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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]

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]

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]

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]

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, 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]

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]

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]

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]

This paper is concerned with some new entire solutions of the reaction–advection–diffusion competition system in periodically varying media. It has been shown in an earlier work that such a system admits some pulsating front-like entire solutions. In this work, we construct some new entire solutions originating from semi-trivial pulsating fronts and investigate some propagation properties of these entire solutions. More precisely, we first construct a family of entire solutions originating from a single front, which propagate with the speed same to that of the front. Then, we construct some entire solutions originating from two different fronts at both sides, which behave as the two fronts propagating from both directions or one direction of the x -axis, and propagate with the speed between those of the two fronts. Finally, we give some numerical simulations to describe the propagation of these entire solutions. These kinds of entire solutions provide some new spreading ways other than pulsating fronts and front-like entire solutions for two competing species in a heterogeneous media. • The novelty lies in the construction of various appropriate sub-super-solutions. • This is the first work to construct semi-trivial entire solutions. • These entire solutions provide new spreading ways for two competing species. [ABSTRACT FROM AUTHOR]

This paper presents Turing–Hopf bifurcation analysis and the resulting spatiotemporal dynamics in a single-species reaction–diffusion model with nonlocal delay. A linear stability analysis is performed to find the conditions for the occurrence of Turing–Hopf bifurcation. The weakly nonlinear analysis is then employed to derive the amplitude equations which describe the slow-time evolution of the critical Turing and Hopf modes. By the utilization of amplitude equations, stability conditions for different pattern formations can be obtained, helping to reveal the affluent spatiotemporal patterns near the Turing–Hopf bifurcation and predict when and where these patterns can emerge. Also, our analysis shows the role of the choice of initial conditions in pattern formations. Numerical simulations are given to verify the theoretical results. • Bifurcation analysis of a single-species model with nonlocal delay is performed. • Amplitude equations near the Turing–Hopf bifurcation are derived. • The existence and stability conditions of spatiotemporal patterns are obtained. • Numerical simulations explore the role of initial conditions in pattern structures. [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]

Based on the previous literature about the random logistic and Gompertz models, the aim of this paper is to extend the investigations to the generalized logistic differential equation in the random setting. First, this is done by rigorously constructing its solution in two different ways, namely, the sample-path approach and the mean-square calculus. Secondly, the probability density function at each time instant is derived in two ways: by applying the random variable transformation technique and by solving the associated Liouville's partial differential equation. It is also proved that both the stochastic solution and its density function converge, under specific conditions, to the corresponding solution and density function of the logistic and Gompertz models, respectively. The investigation finishes showing some examples, where a number of computational techniques are combined to construct reliable approximations of the probability density of the stochastic solution. In particular, we show, step-by-step, how our findings can be applied to a real-world problem. • The generalized logistic model with randomness is fully studied. • Uncertainty considered in all input parameters with arbitrary distributions. • RVT and Liouville PDE are applied to determine the 1-PDF under general assumptions. • Theoretical results are applied to a biological real-world problem. • PSO and Maximum Entropy used to fit the model to real data. • Wavelet-based AMR and lagrangian PDE methods combined to compute the 1-PDF. [ABSTRACT FROM AUTHOR]

In this paper, two modified nonlinear saturation-based controllers and negative velocity feedback controllers are integrated to suppress the horizontal and vertical vibrations of a horizontally supported Jeffcott-rotor system at primary resonance excitation and the presence of 1:1 and 1:2 internal resonances. The second order approximations and the amplitude equations are obtained by applying the integral equation method to analyze the nonlinear behavior of this model. The stability of the steady-state solutions is determined based on the Floquet theory. The necessity of adding a negative velocity feedback to the main system is stated. The effects of different control parameters on the frequency-response curves and the force-response curves are investigated. Time histories of the whole system are included to show the response with and without control. It is shown that the main systems steady-state amplitude under the proposed control is reduced by about 99.8% and the negative velocity feedback can suppress the transient vibrations and prevent the main system having the large amplitude vibration. Finally, numerical simulations are provided to verify the effectiveness of the integral equation method. • Two modified saturation controllers are used for Jeffcott-rotor system. • A integral equation method is used for a four-degree-of-freedom system. • Analytical solutions based on our method agree with numerical simulations. • The saturation controller can avoid the phenomenon of double peaks. • The negative velocity feedback can suppress the transient vibrations. [ABSTRACT FROM AUTHOR]

