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• It is shown that all the nonlinear PDEs and their Lie symmetries derived in the paper "Exact solutions of nonlinear... " CNSNS, 67 (2019), 253-263 have been derived earlier. • The optimal systems of one-dimensional subalgebras of Lie algebras in the above paper are incorrect. • The exact solutions obtained in the above paper follow from earlier works. This comment is devoted to the paper "Exact solutions of nonlinear diffusion-convection-reaction equation: A Lie symmetry approach" (CNSNS, 67 (2019), 253-263) in which several results are not new because were derived much earlier. Moreover, some results in the paper are incorrect or incomplete. [ABSTRACT FROM AUTHOR]

In this corrigendum we correct an error in our paper [T. Caraballo, R. Colucci, J. López-de-la-Cruz and A. Rapaport. A way to model stochastic perturbations in population dynamics models with bounded realizations, Commun Nonlinear Sci Numer Simulat , 77 (2019) 239–257]. We present a correct way to model real noisy perturbations by considering a slightly different stochastic process based, as in the original paper, on the Ornstein-Uhlenbeck process. Namely, we correct the formulae that generates the noisy realizations to ensure the boundedness property to be satisfied with probability one (which turns out not to be true in our original paper even though it was observed in all the simulations). [ABSTRACT FROM AUTHOR]

• Correct an error in our paper published in Communications in Nonlinear Science and Numerical Simulation in 2015. • Construct a new Z 2 -equivariant cubic switching system and prove the existence of 15 limit cycles. • An approach is described for calculating the focus values of systems with two switching lines. In this note, we correct an error in our paper published in Communications in Nonlinear Science and Numerical Simulation in 2015 [5], in which the form of a Z 2 -equivariant cubic planar switching system is not correct. In this note, we add one more switching line and use the same equations given in [5] to construct a correct Z 2 -equivariant cubic system. Then, we prove that the new system can exhibit 15 limit cycles and therefore the conclusion given in [5] still holds. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new modified self-adaptive extragradient method with the inertial technique for solving the variational inequality problem of a pseudomonotone mapping and the fixed point problem of a Bregman relatively nonexpansive operator in a general reflexive Banach space. We prove a strong convergence theorem of our algorithm under some suitable assumptions. Finally, some numerical experiments are provided to demonstrate the effectiveness of the suggested iterative method. • Variational Inequality problem is an important optimization problem. • In this paper, we introduce a new modified self-adaptive extragradient method with the inertial technique for solving the variational inequality problem of a pseudomonotone mapping and the fixed point problem of a Bregman relatively nonexpansive operator in a general reflexive Banach space. • Strong convergence theorem of the presented algorithm are proved under some suitable assumptions. • Finally, some numerical experiments are provided to demonstrate the effectiveness of the algorithm. [ABSTRACT FROM AUTHOR]

This paper considers an improved two-dimensional reaction–diffusion predator–prey model with time delay. Firstly, the distribution of the equilibrium points of the system and the existence conditions are discussed. Secondly, under the assumption of the existence of equilibrium points, the linear approximations of the system in both network and non-network settings are derived. Thirdly, the Turing instability conditions for both non-delay and delay systems are investigated, including cases of diffusion-induced and delay-induced instabilities. Fourthly, amplitude equations are derived based on the non-delay system. Finally, extensive numerical simulations are conducted to validate and illustrate the theoretical results, and to analyze and explain their practical significance. The above results effectively demonstrate that the theoretical findings, simulations, and natural reality are consistent. The numerical modeling conducted on the network structure based on the Monte Carlo method exhibits a more diverse range of dynamic behaviors in the parameter of the Holling III functional response function. • This paper studies the influence of amplitude equation control parameter on the pattern type. • The impact of periodic diffusion is simulated emphatically and explained in practical terms. • Monte Carlo method is employed to study the Turing patterns of network system. [ABSTRACT FROM AUTHOR]

