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• 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]

• 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]

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]

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]

• 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]

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]

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]

• 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 article, we explore the general decay anti-synchronization (GDAS) and general decay H ∞ anti-synchronization (GDHAS) of derivative coupled delayed memristive neural networks (DCDMNNs) with constant and delayed state coupling, respectively. To begin with, on account of the definitions of ψ -type function as well as ψ -type stability, we present the GDAS and GDHAS concepts for the considered DCDMNNs. What is more, several sufficient conditions are derived for reaching GDAS and GDHAS of DCDMNNs with constant and delayed state coupling by selecting correct Lyapunov functionals and devising a proper controller. Moreover, numerical simulations examples are provided to demonstrate the feasibility of the obtained conclusions. • The network model of DCDMNNs is firstly presented in this paper. • This is the first paper towards to studying GDAS of DCDMNNs. • The concept of GDHAS for DCDMNNs is firstly introduced in this paper. • Several criteria are derived to ensure that the DCDMNNs achieve GDAS and GDHAS. [ABSTRACT FROM AUTHOR]

– In this paper, we consider the following stationary two-species chemotaxis-competition system with signal-dependent diffusion and sensitivity (0.1) − d 1 Δ u = λ u − u 2 − b u v , x ∈ Ω , − div [ d (w) ∇ v + χ (w) v ∇ w ] = μ v − v 2 − c u v , x ∈ Ω , − d 3 Δ w = − κ w + τ u , x ∈ Ω , u = v = w = 0 , x ∈ ∂ Ω in a bounded smooth domain Ω ⊂ R N (N ≥ 1) , where λ , μ , b , c , d 1 , d 3 , κ , τ are positive constants, and (d (w) , χ (w)) ∈ [ C 1 [ 0 , ∞) ] 2 with d (w) , χ (w) > 0 for all w ≥ 0. Since there does not exist an immediate change variable that transforms (0.1) into a semilinear system when (0.1) is considered with d (w) , α (w) being arbitrary functions in w , this makes the analysis of system (0.1) much more difficult. By the W 2 , p -estimate and the global bifurcation method, we obtain a bounded connected set of positive steady states joining the semitrivial solution of the form (u , 0 , w) with u , w > 0 and the semitrivial solution of the form (0 , v , 0) with v > 0 , which provides the sufficient condition for the existence of positive steady states. Combining the eigenvalue theory, homogenization technique and various elliptic estimates, we also derive some sufficient conditions for the nonexistence of positive steady states. • Unlike the previous Neumann works, this paper focuses the Dirichlet problem. • The non-existence and existence of coexistence states are studied. • The structure of steady–state bifurcation is described. [ABSTRACT FROM AUTHOR]

Discrete grey forecasting models based on an accumulative operator have been widely used in many practical fields. With the development of grey forecasting models, it is a problem to be solved to further analyze internal mechanisms and unify the structures. This paper aims to reconstruct the model from a perspective of sequence characteristics and simplify the modeling steps under the condition of ensuring the accuracy of the model. First, this paper analyzes dynamic sequence evolution hidden and mines relationship between the structure and original sequence features contained in discrete grey forecasting model. Then, the reconstruction is carried out to prove the equivalence and quantitative relation between reduced-order model and original model. Under order recursive estimation, new parameters are addressed. Finally, theoretical correctness is verified by large-scale numerical simulation. Moreover, the reduced-order model is applied for prediction on the peak of battery incremental capacity and capacity degradation. Results show the effectiveness and superior prediction performance of the reduced-order model, where MAPEs of grey forecasting models have controlled under 4%. • This paper reconstructs grey forecasting models to simplify the unified structure. • A new iteration applied reduces the repeatability and complexity of parameter estimation. • Equivalence relations are revealed between the reconstructed model and traditional one. • Reduced-order reconstruction provides a novel modeling process for grey modeling. • The reconstructed model has good prediction performance for real-world applications. [ABSTRACT FROM AUTHOR]