The asymptotic stability and synchronization of fractional delayed memristive neural networks with algebraic constraints in Riemann–Liouville sense will be investigated in this article. First, algebraic constraints are introduced for the first time into the existing fractional delayed memristive neural networks, and a new fractional singular delayed memristive neural networks (FSDMNNs) model is presented. Then, within the framework of Filippov's solution, a less conservative result for the asymptotic stability of FSDMNNs is obtained by Lyapunov–Krasovskii functional. Subsequently, the appropriate feedback scheme and adaptive scheme are designed to synchronize FSDMNNs and two sufficient conditions are acquired. In addition, the results not only address the influence of delays and algebraic constraints, but can also easily detect and synchronize the actual memristive neural networks. Finally, numerical simulations frankly confirm the correctness and validity of the derived results. • This paper considers the influence of algebraic constraints on FDMNNs. • A new stability criterion of FSDMNNs is attained by a novel Lyapunov functional. • Two easy-to-verify synchronization conditions are gained by suitable controllers. • Our results can be deemed an extension of the existing ones. [ABSTRACT FROM AUTHOR]

Passively mode-locked fiber laser has advantages of compactness, simplicity, flexibility and high quality of the output pulse, which provides an excellent research platform for the analysis of optical soliton characteristics. In this paper, the complex cubic Ginzburg–Landau equation (CGLE) is employed to simulate pulse propagation in passively mode-locked Er-doped fiber laser. We use barycentric interpolation collocation method to numerically solve the dark single soliton, dark two-soliton and bright two-soliton solutions, and study the effects of dispersion, nonlinear effects, gain and loss on optical pulse transmission. We discuss the dynamic characteristics of the bright two-solitons and dark two-solitons for the complex CGLE, respectively. The results is of great significance for the study of optical pulse transmission in fiber laser. • The cubic Ginzburg–Landau equation is employed to simulate pulse propagation in passively mode-locked Er-doped fiber laser. • The dark single soliton solution and bright two-soliton solution are numerically solved by barycentric interpolation collocation method. • The dynamic characteristics of the bright two-solitons and dark two-solitons for the cubic Ginzburg–Landau equation are of great significance for the study of optical pulse transmission in fiber laser. [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]

We are concerned with dynamical behavior of the indirectly and locally memory-damped Timoshenko system. The polynomial/exponential decay results for the long-term dynamical behavior of the Timoshenko system are established when the memory kernels decay polynomially/exponentially. To obtain ideal asymptotic decay rates under basic conditions, we take some analysis processes specially designed for our issues. The obtained long-term dynamical behavior theorems, with the exact uniform decay rates for the solutions to the system, indicate that for the Timoshenko system with indirect memory damping, a local memory effect is enough to produce an entire dissipation mechanism and to ensure the same decay rates as in the case of global memory effect. Moreover, although our conditions on the memory kernels are weaker compared with the ones in the literature, we still derive stronger conclusions. Finally, we present some results of numerical simulation to illustrate quantitatively the behavior of the solution energies of our system, which agree with our theoretical results well. • Timoshenko system in the paper is controlled only by an indirect local memory damping. • Ideal asymptotic decay results are established for the long-term dynamical behavior of the Timoshenko system. • The obtained theorems indicate that for the Timoshenko system with indirect memory damping, a local memory effect is enough to produce an entire dissipation mechanism and to ensure the same decay rates as in the case of global memory effect. • Although our conditions on the memory kernels are weaker compared with the ones in the literature, we still derive stronger conclusions. • Numerical simulation results agree with our theoretical results well. [ABSTRACT FROM AUTHOR]

