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• 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]

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]

• Fixed-time synchronization of a class of memristor-based neural networks were investigated. • Two control schemes are proposed to achieve fixed-time synchronization. • The impulsive effects are overcomed. • Finally, an example and its simulations are given to demonstrate the feasibility of the obtained results. This paper concerns the fixed-time synchronization of a class of memristor-based neural networks with impulsive effects. Since impulses may destroy the synchronization, it is indispensable to design suitable controllers to control its unpredictable effects. In the present paper, two control schemes are proposed to achieve fixed-time synchronization of the impulsive memristor-based neural networks. Finally, two examples and their numerical simulations are presented to show the effectiveness of the theoretical results. [ABSTRACT FROM AUTHOR]

• Sufficient conditions are established for a singular Cucker–Smale model with a general target motion pattern driving force to admit an asymptotic flocking. • A critical value of the exponent in the communication weight is given to lead to global regularity of solutions. • The avoiding collision asymptotic flocking of Cucker–Smale model with a general target motion pattern driving force is obtained for the first time in the literature. • It is the first time to show that the particles finally come to a line-shape formation with collision avoiding by theoretical proof in this paper. This paper is concerned with a singular Cucker–Smale model with a general targeted pattern driving force, which ensures that the particles finally come to the desired spatial configuration. Under some suitable conditions, we prove that the exponent α ≥ 1 in the communication weight leads to global regularity of solutions so that the system has an asymptotic flocking without collision between any agent. As an example, a prescribed driving force is given to demonstrate that the particles finally come to a line-shaped formation with collision avoidance, which can be viewed as a reasonable explanation for the wild geese flying in a line-shape in the sky. Moreover, this mechanism can be applied to the UAVs forming a line-shape to complete some special military missions. These results are novel, which are illustrated by both theoretical analysis and numerical simulations. [ABSTRACT FROM AUTHOR]

This paper proposes a simple model of anomalous diffusion, in which a particle moves with the velocity field induced by a single "dipole" (a doublet or a pair of source and sink), whose moment is modulated randomly at each time step. A motivation to introduce such a model is that it may serve as a toy model to investigate an anomalous diffusion of fluid particles in turbulence. We perform a numerical simulation of the fractal dimension of the trajectory using periodic boundary conditions in two and three dimensions. For a wide range of the dipole moment, we estimate the fractal dimension of the trajectory to be 1.5–1.9 (2D) and 1.6–2.7 (3D). • This paper proposes a simple model of anomalous diffusion, in which a particle moves with the velocity field induced by a single "dipole" (a doublet or a pair of source and sink), whose moment is modulated randomly at each time step. • A motivation to introduce such a model is that it may serve as a toy model to investigate an anomalous diffusion of fluid particles in turbulence. • We perform a numerical simulation of the fractal dimension of the trajectory using periodic boundary conditions in two and three dimensions. • For a wide range of the dipole moment, we estimate the fractal dimension of the trajectory to be 1.5–1.9 (2D) and 1.6–2.7 (3D). [ABSTRACT FROM AUTHOR]

The issue of group consensus for heterogeneous fractional-order multi-agent systems under the cooperation-competition networks with time delays is investigated in this paper. Novel group consensus control protocols with input and communication delays are designed based on cooperative-competitive interaction. The considered multi-agent systems consists of fractional order dynamics with the single integrator and the double integrator, and the speed of agents is not known. The matrix theory, frequency domain approach and graph theory are used to figure out the sufficient conditions for group consensus under the switching and fixed topology, respectively. Finally, numerical simulation examples are given to verify the correctness of the theoretical results. [ABSTRACT FROM AUTHOR]

Highlights • We firstly propose a new periodically intermittent discrete observation control. • Synchronization issue of unified fractional-order chaotic coupled systems is discussed. • The method combined graph theory with Lyapunov method is adopted successfully. • We take nonlinear coupling term into account in our model, which is more general. Abstract In this paper, we propose a novel control technique, named as periodically intermittent discrete observation control (PIDOC), to investigate the synchronization issue of fractional-order coupled systems. Distinguished from periodically intermittent control which is widely applied in control fields, PIDOC proposed in this paper adopts discrete-time state observations in work time during a control period. In this way, PIDOC is more reasonable and available than traditional periodically intermittent control based on continuous-time state observations. Moreover, different from previous work about synchronization of fractional-order coupled systems, coupling term considered in this paper is nonlinear, which is more general. Then, combining graph theory with Lyapunov method, several synchronization criteria are obtained. Next, we successfully employ PIDOC to investigate synchronization of unified fractional-order chaotic coupled systems. Finally, two numerical simulations are presented to verify the validity of our theoretical results and the effectiveness of control scheme. [ABSTRACT FROM AUTHOR]

This paper investigates the issue of fixed-time intralayer synchronization of stochastic switched duplex networks (SSDNs) with limited communication. Firstly, control scheme without sign function is created to explore intralayer synchronization of SSDNs within a fixed time, and the control gains can vary with the error information. By the Lyapunov functional method, some novel synchronization criteria are established. In addition, by means of introducing the channel matrices, this paper considers the limited communication. Moreover, numerical simulation indicates the correctness of our theoretical result. Finally, from the perspective of the application, image encryption is demonstrated to confirm our theoretical implementation. • The switched duplex networks with stochastic disturbances are considered. • Synchronization criteria are obtained via fixed-time and finite-time control. • Limited communication is considered by introducing channel matrices. • Image encryption is realized based on our theoretical results. [ABSTRACT FROM AUTHOR]