In this paper, we introduce some novel concepts within the realm of fuzzy bipolar metric spaces, namely (α η)-contractive type covariant mappings and contravariant mappings, and (β χ)-contractive type covariant mappings. We establish some fixed point theorems that demonstrate both the existence and uniqueness of fixed points for (α η)-contractive type covariant mappings and contravariant mappings, and for (β χ)-contractive type covariant mappings in complete fuzzy bipolar metric spaces utilizing the triangular property. Additionally, to substantiate the findings, some illustrative examples and consequential outcomes are presented. Furthermore, the proven results serve to extend, generalize, and enhance the corresponding outcomes documented in existing literature. A practical application of these findings in the context of non-linear Volterra integral equations is demonstrated, solidifying and reinforcing the credibility of the established results. Overall, this paper contributes to the understanding of fixed point theory in the context of fuzzy bipolar metric spaces and highlights the significance of (α η)-contractive mappings and (β χ)-contractive mappings in this domain. [ABSTRACT FROM AUTHOR]

This paper investigates the problem of secure synchronization in aperiodic intermittent event-triggered pinning control for discrete-time complex networks (DCNs) against multi-pattern link attacks. Firstly, in order to reduce communication burden and control cost, a novel aperiodic intermittent event-triggered control (AIEC) with discontinuous characteristics is designed based on periodic sampling, where triggering instants are determined by the intermittent control (IC). Secondly, multiple pattern attack are modeled, as they can interrupt different edges and change the network topology. In addition, multi-pattern attacks for each link are independent and their impact on the coupling topology is analyzed. Thirdly, under destroyed network topology, this paper designs aperiodic intermittent event-triggered pinning control (AIEPC) by combining pinning control (PC) with the AIEC. Meanwhile, based on PC, the isolated node and pinned nodes form a like-directed spanning tree with an asymmetrical coupling matrix, where rooted at the isolated node and only directed connections between the isolated node and pinned nodes. Fourthly, using the segmentation analysis method and the inequality iteration technique, a sufficient condition for exponential synchronization of error system is obtained by considering the instants neighboring different intermittent control and attack intervals. Finally, a simulation example on Chua's circuit network is provided to verify the validity of the theoretical results achieved in this paper. • Aperiodic intermittent event-triggered control strategy is proposed, the controller operates solely during working intervals and determines if the control signal is transmitted. • The effect of mutually independent multi-pattern link attacks on network topology is investigated, different types of link attacks can break different edges, resulting in different topologies. • Various types of like-directed spanning tree with asymmetric coupling matrix are established based on the pinning control, in which only the isolated node to pinned nodes are directed edges. [ABSTRACT FROM AUTHOR]

In this paper, we employ the Lyapunov theory to generalize the finite time stability (FTS) results from general deterministic impulsive systems to impulsive stochastic time-varying systems, which overcomes inherent challenges. Sufficient conditions for the FTS of the system under stabilizing and destabilizing impulses are established by using the method of average dwell interval (ADT). For FTS of stabilizing impulses, we relax the constraint on the differential operator by allowing it to be indefinite rather than strictly negative or semi-negative definite. Furthermore, the theoretical results are applied to impulsive stochastic neural networks. Finally, two numerical examples are given to validate the reliability and practicability of the obtained results. • This paper extends the results from deterministic systems to stochastic systems. • The differential operator LV can be an indefinite derivative, not just semi-negative. • We first apply the FTS theory of impulsive stochastic systems to study. [ABSTRACT FROM AUTHOR]

This paper continues to explore the membrane potential reconstruction and pattern recognition problem in a neural tissue modeled by Stochastic Dynamic Neural Field (SDNF) equation. Although recent research has suggested an efficient solution based on the state-space approach through nonlinear Bayesian filtering framework, it is becoming extremely difficult to ignore the existence of non-Gaussian uncertainties in the SDNFs as well as the stability problem of neuronal population dynamics to outliers. Motivated by recent events in signal processing and mathematical neuroscience, this paper explores the SDNFs in a presence of non-Gaussian uncertainties, which is the shot noise case, where the corrupted data might appear due to broken sensors. We derive the "distributionally robust" state estimator for the membrane potential reconstruction process that is the Maximum Correntropy Criterion Extended Kalman Filter (MCC-EKF) as well as its fast and numerically robust (to roundoff) implementation method by using the sequential principle of processing the measurement vectors. The numerical experiments are provided to illustrate the performance of the novel estimation methods. • The reconstruction procedure in SDNFs with non-Gaussian uncertainties is designed. • The incomplete measurements and broken sensors case yields shot noise and outliers. • The problem is solved by Maximum Correntropy Criterion Extended Kalman filters. • The MCC-EKFs provide a better estimation quality compared to previous SDNF-EKFs. [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]