This paper concentrates on the almost sure synchronization for a class of stochastic multi-links coupled semi-Markov jump systems through aperiodically intermittent control. For these stochastic switching systems, almost sure synchronization is investigated by employing mode-dependent multiple Lyapunov-like function method and stochastic analysis. Notably, mode-dependent multiple Lyapunov-like function method is designed to provide more flexibility. In addition, when considering the sojourn time distribution of the semi-Markov jump, dependence on both the current state and the next state can effectively reduce unnecessary restrictions compared with previous assumption of the sojourn time distribution function. Ultimately, the Chua's circuit along with numerical simulations are provided to validate the effectiveness of the theoretical results. • This paper proposes aperiodically intermittent control to address almost sure synchronization of stochastic multi-links semi-Markov jump systems. Different from mean square synchronization, almost sure synchronization means noise is abeneficial impact when the system is synchronized. • Compared with the previous assumption of the sojourn time distribution function, semi-Markov jump dependent on both the current and the next states is considered. • We propose a more general multi-links system model to study its synchronization, which makes the results more applicable. [ABSTRACT FROM AUTHOR]

Due to the potential decrease of the computation time for problems with fractional order derivatives, and due to the extension of the range of applicable solvers for a given problem, the approximation of fractional derivatives (e.g., using Oustaloup's method) is an important and frequently discussed topic. A significant problem that can occur (although one that is not discussed very often) when approximations are applied concerns the potential numerical issues that can arise when applying these approximations in a transient analysis, e.g., as shown in the paper, in a time-stepping solver. This paper examines five different methods for approximating the fractional derivative (Oustaloup's method, a refined Oustaloup method, a modified continued fraction expansion method, Matsuda's method and Charef's method). An important preliminary step in the analysis was the establishment of the general form of FDAE (fractional differential-algebraic equations), the general form of the applied approximations and the general form of the DAE (differential-algebraic equations) resulting from the approximation. This allowed for a precise description of the equations according to which the transformation from the FDAE to the approximating DAE takes place. Three problems derived from studies in electrical engineering have been introduced into the analysis (a linear circuit problem, a circuit problem with nonlinear fractional elements, and a problem featuring nonlinearities and pregenerated noise). For these problems, the conditions in which numerical issues appear have been studied in detail. They are the main motivation of this study as these issues are so significant that they often result in zero-solutions or solutions tending to infinity (giving an impression as if the system is unstable). Modifications and alternatives of the transformation into the DAE, that aim at the mitigation of these numerical errors, are mentioned later. The final result is very satisfactory, where the algorithm for the transformation of the FDAE to the approximating DAE practically eliminates the most important barriers (which was the impossibility of using approximations above a certain order due to the mentioned numerical issues). The study presented in this paper is motivated by problems in electrical engineering but due to its generality it is also applicable in other fields where fractional calculus is used. • Five fractional derivative approximation methods are examined. • General FDAE and DAE forms are recalled (for the problem and its approximation). • Numerical issues appearing after applying approximations are demonstrated. • Various methodologies on the mitigation of these numerical issues have been proposed. • The final transformation methodology allows for very high approximation orders. [ABSTRACT FROM AUTHOR]

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]

• The PINN framework proposed in this paper combines the degenerate kernel scheme. • The independent full connection networks model is recommended for solving the IDEs. • Algorithm reliability is verified with numerous examples of forward and inverse problems. • The effects of hyperparameters on the convergence of the PINN method are discussed. In recent years, machine learning has become an interdisciplinary research hotspot in nonlinear science and artificial intelligence. Nonlinear integro-differential equations (IDEs), as an essential mathematical model in science and engineering, often face challenges in forward problem analysis and inverse problem solving due to the complexity of their kernel functions. This paper proposes a machine learning framework that combines degenerate kernel schemes to solve the IDEs of mathematical models in nonlinear science, including forward problems and inverse problems. Herein, the general smooth continuous IDEs are first approximated to the IDEs with degenerate kernels, and the equivalent nonlinear differential equations are obtained by introducing the auxiliary differential operators with new boundary conditions to replace the integral ones. For the functions to be solved in the original IDEs and the new functions in the auxiliary differential operators, the independent full connection deep neural networks (FCDNN) are established. By constructing the loss function based on the equivalent nonlinear differential equations and all boundary conditions and initial conditions, this machine learning is trained to realize the solution of the nonlinear IDEs forward problem by using the Adam optimizer. By constructing new loss components based on the measured values of the function, the IDEs inverse problems can be further solved by machine learning, such as unknown parameters or source items in IDEs. Detailed numerical analysis of the forward problem, inverse problem, and some high-dimensional problems of the nonlinear IDEs shows that the proposed machine learning has high accuracy and universality for such nonlinear problems. In addition, the effects of the characteristics of the solution, the network framework, the forms of activation function and loss function, and physical information distribution points on the convergence of the machine learning method are discussed in detail. The universal machine learning solution for nonlinear IDEs can serve the applications of IDEs in science and engineering. The data and code accompanying this paper are publicly available at https://github.com/Lihui0626/PINN-for-IDEs. [ABSTRACT FROM AUTHOR]