This paper describes a new approach to study the nonlinear dynamical behaviors of a belt-driven two-coupled friction oscillator. First, using Buckingham's π theorem, several dimensionless and physically meaningful parameters are derived. Their relationships with the steady state responses of the system are revealed through numerical simulations. Then a new mapping tool named as sticking phase plot is developed to record the transition process of stick–slip motion. It intuitively shows how bifurcation occurs in sticking region and transforms the dynamical responses. The bifurcation diagrams of the system are obtained by sweeping the dimensionless parameters. Some extraordinary bifurcation phenomena are observed in this system. By using sticking phase plot, it is found that the bifurcations can be broadly divided into four main categories: the Border-Collision Bifurcation, the Grazing-Sliding Bifurcation, the Multi-Sliding Bifurcation, and the Fixed-Point Bifurcation. New nonlinear phenomena are found herein, including chaos caused by the Border-Collision Bifurcation and the Sliding Bifurcations. The local and the global changes caused by the Fixed-Point bifurcation are also observed. In numerical simulation, we adopted the LVIM method proposed by the authors, which can be used to integrate non-smooth systems very accurately and efficiently. • A panoramic view of the dynamics of a belt driven two-coupled nonlinear oscillators is demonstrated. • A new event-driven numerical solver named LVIM is described to efficiently solve the non-smooth vibration system. • A so-called sticking phase plot is proposed to illustrate the complex bifurcations of the sticking–slipping motion. • Interesting phenomena, involving chaotic motions caused by new type of bifurcations are demonstrated. [ABSTRACT FROM AUTHOR]

The design, analysis and numerical simulations of a stabilized variable time-stepping difference scheme, for the Swift–Hohenberg equation, are considered in this paper. The proposed time-stepping scheme is proved to preserve a discrete energy dissipation law. With the help of the new discrete orthogonal convolution kernels, the unique solvability and the unconditional energy stability of the numerical scheme are rigorously proved. In addition, the second-order L 2 norm convergence both in time and in space, of the proposed scheme, is shown under the almost independent of the time-step ratios. To the best of our knowledge, this is the first time that the L 2 norm convergence of the adaptive BDF2 method is achieved for the Swift–Hohenberg equation. Numerical examples are provided to illustrate our theoretical results and to show the computational efficiency of the numerical scheme. • A stabilized variable time-stepping scheme for the Swift-Hohenberg equation is developed. • The theoretical results of the time-stepping scheme are rigorously proved. • The proposed adaptive time-stepping algorithms are suitable for the long time simulation. [ABSTRACT FROM AUTHOR]

Competitive exclusion is a phenomenon that often occurs in ecology when two species compete for the same limited resource, while coexistence of predator and prey is required to sustain a predator–prey system. It is of great interest to examine if coexistence of intraguild predator and prey is possible when they are competing for the same resource. In this paper, we study competitive exclusion and coexistence in an intraguild predation model that incorporates species' mutual interference characterized by a Beddington–DeAngelis functional response. We derive conditions for competitive exclusion and conditions for coexistence of intraguild predator and prey. This is achieved by discussing the existence, local and global stability of all feasible equilibria and uniform persistence. Numerical simulations are carried out to explore effects of intraguild predator interference and intraguild predation strength on the coexistence modes of intraguild predator and prey. • Species' mutual interference is incorporated. • Conditions on competitive exclusion are derived. • Conditions on coexistence are derived. • The model exhibits three modes of coexistence. • Strong intraguild predator interference has a stabilizing effect. [ABSTRACT FROM AUTHOR]

In this paper, the localized solutions of the (2+1)-dimensional B-Kadomtsev–Petviashvili (BKP) equation, which is a useful physical model, are further studied. Firstly, by using the theory of Hirota bilinear operator, the corresponding N-soliton solutions are obtained. Then the localized solutions, which are the M-lump solutions, higher-order breathers and hybrid solutions, are also constructed by taking a long-wave limit and introducing some conjugation conditions. In the meanwhile, the dynamic behaviors of these obtained solutions are analyzed and shown graphically by the corresponding numerical simulations with specific parameters, which can greatly affect the solutions, such as the propagation properties. • Using Hirota bilinear method, N-soliton solutions are obtained. • Using long-wave limit, M-lumps, breathers, hybrid solutions are derived. • Dynamic properties are analyzed and illustrated. [ABSTRACT FROM AUTHOR]

