractional operators within the controller. Firstly, incorporating the fractional-order derivative into the controller facilitates the derivation of a synchronous error system, and a well-suited integral-type sliding switching surface is subsequently constructed, along with the development of an appropriate sliding mode controller, thereby ensuring the existence of sliding movement derived from the principle of sliding mode control. Then through the application of the Lyapunov direct method, it is shown that the sliding mode controller steers the evolution of error system towards the designated sliding switching surface and maintains therein indefinitely, meanwhile, novel criteria for achieving projective synchronization for distinct fractional-order neural networks consist of inconsistent orders is established. Lastly, to exemplify the efficacy and practical applications of the proposed approach, simulations with two types of distinct fractional-order neural networks are carried out utilizing a continuous frequency distribution model. • The paper addresses the issue of projective synchronization for Caputo fractional-order neural networks characterized with inconsistent structures, functions, and orders, which has been overlooked in neural networks. • The Caputo derivative operators are incorporated into the controller to convert the distinct Caputo fractional-order neural networks consist of inconsistent orders into a consistent-order system, enabling the establishment of the synchronous error system for further analysis. • The adoption of the smooth and continuous function t a n h (⋅) as a replacement for s g n (⋅) mitigates the chattering resulting from the discontinuity of the controllor in traditional synovial observer and the sliding mode observer is made to have better estimation accuracy. [ABSTRACT FROM AUTHOR]

Lyapunov stability theory provides a powerful technique for stability analysis of dynamical systems, particularly for the stability of nonlinear dynamical system. This paper deals with the stability of fuzzy dynamical systems using a novel notion called granular fuzzy Lyapunov function. In order to analyze the stability, some new notions are introduced such as fuzzy equilibrium point, the ball of fuzzy state space, granular fuzzy 2-norm and etc. Based on the concept of fuzzy-number-valued function granular differentiability, two theorems are proved for calculation. Moreover, using granular fuzzy Lyapunov function, the granular fuzzy local stability theorem is proposed, and two definitions of exponential stability and like-Lyapunov stability are generalized. Finally, several examples are given to illustrate the proposed theorems. • The granular fuzzy chain rules and granular fuzzy 2-norm are presented. • Granular fuzzy Lyapunov function based on fuzzy-number-valued function granular derivative was defined. • The local stability theorem stability is proposed, and two definitions of exponential stability and like-Lyapunov are generalized. [ABSTRACT FROM AUTHOR]

In this paper, we investigate soliton modulation instability and collision dynamics in the birefringence optical fibers. Soliton transmission in the picosecond or femtosecond range in optical fibers is described by the coupled hybrid nonlinear Schrödinger equations. We focus on the modulation instability of the plane wave solutions and the gain spectrum under different parameters. The three-soliton solutions are used to analyze soliton collisions, and the asymptotic analysis is provided to reveal the properties of soliton collisions. The elastic and inelastic collisions of three-soliton are presented, and there exists the possibility of shape restoration of one or two solitons during the three-soliton inelastic collisions. The relevant results provide not only a new perspective for the realization of optical logic gates, but also theoretical value for the experimental observation of soliton collisions. • Soliton modulation instability and collision dynamics in the birefringence optical fibers are investigated. • Soliton transmission in the picosecond or femtosecond range in optical fibers is described by the coupled hybrid nonlinear Schrödinger equations. • The three-soliton solutions are used to analyze soliton collisions, and the asymptotic analysis is provided to reveal the properties of soliton collisions. [ABSTRACT FROM AUTHOR]

This paper pays attention to the passivity and dissipativity for discrete-time fractional generalized delayed Cohen-Grossberg neural networks. A new fractional passive lemma is firstly proposed for discrete-time system by means of the Lyapunov functional. This facilitates the discussion of system stabilization in terms of input and output energy. Some passive and dissipative conditions are established under comparison principle, reduction to absurdity, inequality techniques and the proposed passive lemma. Finally, two examples are performed to demonstrate the validity of proposed methods. • A novel passive lemma is established to fill the gap of passive criteria for fractional difference system. • The passivity and dissipativity are discussed by the comparison principle, reduction to absurdity and Lyapunov-Krasovskii functional. • The passivity and dissipativity criteria are proposed in terms of linear matrix inequalities. [ABSTRACT FROM AUTHOR]