This paper introduces a distributed adaptive fault-tolerant control scheme for nonlinear multi-agent systems afflicted by actuator failures. The proposed scheme ensures that the consensus-tracking errors of the systems meet the prescribed performance requirements. First, in the backstepping design process, a barrier function is constructed based on the proposed scalar function and normalization function to address the prescribed performance issue. Next, both multi-dimensional Taylor network (MTN) control and robust control techniques are utilized to address unknown nonlinear functions. Continuous switching functions are devised to transition from MTN control to robust control whenever the arguments of the unknown functions surpass the operating range of MTN. This effectively resolves the global problem of the system. Furthermore, a boundary estimation method is proposed to handle actuator failures that may occur infinitely. Stability analysis confirms that all signals in the closed-loop system are globally uniformly ultimately bounded. Additionally, the consensus-tracking errors can always be constrained within prescribed boundaries, achieving global convergence to a prescribed accuracy within a prescribed time. Users can flexibly and independently choose the desired settling time and accuracy, without being restricted by any initial conditions. Simulation results provide evidence of the efficacy of the proposed control method. • This paper considers the possibility of each actuator in each agent of NMASs experiencing an in finite number of failures. • The proposed scheme not only ensures a deterministic settling time but also allows for the flexible design of the settling time based on specific requirements. • The challenge of achieving global tracking in NMASs by redirecting escaping transients back into the active region of the MTN. [ABSTRACT FROM AUTHOR]

In this paper, consensus control and vibration suppression problems are considered for the flexible nonlinear Timoshenko manipulator multi-agent system. The multi-agent system comprises multiple identical flexible Timoshenko manipulators, which can realize cooperation utilizing local information exchange between various agents. The bending deformation and shear deformation generated by the practical robotic manipulators during motion both affect the control effect, so the flexible Timoshenko manipulator is used to describe the dynamic characteristics. The flexible Timoshenko manipulator is a typical distributed parameter system, the deformation states are not only related to time but also to position, and the partial differential equations (PDEs) models are studied in this paper. Based on the PDEs models, the consensus control scheme is developed under the undirected communication topology structure, which can realize joint angle consensus tracking and vibration suppression of all the flexible Timoshenko manipulator agents simultaneously. Finally, the control effect is verified by the numerical simulations. • To describe the infinite-dimensional properties accurately, the partial differential equations (PDEs) models are studied for the flexible nonlinear Timoshenko manipulator. • The designed control scheme can realize joint angle consensus tracking and vibration suppression of all the flexible Timoshenko manipulator agents simultaneously under the undirected communication topology. • Consensus control is the foundation of cooperation control, and the proposed control scheme is designed without any simplification, which can provide theoretical support for flexible robots to complete complex space tasks through mutual collaboration. [ABSTRACT FROM AUTHOR]

• A piezoelectric cantilever-beam-spring-pendulum structure is proposed. • The harvester captures energy from multi-directional vibrations. • The proposed structure has excellent low-frequency energy harvesting performance. • Voltage generated in the x- and z-directions are increased to106.6% and 187.3%. Harvesting energy from environmental vibrations holds significance for powering wireless sensors and transducers. However, the current vibration energy harvesting methods still have room for improvement, especially in the field of low-frequency vibration, where the performance of output voltage is relatively insufficient. This paper proposes a novel piezoelectric energy harvester consisting of a piezoelectric cantilever beam and a spring pendulum in order to address this challenge. Under external excitation, energy exchange is generated between the internal parts of the system, so that the swinging of the pendulum is interchanged with the extension and contraction motion of the spring and the bending motion of the cantilever beam to realize multidirectional energy harvesting. Firstly, the energy equation of system is obtained and the dynamics model of system is derived based on Lagrange's theorem. In addition, the response characteristics of system under harmonic excitation and the effects of different parameters on the system performance are investigated by simulation. With proper parameter selection, the system can produce higher output voltages. Compared to a piezoelectric cantilever-single-pendulum, the proposed structure demonstrates superior performance, especially in low-frequency vibration. The maximum voltage amplitudes in the x- and z-directions were 106.6% and 187.3% of those of the piezoelectric cantilever-single-pendulum, respectively. This paper proves the superior energy harvesting performance of the proposed structure. [ABSTRACT FROM AUTHOR]