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 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 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]

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]

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]

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]

• 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]

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]

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]

• 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]

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 study a first-order Euler semi-implicit finite element scheme for the two-dimensional incompressible Boussinesq equations for magnetohydrodynamics convection with the temperature-dependent viscosity, electrical conductivity and thermal diffusivity. In finite element discretizations, the mini finite element is used to approximate the velocity and pressure, and the piecewise linear finite element is used to approximate the magnetic field and temperature. The unconditional stability of the proposed scheme is proved. By introducing three projection operators with variable coefficients and using the method of mathematical induction, we obtain optimal error estimates under a CFL type condition. Finally, numerical examples are provided to demonstrate these convergence rates. • We studied a linearized Euler finite element scheme for the 2D temperature-dependent MHD-Boussinesq equations. • The proposed scheme is unconditionally stable. • The optimal error estimates are derived. [ABSTRACT FROM AUTHOR]

In this paper, we consider a kind of multi-term Caputo tempered fractional stochastic differential equations and prove the existence and uniqueness of the true solution. Then we derive an Euler–Maruyama (EM) scheme to solve the considered equations. In view of the huge computational cost caused by the EM scheme to achieve reasonable accuracy, a fast EM scheme is proposed based on the sum-of-exponentials approximation to improve its computational efficiency. Moreover, the strong convergence of our two numerical schemes are proved. Finally, several numerical examples are carried out to support our theoretical results and demonstrate the superior computational efficiency of the fast EM scheme. • We study SDEs with multi-term Caputo tempered fractional derivatives. • We construct a fast EM scheme based on the sum-of-exponentials approximation. • The numerical scheme has been proven to be strong convergent and efficient. [ABSTRACT FROM AUTHOR]

In this paper, we consider a first-order projection finite element scheme for the three dimensional incompressible magnetohydrodynamics system based on a magnetic vector potential formulation by writing the magnetic induction B = curl A , where A is a magnetic potential. The main advantage of this projection scheme has two-fold. One is that numerical solutions of velocity field and magnetic induction both satisfy the divergence-free condition in fully discrete level. Another is that the proposed scheme is unconditionally stable for any mesh size and time step size. Under a reasonable regularity assumption, we derive spatial–temporal error estimates of the velocity and magnetic vector potential. Finally, numerical results are displayed to illustrate convergence rates. • The first-order projection finite element scheme was proposed for the incompressible vector potential MHD system. • The proposed projection scheme is a linearized semi-implicit scheme and is unconditionally stable. • The optimal error estimate was proved. [ABSTRACT FROM AUTHOR]

The periodic and quasi-periodic solutions of the integrable system have been studied for four decades based on the Riemann theta functions. However, there is a fundamental difficulty in representing the solutions numerically because the Riemann theta function requires several transcendental parameters. This paper presents a novel method for the numerical representation of such solutions from the algebraic treatment of the periodic and quasi-periodic solutions of the Baker–Weierstrass hyperelliptic ℘ functions. We demonstrate the numerical representation of the hyperelliptic ℘ functions of genus two. • The hyperelliptic solutions of the KdV equation are displayed numerically. • It is proved algebraically that hyperelliptic p-functions obey the KdV equation. • The p-function is numerically expressed without theta functions. • The structure of the symmetric product of curves is expressed graphically. [ABSTRACT FROM AUTHOR]