This paper deals with the modeling of the management of the energy-demand in a microgrid network connected to a macrogrid network by means of a generalized kinetic theory model. Specifically the microgrid network N , composed of three energy sources, is connected to an external energy distribution network N e. The role of the network N e is to manage the absorption of the energy in the network N. The numerical investigations are addressed to the ability of the proposed model to reach the energy-demand. Specifically, a constant energy-demand and a time-dependent energy-demand are analyzed. The numerical simulation shows that the model is able to reproduce two different behaviors: on the one hand the three energy sources are able to satisfy the related energy-demand; on the other hand the three energy sources are not able to supply the prescribed amount of energy and the action of the external network N e is required. The operating mode of the network N , namely isolated (stand-alone) or connected to the external network N e , is shown by performing a sensitivity analysis on the functional parameters. Discussions and research perspectives are postponed to the last section of the paper. • Management of the energy-demand. • Microgrid network connected to a macrogrid network. • Generalized kinetic theory model. • Constant energy-demand and a time-dependent energy-demand. • Sensitivity analysis on the functional parameters. [ABSTRACT FROM AUTHOR]

In this paper, we construct and analyze a reaction–diffusion hybrid model incorporating impulsive control and periodic evolution domain in Wolbachia -infected and wild mosquito populations to investigate the joint impact of spatial diffusion, impulsive control and the evolution of a domain on the control of mosquitoes. The explicit formulas of the two ecological reproduction indexes R U and R I are introduced, which relate to the evolution rate of domain and the impulsive function. Then we establish the threshold-type dynamics for the impulsive problem in terms of R U and R I by the method of upper and lower solutions, which present the extinction of mosquitoes, and competition and the coexistence of Wolbachia -infected and wild mosquitoes under impulsive control and periodic evolution domain. Numerically, we perform simulations to certify the analytical results, determine the importance of parameters on the persistence and extinction of mosquito populations, and expound how impulsive control, the evolution domain and fitness effect caused by Wolbachia affect the mosquito population evolution. • A model including impulsive control, Wolbachia and evolving domain is developed. • The formulas of the two ecological reproduction indexes (R U and R I) are derived. • The threshold dynamics in terms of R U and R I are fully solved. • Numerical simulations expound the effect of key parameters on the mosquitoes. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new numerical simulation iterative algorithm to solve the split feasibility problem and the fixed point problem with demicontractive mappings. Our algorithm mainly involves primal-dual iterative, relaxed projection, inertial technique and self-adaptive step size. Under reasonable conditions, the strong convergence of our algorithm is established. Moreover, we provide some numerical simulation examples to demonstrate the efficiency of our iterative algorithm compared to existing algorithms in the other literature. [ABSTRACT FROM AUTHOR]

A novel messages spreading model is suggested in this paper. The model is a natural generalization of the SIS (susceptible-infective-susceptible) model, in which two relevant messages with same probability of acceptance may spread among nodes. One of the messages has a higher priority to be adopted than the other only in the sense that both messages act on the same node simultaneously. Node in the model is termed as supporter when it adopts either of messages. The transition probability allows that two kinds of supports may transform into each other with a certain rate, and it varies inversely with the associated levels which are discretely distributed in the symmetrical interval around original point. Results of numerical simulations show that individuals tend to believe the messages with a better consistency. If messages are conflicting with each other, the one with higher priority would be spread more and another would be ignored. Otherwise, the number of both supports remains at a uniformly higher level. Besides, in a network with lower connected degree, over a half of the individuals would keep neutral, and the message with lower priority becomes harder to diffuse than the prerogative one. This paper explores the propagation of multi-messages by considering their correlation degree, contributing to the understanding and predicting of the potential propagation trends. [ABSTRACT FROM AUTHOR]

This paper describes an updated exponential Fourier based split-step method that can be applied to a greater class of partial differential equations than previous methods would allow. These equations arise in physics and engineering, a notable example being the generalized derivative non-linear Schrödinger equation that arises in non-linear optics with self-steepening terms. These differential equations feature terms that were previously inaccessible to model accurately with low computational resources. The new method maintains a 3rd order error even with these additional terms and models the equation in all three spatial dimensions and time. The class of non-linear differential equations that this method applies to is shown. The method is fully derived and implementation of the method in the split-step architecture is shown. This paper lays the mathematical ground work for an upcoming paper employing this method in white-light generation simulations in bulk material. [ABSTRACT FROM AUTHOR]

This paper formulates a single-species stochastic chemostat model with periodic coefficients due to seasonal fluctuation. When the noise is small, a modified break-even concentration is identified, whose value below or above the averaged concentration of the input nutrient can completely determine whether the microorganism will persist or not, where an accuracy decay rate is given for extinction. In case of persistence, existence of the random positive periodic solution is proved for the considered model. Further, the random periodic solution is shown to be globally attractive under some mild extra condition. The periodic dynamics obtained in this paper are supported by computer simulations. [ABSTRACT FROM AUTHOR]

In this paper, the global exponential stability and stabilization problems are investigated for memristive neural networks (MNNs) with stochastic disturbances, spatial diffusions and distributed delays. The spatial diffusions are not assumed to be symmetric and distributed delays are relaxed to be unbounded. Then, the presented MNNs are modeled as a class of stochastic partial systems with hybrid delays. Based on nonsmooth analysis, the Lyapunov–Krasovskii functional and inequality approach, some simple algebraic criteria are derived for the global exponential stability as well as the stabilization via two kinds of designed feedback controllers. The derived criteria are easily verified and the obtained results are available for other delayed systems with or without stochastic disturbances and spatial diffusions. Finally, the stability and stabilization results are validated by numerical simulations. • The stochastic model with spatial diffusions and distributed delays is discussed. • The derived results are generalized and they contain previous ones as special cases. • The method is available for other stochastic delayed partial differential systems. • The space set of the spatial diffusions in our model is not symmetrical. • The distributed delays are unbounded. [ABSTRACT FROM AUTHOR]