This paper investigates the reinforcement learning (RL) adaptive tracking control design problem for a class of mismatched stochastic nonlinear systems with non-affine structure. The stochastic system studied in this paper is more generally representative due to the presence of non-affine inputs, internal uncertainties, and mismatched external disturbances. Firstly, in order to solve the non-affine structure of stochastic systems, an extended stochastic differential equation is constructed. Based on the actor–critic framework, by generating reinforcement signals, the network is stimulated to evolve more quickly towards the desired direction, while compensating internal uncertainties and induced uncertainties in the stochastic system. Furthermore, aiming at the approximation errors and external disturbances, while satisfying stochastic stability, the adaptive laws for disturbances boundary estimator are established with the usage of higher powers of tracking errors. As a result, a novel non-affine adaptive tracking controller has been proposed by integrating RL, disturbances boundary estimation, and dynamic surface method. Through the stability analysis, it is proved that all closed-loop signals are bounded in probability, and the system output converges to a small neighborhood near the desired trajectory. Numerical simulations demonstrate the effectiveness and superiority of the proposed controller. • Auxiliary integration obtain extended stochastic differential equation systems. • Actor-critic network reduces fitting error and improves environmental adaptability. • All signals Inclosed-loop are bounded in probability. [ABSTRACT FROM AUTHOR]

This paper presents an adaptive fuzzy event-triggered control (ETC) method for a class of time-delay nonlinear multi-agent systems (MASs) with dead zone (DZ) and partial state constraints (PSCs). It should be pointed out that the states considered in this paper are unmeasurable and part of the system states are constrained by time-varying boundary. By utilizing fuzzy logic systems (FLSs) and backstepping design framework, a fuzzy state observer is constructed to estimate the unmeasured states, while the introduction of a time-varying barrier Lyapunov function (BLF) addresses the partial state constrained problem. Furthermore, during the design process of the state observer, compensation for the influence of the unknown DZ input on the system is achieved through utilization of known designed event-triggered input signal and adaptive parameter. Building upon this foundation, employing the Lyapunov stability theory and adaptive backstepping technique with ETC strategy, the developed distributed control mechanism demonstrates that all variables of the closed-loop system are bounded, the consensus error of the system can converge to a small region of origin and Zeno behavior can be ruled out. Finally, two simulation examples further illustrate the proposed control strategy and theorem. • Observer-based adaptive tracking control for nonlinear time-delay MASs is studied. • Dead-zone input and partial state constraints are considered simultaneously. • A novel adaptive ETC scheme is proposed for nonlinear MASs with unknown dead zone. • By constructing a novel fuzzy state observer, a new control method is proposed. • The tracking control problem of MASs with partial state constraints is studied. [ABSTRACT FROM AUTHOR]

In this paper, a novel framework for multi-agent systems is established to explore the swarm phenomenon of individuals with opinions in nature. Unlike the conventional researches in which agents in the system rigidly follow the predefined protocol, this paper combine opinions evolution with dynamic behaviors into multi-agent systems. To study the effect of opinions on behaviors, a protocol incorporating with opinions is introduced. The opinion of an agent is influenced by opinions of its neighbors and itself, while the behavior of the agent is influenced by both opinions and behaviors of agents. The existence and distributions of opinion equilibrium points are analyzed, and a necessary and sufficient condition of behavior stability is proposed. Moreover, the implementation of opinions leads to collective behavior, and the steady state of the system is analyzed. Finally, numerical simulations illustrate the effectiveness of the theoretical results. • A novel framework for multi-agent systems with dynamical opinions is established in the paper, by which agents will exhibit collective behavior. • Design an opinion updating strategy, and analyze the existence and distributions of opinion equilibrium points. • The behavior stability of the system is analyzed, and the steady states of the system are influenced by both opinions and behaviors of agents. [ABSTRACT FROM AUTHOR]