In this paper we simulate numerically different types of positive solutions of diffusive Lotka–Volterra competition systems: semitrivial positive solutions, semitrivial positive metasolutions, coexistence states and meta-coexistence states , in the case in which one of the species has two refuges. The existence of these solutions has been demonstrated in López-Gómez (2003). Once the location and extension of both refuges and the distribution of the resources in the territory are known, the competition strategies of each of the species can be designed and incorporated into the model. Or vice versa, if the competition strategies are known, the resources into the territory can be distributed in a way to obtain better results. • Numerical simulation of the results in J. López-Gómez, Coexistence and meta-coexistence for competing species (2003). • Numerical simulation of semitrivial classical positive solutions and semitrivial positive metasolutions. • Numerical simulation of coexistence states and meta-coexistence states. • Spatial competition strategies in the case of two refuges. [ABSTRACT FROM AUTHOR]

This paper deals with the asymptotic spreading for a class of diffusion equations with degenerate monostable nonlinearity, where the initial value is asymptotically front-like. By the method of squeezing technique and super- and sub-solutions, we prove that the speed of asymptotic spreading may be finite or infinite, which depends on the degeneracy of nonlinearity as well as the decaying rate of the initial value. Different from the non-degenerate case, the traveling wave solution with the critical speed is globally asymptotically stable if the initial value has light tails, while an acceleration propagation may occur with the initial value having heavy tails in the degenerate case. Furthermore, if the initial value has heavy tails but lighter than algebraic, then the speed of the asymptotic spreading is finite, and the balance between finite or infinite speed is established if the initial value has algebraic tails. Our results reflect that the degeneracy may be harmful to the acceleration propagation. • A time periodic diffusion equation with degenerate nonlinearity is considered. • The asymptotic spreading of the initial value has light tails and heavy tails are investigated. • Some new and interesting phenomenon about time periodic diffusion equation with degenerate nonlinearity are given through numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper, we present a systematic renormalization group method to investigate a class of singularly perturbed delay differential equations. The uniformly valid approximate solution can be obtained, and we give a rigorous proof of the error estimate of the approximate solution. In addition, some numerical comparisons between the exact solution, the result by the averaging method and our result are given for two examples. • We present a systematic RG approach to deal with the singularly perturbed DDEs. • We complete the entire task by unifying the piecewise RG equations into a delay one. • By the numerical simulations, the uniformly valid of our result be proved. [ABSTRACT FROM AUTHOR]

• Convergence dynamics of a pancreatic cancer model is studied. • A multi-step iterative localization process is used. • Ultimate sizes of interacting cell populations are computed. • Tumor eradication conditions are derived. • Tumor persistence conditions are obtained. In this paper, we study various types of ultimate dynamics for the 5-dimensional pancreatic cancer model without treatment. This model has been created by Hu et al. and describes interactions between pancreatic cancer cells (PCCs); pancreatic stellate cells (PSCs); immune cells and two types of cytokines, tumour–promoting and tumour–suppressing. Using iteratively the localization theorem of compact invariant sets we derive ultimate upper bounds for all cell populations and establish the property of the existence of the attracting set. Next, we find several conditions under which our system demonstrates various types of convergence dynamics: to equilibrium points located in the intersection of PCC-free plane with the PSC-free plane; equilibrium points with positive values of PSC density located in PCC-free plane; ω -limit sets with positive values of PCC density. Further, we explore the dynamics within the PSC-free plane and find attraction conditions to ω -limit sets with positive values of PCC density. Our assertions are expressed by means of iterative sequence of algebraic inequalities respecting parameters of the model. Our theoretical studies are supplied by results of numerical simulation. Locations of attracting equilibrium points have been found with help of the iteration procedure containing from 2 steps up to 23 steps in the dependence of parameter conditions and the location with respect to the PSC-free plane. [ABSTRACT FROM AUTHOR]

• A negative answer is given for the basic open question of the inverse spectral theory for differential operators with constant delay. • The obtained results essentially modify the future strategy of studying nonlinear inverse problems for the operators with delay. • A general approach is developed for constructing iso-bispectral potentials. • Numerical simulations have led to a fundamental theoretical result. The inverse spectral theory was developed most widely for purely differential operators. Meanwhile, various their nonlocal perturbations are often more adequate for modelling various real-world processes frequently possessing a nonlocal nature. Thus, there recently appeared a growing interest in the nonlinear inverse problems for differential operators with constant delay. However, because of their peculiarity, such problems are still insufficiently studied. In particular, there has been a long-term basic open question about the unique solvability of such inverse problems for small values of the delay. In this paper, we address that question by giving a negative answer in an important and illustrative situation. Our study is based on numerical simulations, which have finally led to a deep theoretical result. Moreover, we develop a general approach for constructing relevant iso-bispectral potentials. The obtained results essentially change the future strategy of studying inverse problems for differential operators with delay. [ABSTRACT FROM AUTHOR]