In Electro-Osmotic Pumps (EOPs), a micro fluid flow is formed by applying an external electric field via two electrodes immersed in the electrolyte. The immersed electrodes, due to lying in the fluid flow path, can consider an obstacle that can negatively affect the flow inside the microchannel. This issue is usually neglected in studies performed on the electroosmotic flows. Concerning this motivation, in this paper, a novel concept of a multi-stage EOP (introduced earlier by the researchers) was investigated numerically where the electrodes were attached to the wall of the micropump so as not to obstruct the fluid flow and to facilitate the serialization of micropumps. A two-dimensional numerical code is developed to analyze the steady-state performance of the multi-stage EOP. The governing equations for the fluid flow, internal and external electric fields, and ion concentration distribution are solved using the Finite Volume method. The results showed that in the multi-stage EOP consisting of multiple micropumps connected in series, the maximum pressure of EOP will enhances by increasing the number of micropumps. In contrast, the maximum flow rate remains approximately constant. The relationship between the maximum pressure, and the strength of the applied external electric field, was also investigated based on the electric field strength, and the results were expressed as a mathematical correlation. [ABSTRACT FROM AUTHOR]

Continuous stirred tank reactor (CSTR) is a common reactor in the chemical industry. The accurate observation of the concentration conversion rate of the mixture and the internal temperature of the reaction vessel is a prerequisite for obtaining the desired mixture. This paper proposes a novel observer based on residual neural networks for CSTR systems. Firstly, the mathematical model of the CSTR reaction is given, as well as a detailed description of the structure and equations of the residual neural networks and the designed observer. Then the matrix method is used for the nonlinear isolation of the residual neural networks and the theory of quadratic constraints for nonlinear activation functions of the neural networks is applied. Thus, the convergence of the proposed observer is analyzed theoretically in detail. Finally, the numerical simulations are implemented to demonstrate that the proposed residual neural network-based observer can quickly and accurately observe the state changes during the CSTR reaction. • A novel residual neural network-based observer for CSTR is proposed. • The convergence analysis of the proposed observer is presented. • The selection of hyperparameters of the residual neural networks is given. • The effectiveness of the proposed observer is verified by numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper, we study a predator–prey model assuming that the prey population gathers together in herd and considering feeding satiation for the predator population as well. After analyzing the equilibrium points of the model, their stability and the existence of bifurcations we show the existence of multistability for three different equilibrium points via numerical simulations. This last analysis is performed using the bSTAB software and its extensions. It allows to compute the basin of stability values and to plot bifurcation diagram surfaces with respect to the model parameters. • A predator-prey model is studied assuming that the prey population gathers in herd. • The equilibrium points of the model are analyzed, as well as their stability. • We show the existence of multistability for three different equilibrium points. [ABSTRACT FROM AUTHOR]

In this paper, we construct two kinds of exponential SAV approach with relaxation (R-ESAV) for dissipative system. The constructed schemes are linear and unconditionally energy stable. They can guarantee the positive property of SAV without any assumption compared with R-SAV and R-GSAV approaches, preserve all the advantages of the ESAV approach and satisfy dissipation law with respect to a modified energy which is directly related to the original free energy. We also give the rigorous consistency estimates of the constructed schemes for the L 2 gradient flows. Moreover, the second version of R-ESAV approach is easy to construct high-order BDF k schemes. Especially for Navier–Stokes equations, we construct two kinds of novel schemes based on the R-ESAV method. Finally, ample numerical examples are presented to exhibit that the proposed approaches are accurate and effective. • R-ESAV schemes can improve the accuracy of the solution significantly compared with the original ESAV approach. • We construct two kinds of novel schemes based on the R-ESAV method for Navier–Stokes equations. • Only one linear system with constant coefficients at each time step needs to be solved. • The positive property of SAV without any assumption can be guaranteed. • Numerical simulations are tested to show that the modified energy equals to the original free energy at almost all times. [ABSTRACT FROM AUTHOR]

This paper aims to incorporate a high order diffusion term into a SIQR epidemic model with transient prophylaxis and lasting prophylaxis. The existence and uniqueness of the global positive solution is proven and we find a condition ensuring the extinction of an infectious disease. The existence of a stationary distribution for the stochastic epidemic model is investigated as well. Numerical simulations are conducted to support our theoretical results and an example of application with COVID-19 data from Canada is used to estimate the transmission rate and reproduction number associated with the stochastic model, while constructing a model fitting the data. • A stochastic SIQR epidemic model with higher-order perturbation is investigated. • The existence and uniqueness of the global positive solution are studied. • A stochastic threshold is established for extinction. • COVID-19 data from Canada are used to estimate the transmission rate. [ABSTRACT FROM AUTHOR]