The ultimate boundedness and stability of highly nonlinear neutral stochastic delay differential equations (NSDDEs) are investigated in this paper. Different from many previous works, the highly nonlinear NSDDEs with semi-Markov switching signals are considered for the first time in this paper. Meanwhile, the time delay function in this paper is only required to meet much more relaxed restrictions, which can invalidate plentiful methods with requirements on its derivatives for the stability of NSDDEs. To overcome this difficulty, several novel techniques to tackle NSDDEs with such a delay are developed. Furthermore, a crucial property on the ergodic semi-Markov process is established, which plays a key role in the proof on the existence and boundedness of global solution. The generalized Khasminskii-type theorems are established for the existence and uniqueness of global solution by applying new methods and this property, and the criteria for boundedness and stability are also provided. In particular, feasible measures are provided to deal with the neutral term in our stability criteria. Finally, the effectiveness is verified by a numerical example. • The stability of highly nonlinear neutral stochastic delay differential equations is first solved. • A new lemma is established to present a favorable result on the average switching number of semi-Markov switching. • A more general Khasminskii-type condition is introduced in this paper. • The delay function is not required to be differentiable or continuous. • An example is provided to check the theory results. [ABSTRACT FROM AUTHOR]

In this paper, the unsteady incompressible thermomicropolar fluid (UITF) equations are considered. Theoretically, some a priori regularity conclusions are presented firstly, which seem to be not available in the literatures. Numerically, a penalty finite element method (PFEM) for the UITF equations is studied, the Euler semi-implicit temporal semi-discrete method of the penalty UITF equations is proposed, the stability and the L 2 - H 1 error estimates of the temporal discrete solutions are proved. Finally, the stability and the L 2 - H 1 error estimates for the finite element fully-discrete approximation of the penalty UITF equations are rigorously proved. The accuracy and efficiency of the fully-discrete PFEM are demonstrated by some numerical examples. • This paper focus on penalty method for unsteady incompressible thermomicropolar fluid equations. • Some a priori regularity conclusions are presented. • Stability and L 2 − H 1 error estimates of Euler semi-implicit method are proposed. • Stability and L 2 − H 1 error estimates of fully-discrete method are proved. [ABSTRACT FROM AUTHOR]

• From the network level, we systematically investigate the impact of inhibitory synaptic loss and inhibitory proportion on the generation of theta oscillation from a simulation model. The paper visually demonstrates the effect of each factor and the co-effect of the combined factors on the generation of the theta rhythm. • From the neuronal level, we explore the role of the neuronal intrinsic characteristic spike frequency adaptation (SFA) and its steady-state gating variable dynamics on the generation of theta oscillation. The paper proposes a novel possible pathogenesis of theta oscillation by exploring the dynamics of SFA current and its gating variation. • We dig the implicit variations of (a) and (b) by the means of Cubature Kalman filter (CKF) and further observe the simulation results in (a) and (b) via utilizing electroencephalogram data. We verify the implicit dynamics of inhibitory proportion, inhibitory synaptic loss and the gating variable of SFA via CKF. • Utilizing the mean field theory to simplify the simulation model, we give the theoretical deduction to verify the effect of inhibitory proportion, inhibitory synaptic loss and the gating variable of SFA on the generation of pathological theta oscillation. The theoretical variations are consistent with the simulation results in (a) and (b). • In summary, our work is based on previous experiments to investigate a possible mechanism of the pathological theta oscillation from simulation model, data-driven verification and theoretical deduction. The brain rhythm is strongly associated with the brain function. Alzheimer's disease (AD) is characterized reflected by the brain rhythm switching from the alpha band (9–12 Hz) to the theta band (4–8 Hz), accompanied with the loss of brain function. However, extracting the implicating intrinsic characteristic variations of the brain network by utilizing the Electroencephalogram (EEG) information is challenging. Kaman observer, serving as an effective Bayesian technique, can provide a visualization service for probing the intrinsic characteristics underlying the pathological theta oscillations. This work first establishes an excitation-inhibitory neural network model and explores the role of the fraction of the inhibitory neurons and inhibitory synapses in the pathological theta oscillation. The results indicate that the reduced inhibitory neuronal proportion and attenuated inhibitory synaptic weight are the main neural bases of the frequency reduction of neural oscillation. Then, we further explore the intrinsic spiking characteristic by considering spike frequency adaptation (SFA) to the inhibitory neurons. The results show that the SFA reduces the firing rate of neurons, which facilitates the theta rhythm. The enhancement of SFA current by increasing the time constant of its gating variable can further decrease the theta frequency from 7 Hz to 4 Hz. Furthermore, for this high-dimensional nonlinear excitation-inhibitory neural network model, cubature Kalman filter (CKF) is employed to estimate the above potential variations from the EEG data. The observation results show that the attenuated trends of the inhibitory neuronal proportion and the decreased inhibitory SFA current result in the descending brain rhythm. Finally, the theoretical simulation is deduced by utilizing the mean field theory for simplifying high-dimensional model to verify the simulation results. The theoretical variations of inhibitory parameters and adaptation gating variable are consistent with the simulation results. In summary, we investigate the multi-origin factors related to inhibitory neuronal intrinsic characteristics from forward model simulation and inverse EEG estimation process. And we further verify the simulation and data-driven results by theoretical derivation. This work enhances the understanding of the systematic function of inhibitory intrinsic characteristics on pathological theta oscillation and provides an effective method to decode the dynamics underlying neural activities. [ABSTRACT FROM AUTHOR]