The primary objective of this research paper is to present a novel and effective meshless numerical approach for solving the 2D time fractional reaction diffusion system with distributed order on an arbitrary domain. Gauss–Legendre quadrature formula is applied to discretize distributed-order derivative integral. We establish the piecewise parabolic fractional interpolation theory and with its assistance, the proposed approach can proficiently solve the non-smooth solutions of the equations and more accurately approximate the Caputo fractional derivative. This meshless method based on improved neural network bases combines the high accuracy approximation advantage and strong express ability of neural network to construct the basis functions set on arbitrary domains, which significantly reduces a computational consumption. The bases constructed based on the neural network allow the selection of 12 bases numbers to achieve an approximation equivalent to that of 1000 ordinary bases. The theoretical analyses of error and convergence order for the meshless approach are carried out. Numerical examples are implemented to validate the high precision and capability of the meshless numerical approach. • A meshless based on piecewise parabolic fractional interpolation theory and improved neural network bases is proposed. • The method effectively solves the non-smooth solutions and achieves high-precision approximation. • The approach overcomes the difficulty of finding appropriate bases functions and reduces the amount of calculation. [ABSTRACT FROM AUTHOR]

A convex rectangular pentagon, also called house-shaped , is a pentagon with the added restriction that two non-adjacent sides have equal lengths, each of which forms a right angle with the intervening side. In this paper, we focus on the existence of central configurations of the 5-body problem, where the five bodies are in a symmetric house-shaped configuration. That is, when the five bodies are located at the vertices of a convex rectangular pentagon, and moreover, the body not belonging to the two parallel sides lies along the symmetry line. • We consider a convex, house-shaped rectangular pentagon, which is a symmetric pentagon. • The central configurations of five masses forming a house-shape are characterized. • We prove the existence of convex central configurations for 5 bodies arranged in a house-shaped. • We provide numerical evidence supporting the existence of a concave central configuration in a house-shaped arrangement. [ABSTRACT FROM AUTHOR]

In solving real world systems for higher-codimension bifurcation problems, one often faces the difficulty in computing the normal form or the focus values associated with generalized Hopf bifurcation, and the normal form with unfolding for higher-codimension Bogdanov–Takens bifurcation. The difficulty is not only coming from the tedious symbolic computation of focus values, but also due to the restriction on the system parameters, which frequently leads to failure of the conventional approach used in the computation even for simple 2-dimensional nonlinear dynamical systems. In this paper, we use a simple 2-dimensional epidemic model, for which the conventional approach fails in analyzing the stability of limit cycles arising from Hopf bifurcation, to illustrate how our method can be efficiently applied to determine the codimension of Hopf bifurcation. Further, we apply the simplest normal form theory to consider codimension-3 Bogdanov–Takens bifurcation and present an efficient one-step transformation approach, compared with the classical six-step transformation approach to demonstrate the advantage of our method. [ABSTRACT FROM AUTHOR]

In this paper we develop a systematic reduction procedure for determining intermediate integrals of second order hyperbolic equations so that exact solutions of the second order PDEs under interest can be obtained by solving first order PDEs. We give some conditions in order that such a procedure holds and, in particular, we characterize classes of linear second order hyperbolic equations for which the general solution can be found. [ABSTRACT FROM AUTHOR]

In this study, we develop a phase-field model to describe the solid–liquid phase changes, heat conduction phenomena, during the selective laser melting process. This model is based on the variational principle of minimizing the free energy functional. The proposed model integrates the phase-field equation and the energy equation, which are used to capture the dynamical behavior of the interfacial evolution. We use the semi-implicit Crank–Nicolson scheme and central difference to ensure second-order accuracy in time and space. The numerical scheme is unconditional energy stable. This paper rigorously proves the energy stability of the phase-field model of the Selective Laser Melting process, which confirms the numerical stability and the physical rationality of the solution. Various numerical experiments are performed to verify the robustness of our proposal model. This model can effectively simulate the energy transfer and shape structure changes of the products during the selective laser melting manufacturing process, which provides a reliable guarantee for predicting and optimizing the quality and performance of the selective laser melting process additive manufacturing process. • A phase-field based modeling for the selective laser melting process is established. • A Crank–Nicolson scheme is adopted to ensure second-order accuracy in time and space. • A digital model reflecting the actual manufacturing process is developed. • Various experiments are performed to verify the robustness and efficiency of our model. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of doubly perturbed uncertain differential equations driven by canonical process that is the counterpart of Wiener process in the framework of uncertain theory. We give some sufficient conditions for the existence and uniqueness of solutions of the present problem under some weak conditions, where a convex combination of the Nagumo and Osgood conditions is considered. Subsequently, we consider the stability properties with regard to the initial value and coefficients. Finally, an example is provided to illustrate the effectiveness of our main results. • Existence and uniqueness are obtained under some weak conditions. • The stability properties of doubly perturbed UDEs are first considered. • Extending the classical hypothesis on the perturbed intensity to the generic case. [ABSTRACT FROM AUTHOR]