This paper explores the integration and harmonization of an epidemic dynamics model and a resource production–inventory management model, aiming to identify optimal control strategies of epidemic containment. We begin with establishing a generalized optimal control model for epidemic dynamics, which offers a unified framework to verify the sufficient conditions for the existence of the optimal solutions to a class of epidemic optimal control models. Based on the model, a resource production–inventory dynamic model with learning ability is proposed to find optimal strategies of the medical resource management. Subsequently, we propose the optimal control model for a dynamics system of COVID-19 and explore the optimal strategies of the combined applications of vaccine and recommended treatment in comprehensive control of the epidemic. Finally, the viability of effective integration and coordination between the optimal control models of the epidemic dynamics and the resource production–inventory system is verified by numerical simulation and analytical analysis, and a new method is implemented for finding the optimal management strategy for epidemic control. Therefore, this study advances a more comprehensive understanding of the interplay between epidemic dynamics and resource management, illustrating the feasibility and efficacy of integrating multiple models to achieve optimal decisions in both theoretical and practical aspects of epidemic control. • Sufficient condition of optimal solution to a class of optimal control models is set. • New optimal control models of COVID-19 and production–inventory dynamics are set. • Vaccine and treatment based medical resource optimal management way is implemented. • A multi-model integrated approach of optimal epidemic control strategies is explored. [ABSTRACT FROM AUTHOR]

In this paper, we design two totally decoupled, linear and unconditionally energy stable schemes with first-order time accuracy for the multi-physics diffuse interface model of two-phase magnetohydrodynamic (MHD) problem. These schemes combine the semi-implicit stabilization method/invariant energy quadratization (IEQ) method for the phase-field equations, Gauge–Uzawa method for the MHD equations, with some subtle implicit–explicit treatments for nonlinear coupled terms. Then this strong nonlinear and coupled complex system is split into a series of small linear elliptic equations, which makes the calculation more efficient. Additionally, compared to the standard projection method, the given schemes can overcome the selection of the initial value about pressure p and the treatment of artificial boundary condition on p which can lead to boundary layers and lose accuracy. Finally, rigorous proof of unconditional energy stability and several simulations of numerical experiment are conducted to demonstrate the validity of the decoupled schemes. • Our schemes split nonlinear coupled MHD into elliptic type problems. • We establish the unconditional energy stabilities for the schemes. • Numerical simulations demonstrate the validity of the given schemes. [ABSTRACT FROM AUTHOR]

This paper considers hybrid systems — dynamical systems that exhibit both continuous and discrete behavior. Usually, in these systems, interactions between the continuous and discrete dynamics occur when a pre-defined function becomes equal to zero, i.e., in the system occurs a zero-crossing (the situation where the function only "touches" zero is considered as the zero-crossing, as well). Determination of zero-crossings plays a crucial role in the correct simulation of the system in this case. However, for models of many real-life hybrid systems, such interactions may lead to the so-called Zeno executions, i.e., situations where the system undergoes an unbounded number of discrete transitions in a finite and bounded length of time. In this case, standard numerical methods of simulating the systems may fail, since the time between two transitions can decrease significantly leading to ill-conditioning of the simulation. Correct determination of zero-crossings for a complex real-life system can require a lot of computational resources and, as a consequence, slow down the simulation significantly. This paper presents a new way to execute the simulation generating time observations of the hybrid system dynamically using numerical infinitesimals introduced recently, allowing thus to determine zero-crossings more accurately. The proposed method allows to automatically detect zero-crossings with predefined accuracy and to analyze better the behavior of the system around the zero-crossings generating observations more densely, where it is necessary. Moreover, the search for zero-crossings is performed efficiently without re-evaluation of the whole system at each observation. To show the validity of the proposed algorithm, the well-known Bouncing Ball hybrid system has been studied and the obtained simulation results were compared with the standard method. • Hybrid systems exhibiting both continuous and discrete behavior are considered. • Standard simulation methods of hybrid systems can fail due to the Zeno effect. • A new method for simulating hybrid systems in presence of Zeno behavior is proposed. • Numerical infinitesimals are used for an accurate finding of zero-crossing points. • The Infinity Computer used in simulations increases the accuracy significantly. [ABSTRACT FROM AUTHOR]

With the requirements of users enhanced for wireless communication, the cooperative communication will become a development in tendency future. In this paper, a model based on complex networks with both preferential and random attachments is researched to solve an actual network-CCN (cooperative communication network). Firstly, the evolution of CCN is given by four steps with different probabilities. At the same time, the rate equations of nodes degree are presented to analyze the evolution of CCN. Secondly, the degree distribution is analyzed by calculating the rate equation and numerical simulation. Finally, the robustness of CCN is studied by numerical simulation with random attack and intentional attack to analyze the effects of degree distribution, average path length and average degree. The results of paper are more significant for building CCN to program the resource of communication. [ABSTRACT FROM AUTHOR]

Highlights: [•] A new theory of fractional-order heat transport is introduced in the paper. [•] Fractional operators involve non-local thermal exchanges in the conductor. [•] Non-local effects depend of generalized measures of relative temperatures. [•] Thermodynamic consistency has been assessed for diffusive/ballistic heat transfer. [•] Numerical simulations have been reported in the paper. [Copyright &y& Elsevier]

As the spread of diseases among species becomes more and more common, our society pays increasingly more attention to the harm caused by the disease. Research on eco-epidemiological models will not only promote the development of biology and ecology, but also have important impacts on the economy and health of the society. This paper deals with a stochastic eco-epidemiological model which contains three species and is governed by Ornstein–Uhlenbeck (OU) process. Two common approaches to incorporate the effects of environmental perturbations in stochastic systems are first discussed and compared analytically and computationally. Based on the existence and uniqueness of the global solution to the model, several sharp conditions for the persistence and extinction of the species are established. Then, a criteria for the existence of the stationary distribution to the system is established by constructing suitable Lyapunov functions. Furthermore, the analytical expression of the probability density function of the model around the quasi-equilibrium is obtained. Finally, the impact of the main parameters on the dynamical behaviors and the noise perturbation effect are analysed in the numerical simulations section to illustrate the theoretical results. • A stochastic eco-epidemiological model with Ornstein–Uhlenbeck process is studied. • Some conditions for the persistence and extinction of species are established. • A criteria for the existence of stationary distribution is given. • The probability density function near the quasi-equilibrium is obtained. [ABSTRACT FROM AUTHOR]