Given the critical role of time delays in epidemic modeling, this paper delves into the dynamics and finite-time optimal stabilization of a novel epidemic system characterized by such delays. Our findings reveal that time delays significantly influence both the system's dynamics and the formulation of an optimal control strategy. Specifically, the system's endemic equilibrium point remains locally asymptotically stable under mild conditions for small time delays. However, exceeding critical delay thresholds induces Hopf bifurcation. To closely regulate the system's state towards the equilibrium point and minimize control costs, we propose an optimal control problem. We derive the explicit form of the optimal control strategy employing Pontryagin's Maximum Principle. Finally, numerical simulations further confirm the theoretical results obtained in this paper. • A time-delayed SEAIR epidemic model with a general incidence function is studied. • We focus on discussing the effects of time delays on dynamics and optimal control. • The theoretical results are well verified by the numerical simulation results. [ABSTRACT FROM AUTHOR]

Based on m -norm, this paper investigates global exponential synchronization (GES) for BAM memristive neural networks (BAMMNNs) with mixed delays and reaction–diffusion (RD) terms. Different from the existing literatures, this paper discusses the GES of the NNs based on a new integral inequality with infinite distributed delay. This method is based on inequality technique and comparison principle, which makes the form of Lyapunov function and controller more simple. Next, by introducing two different control strategies and the concept of driven response, two sufficient conditions are got to ensure GES of the proposed system. It is noteworthy that the results obtained by algebraic inequality are extension of the previous conclusions. Finally, two instances verify the correctness of the conclusions. • In this paper, a new integral inequality involving infinite distributed delays is established by using inequality technique. By using this inequality, a new conclusion on GES of BAMMRDNNs is obtained. • By using Lyapunov functions and inequality methods, two different control methods are proposed, then the GES criteria based on m norm are derived. Compared with the existing results [41,47,56], some strong assumptions are eliminated, which makes the results easier to verify. [ABSTRACT FROM AUTHOR]

• A new car-following model is proposed to describe the dynamic behavior of autonomous vehicle. • The bifurcation characteristic of traffic flow composed of autonomous vehicle is analyzed. • A controller that considers multi-step memory and multi-step prediction effect is designed. • A hybrid control strategy is proposed, which combines multi-step prediction and memory mechanism with PID. This paper is committed to capturing the dynamic behaviors of homogenous flow of autonomous vehicles (AVs), and exploring the control strategies to improve traffic conditions, which can alleviate traffic congestion and improve traffic efficiency. Firstly, a car-following model of AVs considering real-time driving state is established. Secondly, based on bifurcation theory and stability theory, bifurcation analysis is carried out and the relationship between bifurcation and stability is revealed. In order to suppress the bifurcation and improve the stability, a controller considering multi-step prediction and memory mechanism (MPM) is designed, and the root trajectories for eigenvalues and stable time length of the model controlled by MPM controller are calculated. In response to the limitations of the MPM controller, a hybrid controller including the MPM controller and PID controller is further proposed, and it is found that the model controlled by hybrid controller has greater range of stable bifurcation parameter and stable time length, which means better ability of bifurcation suppression. Finally, the capabilities of the controller proposed in this paper are effectively demonstrated by numerical experiments in MATLAB and simulation experiments in the ROS-Gazebo environment. [ABSTRACT FROM AUTHOR]