Scientists are aware of the importance of studying interdependent networks, as they represent real-world scenarios better than isolated networks, such as the interdependent communication and power networks. However, in real scenarios, higher-order interactions exist in these networks. Recent research has shown that higher-order interactions that cannot be reflected in simple networks (i.e., a network only has pairwise interaction) can be reflected through hypergraph. This paper proposes an interdependent hypergraph model that considers the dependencies of interlayer nodes. We studied the cascading failures in the hypergraph with different interlayer dependencies. Through theoretical analysis, we have determined the maximum attack intensity the network can withstand and how its robustness changes under different attack intensities. In completely interdependent hypergraph, the network becomes increasingly fragile as the attack intensity increases, and the final network size suddenly jumps to zero, indicating a first-order phase transition in the network. We conducted experiments on multiple artificially synthesized hypergraph. The experimental results indicate that our analytical solution is consistent with the simulated solution. These findings will help us better understand the impact of different topologies on network robustness in interdependent hypergraph, enabling us to take more effective measures to address potential network failures and attacks. • Two interdependent hypergraph models are proposed based on layer dependencies. • Consider the impact of topology on the robustness of interdependent hypergraphs. • The critical value of hypergraph collapse is determined through theoretical analysis. • There is a first-order phase transition for completely interdependent hypergraphs. [ABSTRACT FROM AUTHOR]

• Novel Nanocomposite Design: Dual-FG circular plate with FG polymeric matrix and FG-GPL reinforcement distribution. • Different effect of GPL distribution patterns when using isotropic and FG matrix. • Kerr's nonlinear elastic foundation as a four-parameter foundation has a great power in adjusting system frequencies. • Asymmetric GPL distributions in FG matrix have dual behavior in low and high volume fractions. This paper studies the geometrically nonlinear free vibration of a novel nanocomposite circular plate. The matrix of the nanocomposite structure is made of functionally graded (FG) polymer, and the distribution of graphene platelets (GPLs) as the reinforcement is assumed based on the five various linear FG models. Therefore, this novel nanocomposite is called dual-FG. The Voigt rule of mixture and Halpin-Tsai method are employed to homogenize FG polymer, and distribution of the GPLs within the matrix, respectively. Moreover, the whole structure is fixed on a nonlinear Kerr elastic foundation. The displacements of the plate are estimated by means of the first-order shear deformation theory (FSDT). Von-Kármán type of geometrically nonlinear relations expresses the plate's strains. The generalized differential quadrature (GDQ) technique, bilateral iterative displacement control method (BIDCM), and the weighted residual Galerkin procedure are implemented to extract the free vibration characteristics of the structure. The developed structure is validated with those reported in the open literature. A comprehensive parametric study is conducted to illustrate the effect of various parameters on the dynamic response of the dual-FG nanocomposite circular plate. It will be observed that the influence of the volume fraction and distribution model of GPL is more pronounced on the vibration frequency in the FG matrix. It was noted that the volume fraction of graphene platelets (GPL) and the power-law index of the matrix exhibit a direct and inverse relationship, respectively, with the natural frequency of the structure. Furthermore, the V-GPL model exhibits the highest percentage growth in nonlinear frequencies, while the Λ-GPL model shows the least growth. This observation is particularly pronounced for plates with hinged boundary conditions. [ABSTRACT FROM AUTHOR]