• Numerical construction of smooth self-similar solutions to the Euler equations. • First construction of smooth linear perturbations. • Numerical evidence that the singularity formation is unstable. • Development of a numerical strategy to solve technical mathematical problems. • Our techniques can be adapted to other PDEs. This paper addresses the construction and the stability of self-similar solutions to the isentropic compressible Euler equations. These solutions model a gas that implodes isotropically, ending in a singularity formation in finite time. The existence of smooth solutions that vanish at infinity and do not have vacuum regions was recently proved and, in this paper, we provide the first construction of such smooth profiles, the first characterization of their spectrum of radial perturbations as well as some endpoints of unstable directions. Numerical simulations of the Euler equations provide evidence that one of these endpoints is a shock formation that happens before the singularity at the origin, showing that the implosion process is unstable. [ABSTRACT FROM AUTHOR]

• We developed a powerful simulation framework to desynchronize the distribution of a network of uncoupled and noise-free (or noisy) neural oscillator populations. • The control model works based on the neurons phase distributions. • The PDEs of the control model is solved by fast and precise numerical methods. • Simulation carried out show the methods implemented in this paper are highly accurate, fast, and reliable in the presence of noise. Since control of synchronization at the onset of seizure can be considered an effective method of preventing or treating epilepsy, developing an advanced and accurate numerical simulation approach to implement control on noise-free (or noisy) population of synchronized neurons makes the control performance more effective. In this paper, we simulate a control by two powerful, fast and accurate meshless methods, i.e. compactly supported radial basis function collocation (CS-RBF) and radial basis function generated finite difference (RBF-FD) approaches, which enable us to realize new control objectives. It should be noted that the challenges faced by this model include the diverse behavior of neuronal populations, the speed of action in synchronization control, the minimization of energy consumption and the applicability of controlling real data, we show that the both proposed meshfree methods solve all these challenges. In addition, the evaluation of both methods by different phase response curves (PRC) as well as in the presence of Gaussian white noise shows that these methods can be implemented in experimental cases. The analyses and numerical results presented eventually confirm these claims. [ABSTRACT FROM AUTHOR]

Highlights • Modeling and numerical simulations of a nonlinear system subjected to stochastic excitations. • Harmonic Balance Method (HBM) and shooting method for mechanical systems with stochastic excitations. • Comparisons between experiments and numerical simulations. Abstract This paper presents experiments and numerical simulations of a nonlinear clamped-clamped beam subjected to stochastic broadband excitations. Broadband excitations are performed experimentally in order to show the hardening effect and the enlargement of the response peak in the vicinity of the primary resonance, and to detect the presence of secondary peaks resulting from the harmonics generated by the primary resonance. Using the decomposition of a random signal into a multi-harmonic periodic equivalent signal, the shooting method and the Harmonic Balance Method (HBM) are used to simulate the response of the nonlinear clamped-clamped beam subjected to a stochastic excitation. The latter is modeled as a multi-harmonic periodic equivalent signal in order to be used by both the shooting method and the HBM. A periodogram strategy is used to ensure a good estimate of the resulting Power Spectral Density (PSD). The two nonlinear methods are compared in terms of computation time and precision. A specific attention is paid on correct use of these methods. Finally, comparison between experiments and numerical simulations are performed for different levels of stochastic excitations. Good correlations are observed thus validating the global nonlinear proposed strategy. [ABSTRACT FROM AUTHOR]

Highlights • We propose a novel discount strategy for WOM marketing (the IBD strategy). • We model the WOM spreading process as a dynamic model (the DPA model). • We investigate the dynamics of the DPA model. • We examine the effect of different factors on the expected net profit of the IBD strategy. • Based on the above study, we suggest some promotional strategies. Abstract This paper addresses the issue of discount pricing in word-of-mouth (WOM) marketing. First, we propose a novel discount strategy we refer to as the Influence-Based Discount (IBD) strategy, in which each buyer would enjoy a discount that is linearly proportional to his/her influence in the WOM network. For the purpose of evaluating the performance of an IBD strategy, we model the associated WOM spreading process as a higher-dimensional nonlinear differential dynamical system we refer to as the Dormant-Potential-Adopting (DPA) model. Next, we examine the dynamics of a DPA model through computer experiments. Thereby, we estimate the expected net profit of an IBD strategy and inspect how different factors influence this profit through numerical experiments. On this basis, we suggest some feasible promotional measures. [ABSTRACT FROM AUTHOR]

Highlights • Nonlocality is an essential feature in fractional calculus. • Methods neglecting nonlocality lead to wrong results. • Numerical evidence of poor results obtained by neglecting nonlocality is given. Abstract In the paper titled "New numerical approach for fractional differential equations" by Atangana and Owolabi (2018) [1], it is presented a method for the numerical solution of some fractional differential equations. The numerical approximation is obtained by using just local information and the scheme does not present a memory term; moreover, it is claimed that third-order convergence is surprisingly obtained by simply using linear polynomial approximations. In this note we show that methods of this kind are not reliable and lead to completely wrong results since the nonlocal nature of fractional differential operators cannot be neglected. We illustrate the main weaknesses in the derivation and analysis of the method in order to warn other researchers and scientist to overlook this and other methods devised on similar basis and avoid their use for the numerical simulation of fractional differential equations. [ABSTRACT FROM AUTHOR]