In this paper, we develop a nonconservative kinetic framework to be applied to the study of immune system dysregulation. From the modeling viewpoint, the model regards a system composed of stochastically interacting agents, under the action of an eternal force field. According to the application perspectives of this paper, the external force field has a specific analytical shape. In this case, some analytical results are proved, i.e. existence, uniqueness, positivity, and boundedness of solution of the related Cauchy problem, at least locally in time. Then, the model is refined to be implemented for the study of treatment strategies in case of autoimmune response. Specifically, we distinguish the autonomous case from the nonautonomous one, representing the absence or delivery of drugs, respectively. The former allows us to gain some stability results. Whereas, the latter is qualitatively studied. Numerical simulations are provided for both schemes. • A nonconservative kinetic model with external force field. • Solution existence, uniqueness, positivity, boundedness for nonautonomous system. • Kinetic system applied to autoimmune treatment strategy. [ABSTRACT FROM AUTHOR]

This paper focus on the design issue of event-based dissipative filter for quantized nonlinear singular systems. To save the communication resource, we employ a dynamic quantizer to quantized the measurement output signal prior to transmitting it to the filter via digital communication. Furthermore, the paper also presents an event-triggered mechanism for determining the transmission of the quantized signal. The intention of this paper is to try to propose an event-triggered filter to ensure that the filtering error system (FES) is admissible and satisfies strictly dissipativity performance, in the presence of dynamic quantization and occurring uncertainties. The Lyapunov function is utilized to establish a sufficient condition for the FES to meet the strictly dissipative performance. Depending on the obtained conditions, the required filter parameters are proposed using the standard linear matrix inequality (LMI) technique. Ultimately, the effectiveness of the developed method is verified by an actual circuit simulation. • The problem of dissppativity-based robust filter design for singular systems is investigated. The admissibility and the dissipative performance of the filtering error system are discussed. • The event triggering mechanism, dynamic quantization and parameter uncertainty are also studied in the filtering error system. By introducing strict LMI, the design conditions are given to ensure that the filtering error system is admissible and possesses strictly dissipative performance. • The quantizers dynamic parameter is designed within the quantization range based on the LMI conditions. [ABSTRACT FROM AUTHOR]

• Handling noninteger systems with constraints, unknown directions and nonlinearities. • New method approximates actuator nonlinearities using caputo fractional derivatives and continuous functions. • Dynamic surface control techniques to prevent complexity explosion in backstepping. • Event-triggered control to conserve resources, reduce communication burden and boost efficiency. • Time-varying asymmetric barrier lyapunov to ensure constraints are not violated. This paper addresses issues and challenges associated with approximation-based adaptive neural event-triggered output feedback control schemes for a group of non-integer order non-strict feedback systems subject to asymmetric time-varying pseudo-state constraints, unknown control directions, and input nonlinearities. The actuator nonlinearities are first approximated using Caputo fractional derivative definitions and novel continuous functions. After that, by introducing auxiliary non-integer order integrators, the original non-affine plant is transformed into an augmented affine system. Furthermore, neural networks, a high-gain observer, and Nussbaum-type functions are applied to deal with the unknown functions, the immeasurable pseudo-states, and the unknown control directions, respectively. In parallel, by utilizing non-integer order filters, the dynamic surface control technique is incorporated to eliminate the "explosion of complexity" that is frequently encountered with backstepping approaches. In addition, an event-triggered mechanism is integrated into the control design procedure to save computational resources and reduce communication burden. Time-varying asymmetric barrier Lyapunov functions are built with error variables to guarantee that the full pseudo-state constraints are not violated. Based on the Lyapunov stability theory, theoretical analysis indicates that the proposed control scheme ensures that (1) the Zeno behavior is excluded, (2) all the closed-loop signals are bounded and (3) the tracking errors converge to the origin asymptotically in finite time. This paper makes the following contributions: (1) New fractional differential inequalities are developed to extend traditional approaches for the stability analysis and the controller design procedure of integer order systems to fractional-order systems; (2) Compared with existing achievements, the proposed adaptive event-triggered control strategy is more advantageous in practice. Finally, simulation studies are worked out to prove the validity of the proposed approach and corroborate the theoretical results. [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]