Highlights • We discuss the decay synchronization problem of CMTDNs. • We design a novel nonlinear feedback controller and a Lyapunov function. • Some sufficient conditions are derived to ensure the decay synchronization of CMTDNs/CMDNs. • The main results can easily extended to exponential synchronization, polynomial synchronization and logarithmic synchronization. • Two simulations are shown to present the correctness of the main results. Abstract In this paper, we discuss the decay synchronization of complex multi-links time-varying dynamic networks (CMTDNs) for the first time. The exponential synchronization, polynomial synchronization and logarithmic synchronization can be seen as the special cases of the decay synchronization. CMTDNs are more general and realistic than the complex network with the single-link. By virtue of the definitions of the ψ -type function and ψ -type stability, we give the definition of the decay synchronization of the drive-response CMTDNs. Some novel and easily validated sufficient conditions are deduced to ensure the decay synchronization of drive-response CMTDNs based on the Lyapunov stability theory with the help of a novel nonlinear feedback controller and a new Lyapunov–Krasovskii function. Finally, two numerical simulations have been given to verify the correctness of main results. [ABSTRACT FROM AUTHOR]

Variable-pitch cutter and spindle speed variation are two well-known methods to suppress milling chatter by disrupting regenerative effect. Considering their compatibility in practical application, their combined system perhaps has better potential for chatter suppression. However, to authors’ knowledge, there has been little research associated with this issue. In this paper, a milling dynamic model, which can simultaneously take into account the effect of the variations of cutter pitch angles and spindle speed, is constructed and linear stability analyses are carried out via an updated semi-discretization method, which has great capability to predict the stability lobes of milling process with multiple time-periodic delays. Then, through comparisons with previously associated works along with time-domain simulations, the effectiveness of the proposed model is confirmed. Moreover, the combined influences of the new system and the main system parameters on stability are deeply evaluated and discussed by conducting lots of simulations. Results show that the combined milling process exhibits a great capability to avoid the onset of milling chatter and can result in nearly 3-fold increase in depth of cut than that of a tradition one, but 1-fold increase for the process with variable pitch cutter or variable spindle speed in some special cases. [ABSTRACT FROM AUTHOR]

Few literatures apply complex oligopoly dynamics theory in games of incomplete information. This paper aims at analyzing dynamic behaviors of Bayesian game. A dynamic Cournot model with asymmetric information is proposed based on adaptive expectation and bounded rationality. Theoretical analysis draws two important conclusions: firstly, Bayesian Nash equilibrium of dynamic Cournot duopoly model with two players of adaptive expectation is always globally asymptotically stable. Secondly, Bayesian Nash equilibrium of dynamic Cournot duopoly model with players of adaptive expectation and gradient rule based on marginal profit is locally asymptotically stable only when parameters satisfy certain conditions. In our model, a firm of uncertain cost function is designed. A probability parameter θ of private type which differentiates high cost and low cost is introduced. Bifurcation, or even chaos with respect to θ , is performed by simulation which implies that large possibility of high-cost production yields easier chaos in duopoly market. High adjustment speeds of output form a three-dimensional strange attractors region. The unstable system's negative impact on equilibrium output and profit highlights the importance of system stability. Chaos control is in order to stabilize the equilibrium of the improved dynamic Cournot model with asymmetric information. [ABSTRACT FROM AUTHOR]

This paper is concerned with a backward heat conduction problem with time-dependent thermal diffusivity factor in an infinite “strip”. This problem is drastically ill-posed which is caused by the amplified infinitely growth in the frequency components. A new regularization method based on the Meyer wavelet technique is developed to solve the considered problem. Using the Meyer wavelet technique, some new stable estimates are proposed in the Hölder and Logarithmic types which are optimal in the sense of given by Tautenhahn. The stability and convergence rate of the proposed regularization technique are proved. The good performance and the high-accuracy of this technique is demonstrated through various one and two dimensional examples. Numerical simulations and some comparative results are presented. [ABSTRACT FROM AUTHOR]

The discrete-time (map-based) approach to modeling nonlinear dynamics of spiking activity in neurons enables highly efficient numerical simulations for capturing realistic neurobiological behavior by utilizing a large time interval between computed states (samples) of neuron activity. The design and parameter tuning of these models assumes a fixed and preset sampling rate. When change of the time step is needed, it requires revisiting stages of the model design and parameter tuning. This paper presents an approach to the design of map-models in a new form where time step is added as a control parameter and can be easily changed to vary the time scale of the model behavior, i.e. sampling rate, essentially preserving the model behavior. It also discusses modification of the noise generator models needed to support simulation of map-based neurons with the modified sampling rate. The effects caused by direct control of time scale on model dynamics and limitations of this approach are discussed. [ABSTRACT FROM AUTHOR]

• Numerical analysis of the shear banding phenomenon in two dimensions is performed. • Finite volume method for numerical simulation of shear banding processes is proposed. • The governing equations describe the shear banding process are derived on the basis of the modification of a hypoelastic Wilkins model. • The efficiency and accuracy of the proposed numerical algorithm is shown on three benchmark problems. • Self-organization processes of shear banding is studied. The aim of this paper is to develop an effective finite volume method for numerical simulation of the adiabatic shear bands (ASB) formation processes. A formation of ASB happens at high-speed shear strains of ductile materials. A numerical simulation of such problems using Lagrangian approach is associated with some problems, the main one of which is a mesh distortion at large deformations. We use Eulerian approach to describe a motion of the non-linear elasto-plastic material. More specifically, we consider a modification of a well-known hypoelastic Wilkins model. In this paper we suggest a numerical method for modeling of high-speed shear deformations on two-dimensional meshes. The method is verified on the three test problems suggested by other authors. [ABSTRACT FROM AUTHOR]

• The concepts of average control rate and average control period are proposed to characterize the distribution of work intervals and rest intervals of aperiodically intermittent control. • The constraints on the control rate, control intervals and rest intervals are reduced compared with previous results. • Average control width of AIC instead of the lower bound of all control intervals of AIC, is used to measure the small time delays and large time delays. • For the case of small time delays, the upper bound of time delays can be larger than the lower bound of control widths. This paper focuses on the stabilization problem of complex-valued hybrid stochastic delayed systems via aperiodically intermittent control (AIC). As for AIC in existing literature, strict conditions on the lower bound of control intervals and upper bound of control periods or the maximum proportion of rest intervals are required. In this paper, we relax these constraints by proposing average control ratio and average control period to describe the distribution of control and rest intervals of AIC, i.e., the lower bound of certain control widths can be arbitrarily small, upper bound of certain control periods can be very large and the proportion of rest widths can be any value in (0 , 1). Thus, the conservativeness is reduced compared with the existing related results. Then based on the complex generalized Itô's formula, several novel stabilization criteria are obtained for small time delays and large time delays, respectively, which avoid splitting the real and imaginary parts. Especially, for the case of small time delays, the upper bound of time delays can be larger than the lower bound of all control widths, which has wider applications than previous results. Finally, two examples along with their numerical simulations are given to show the effectiveness and less conservativeness of our results. [ABSTRACT FROM AUTHOR]

This paper is devoted to studying the dynamical behaviors and stationary patterns of a diffusive modified Leslie–Gower predator–prey model with density-dependent motion in the predator population. We establish the existence of classical solutions with the uniform-in time bound and then analyze the local and global stability of the spatially homogeneous co-existence steady state under certain parametric conditions. By choosing the prey diffusion rate d 2 as the bifurcation parameter, the steady state bifurcations from the positive constant equilibrium solution are investigated. Numerical simulations are performed to corroborate our analytical findings. • We establish the existence of classical solutions for a diffusive modified Leslie–Gower predator prey model. • We analyze the local and global stability of the spatially homogeneous co-existence steady state. • The steady state bifurcations from the positive constant equilibrium solution are investigated. [ABSTRACT FROM AUTHOR]

In this paper, the influence of the fear effect and Leslie–Gower function on the dynamic behavior of the predator–prey model is considered. First, the well-posedness analysis of this model is demonstrated, and the existence and local stability of the equilibrium point are given. Then by using the bifurcation theory and selecting the appropriate bifurcation parameters, many types of bifurcation phenomena in the system are discovered, including transcritical bifurcation, Hopf bifurcation and Bogdanov–Takens bifurcation. Meanwhile, the numerical simulations on the basis of theoretical analysis are further carried out to intuitively show the influence of fear effect on population. As the degree of fear increases, the system will undergo multiple dynamic behaviors switching until the final extinction of prey population, while the predator population will survive due to the presence of substitute prey. In addition, the initial density of predator and prey can determine whether the solution of system tends to a coexisting steady state or a periodic oscillation. • A novel prey–predator model with the impact of fear is introduced. • The system can experience many types of bifurcation phenomena. • Numerical simulations and sensitivity analysis are performed. • Our results show that fear effect plays an important role on deriving the complex dynamics. [ABSTRACT FROM AUTHOR]

This paper mainly studies the bounded real lemma for inertial neural networks with unbounded time-varying transmission delays, unbounded distribution delays and state-dependent switching. Bounded real lemmas of inertial neural networks under consideration are presented by proposing a parameterizing approach based on the system solutions. The advantage of this approach is that it neither decomposes the model into two first-order differential equations nor constructs any Lyapunov–Krasovskii functional, thus reducing computational effort and complexity. Furthermore, the obtained sufficient condition contains only a few simple linear scalar inequalities, which can be easily solved by using MATLAB. Finally, a numerical example and its numerical simulation are used to demonstrate the validity of the theoretical results. • A novel approach based on the system solutions is proposed for the first time. • The proposed method reduces computational effort and complexity. • The obtained sufficient condition can be easily solved by using MATLAB. [ABSTRACT FROM AUTHOR]

In this paper, the global exponential bipartite synchronization (GEBS) for neutral-type memristive inertial coupling mixed time-varying delays neural networks (NTMI-CMTVDNNs) with synergistic and antagonistic interactions (SAI) is investigated. Firstly, the NTMI-CMTVDNNs with SAI between neurons are modeled by a signed graph. Inertial network is equivalently transformed into the first-order differential equations via variable substitution. Secondly, a novel lemma is proposed to deal with parameter mismatch of the above modeled network. By designing delay-dependent and delay-independent discontinuous control laws, the delay-dependent sufficient conditions under the bounded and unbounded activation functions are obtained to ensure the GEBS of the NTMI-CMTVDNNs with SAI with or without a leader neurons, respectively. Finally, two numerical simulation examples illustrate the validity of our theoretical results. [ABSTRACT FROM AUTHOR]

• An efficient numerical approximation via parameter switching with convexly combining attractors. • Two binary operations formally express the convex combination providing analytic description. • Extensive numerical simulations demonstrating the effectiveness of the numerical approximation. This paper presents an effective approach to constructing numerical attractors of a general class of continuous homogenous dynamical systems: decomposing an attractor as a convex combination of a set of other existing attractors. For this purpose, the convergent Parameter Switching (PS) numerical method is used to integrate the underlying dynamical system. The method is built on a convergent fixed step-size numerical method for ODEs. The paper shows that the PS algorithm, incorporating two binary operations, can be used to approximate any numerical attractor via a convex combination of some existing attractors. Several examples are presented to show the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

• Fluid flows must depend on the geometries of given domains or irregularities of given data. In particular, when the datum has a jump discontinuity at a point, a curve started at the point can be generated into the domain and the jump discontinuity is propagated along the curve. The curve is directed by the fluid velocity vector. So the pressure gradient given in the momentum equations is not well-defined across the curve. • Recently we have shown that, when the inflow boundary data has a jump, there is a curve contained in the domain such that the fluid flows have jump there. By constructing a lifting vector field for the pressure jump value on the curve we decompose the solution into the jump part, the contact singularity part and the smoother one. In fact the solutions of the nonlinear compressible Stokes system are (0.1) u = K + Φ + w , p = p b + k + τ + σ , where p b has a jump discontinuity on inflow boundary, the pair (K , k) denotes the jump part, (Φ , τ) the contact singularity and (w , σ) the smoother part. The contact singular part is due to the intersection of the interface curve to the boundary of the domain. It is assumed that we have two contact points, denoted by a j , j = 1 , 2. • We call the curve the interface or jump curve and denote by C. Since the curve is directed by the velocity vector it is important to have a precise structure of the velocity vector. • In this paper we design a finite element numerical scheme based on the decomposition and try to address and confirm the important roles of each component in the decomposition numerically. It has been shown in [8] that the solutions of compressible Stokes flows with inflow jump condition can be decomposed into the jump discontinuity part (due to the pressure jump) plus the contact singularity (to the boundary) plus the smoother one, which is twice differentiable. In this paper we design a numerical scheme of each part in the decomposition and numerically demonstrate its essential role for capturing the jump discontinuity behaviors of the solutions. Several numerical simulations are presented, describing the critical role of each part. It is thought that such algorithm is new in constructing the jump discontinuity solutions. [ABSTRACT FROM AUTHOR]

This article considers a Cournot duopoly model in a continuous-time framework and analyze its dynamic behavior when the competitors are heterogeneous in determining their output decision. Specifically the model is expressed in the form of differential equations with discrete delays. The stability conditions of the unique Nash equilibrium of the system are determined and the emergence of Hopf bifurcations is shown. Applying some recent mathematical techniques (stability switching curves) and performing numerical simulations, the paper confirms how different time delays affect the stability of the economy. [ABSTRACT FROM AUTHOR]

The fractional-order discrete maps with chaotic behaviors based on the theory of “fractional difference” are proposed in recent years. In this paper, instead of using fractional difference, a new fractionalized logistic map is proposed based on the numerical algorithm of fractional differentiation definition. The bifurcation diagrams of this map with various differential orders are given by numerical simulation. The simulation results show that the fractional-order logistic map derived in this manner holds rich dynamical behaviors because of its memory effect. In addition, new types of behaviors of bifurcation and chaos are found, which are different from those of the integer-order and the previous fractional-order logistic maps. [ABSTRACT FROM AUTHOR]

In this paper, the optimal control problem of epidemic spreading on random surfer heterogeneous networks is considered. An epidemic spreading model is established according to the classification of individual’s initial interaction radii. Then, a control strategy is proposed based on adjusting individual’s interaction radii. The global stability of the disease free and endemic equilibrium of the model is investigated. We prove that an optimal solution exists for the optimal control problem and the explicit form of which is presented. Numerical simulations are conducted to verify the correctness of the theoretical results. It is proved that the optimal control strategy is effective to minimize the density of infected individuals and the cost associated with the adjustment of interaction radii. [ABSTRACT FROM AUTHOR]

In this paper, the reliability of a non-linearly damped friction oscillator under combined additive and multiplicative Gaussian white noise excitations is investigated. The stochastic averaging method, which is usually applied to the research of smooth system, has been extended to the study of the reliability of non-smooth friction system. The results indicate that the reliability of friction system can be improved by Coulomb friction and reduced by random excitations. In particular, the effect of the external random excitation on the reliability is larger than the effect of the parametric random excitation. The validity of the analytical results is verified by the numerical results. [ABSTRACT FROM AUTHOR]

Multi-time scales modeling of hydro-turbine governing system is crucial in precise modeling of hydropower plant and provides support for the stability analysis of the system. Considering the inertia and response time of the hydraulic servo system, the hydro-turbine governing system is transformed into the fast-slow hydro-turbine governing system. The effects of the time-scale on the dynamical behavior of the system are analyzed and the fast-slow dynamical behaviors of the system are investigated with different time-scale. Furthermore, the theoretical analysis of the stable regions is presented. The influences of the time-scale on the stable region are analyzed by simulation. The simulation results prove the correctness of the theoretical analysis. More importantly, the methods and results of this paper provide a perspective to multi-time scales modeling of hydro-turbine governing system and contribute to the optimization analysis and control of the system. [ABSTRACT FROM AUTHOR]