• A reverse-time nonlocal nonlinear self-dual network equation is proposed. • Discrete nonlocal version of the N-fold Darboux transformation is constructed. • Unstable and stable soliton solutions are derived and shown graphically. • It is shown that individual components show instability whereas the potential terms present stable structures. Inspired by the idea of Ablowitz and Musslimani, a reverse-time nonlocal nonlinear self-dual network equation is proposed under certain nonlocal symmetry reduction. N -fold Darboux transformation technique is used to construct discrete unstable and stable multi-soliton solutions in zero seed background for this system. Based on the asymptotic and graphic analysis, structures of one-, two- and three-soliton solutions are shown graphically. It is found that the individual components in this nonlocal equation show unstable temporal soliton structures whereas the combined potential terms exhibit stable soliton structures. Dynamical behaviors for one-soliton solutions are investigated via numerical simulations. It is worth noting that the solutions of this nonlocal equation are clearly different from those of its corresponding local part. Results obtained in this paper may be helpful for the explanation of the propagation of electrical signals. [ABSTRACT FROM AUTHOR]

• A Cournot-Bertrand duopoly mixed competition model is established. • The stability conditions of the system are analyzed by numerical calculation and numerical simulation. • The Arnold's tongues are studied. • The coexistence of attractors is analyzed by using the basin of attraction. • The contact bifurcation in the basin of attraction is studied by using the theory of critical curve and noninvertible map. In this paper, we consider the Cournot-Bertrand duopoly mixed competition model, which is characterized by different decision-making variables and different objective functions of the two enterprises. This form of competition is more consistent with the complex actual economic market. The general stability conditions of the four equilibria are given and analyzed with eigenvalues and Jury criterion, so that enterprise decision makers could choose appropriate parameters to determine the development of the enterprise. Under the specific parameter conditions, the stability conditions are analyzed and demonstrated in detail by using two-dimensional bifurcation diagrams, stable region diagrams and bifurcation curves. Many fractal Arnold's tongues are found in two-dimensional bifurcation diagrams. These tongues are associated with the Neimark-Sacker bifurcation of the fixed point and are arranged in accordance with the organization law of the periodic tree of the Stern-Brocot tree, which is helpful for us to analyze the transitions of system between period and chaos. With the help of the basin of attraction, the coexistence of attractors is analyzed, on which the decision maker can choose the initial conditions that are beneficial to the development of the enterprise. We also analyze the topology structure of basin of attraction and the formation mechanism of holes in the basin of attraction by using the theory of critical curve and noninvertible map. [ABSTRACT FROM AUTHOR]

• The energy of the stochastic space-fractional nonlinear wave equation is obtained under the two different noise case. • We use Crank-Nicolson difference method based on second-order fractional differences to simulate the different behavior of stochastic space-fractional nonlinear wave equation in additive noise and multiplicative noise. • The discrete averaged energy of the stochastic space-fractional nonlinear wave equation coincides with theoretical results. • The energy of system possesses dissipation-preserving or energy-conserving property under the suitable conditions. • The space-time convergence order of our numerical schemes in time and space is presented in the sense of expectation. In this paper, we numerically solve a stochastic space-fractional nonlinear wave equation which includes fractional derivative, nonlinear term, damping term and noise term. The energy of system in the continuous case is derived in detail and discrete energy preserving physical characteristic is also proved. We propose a numerical method that uses Crank-Nicolson difference discretization based on second-order fractional differences. The numerical schemes for stochastic space-fractional nonlinear wave equation with two different noise possess dissipation-preserving energy or energy-conserving under suitable conditions. Moreover, the convergence order of our numerical schemes in time and space is presented in the sense of expectation. Numerical experiments with various types of noise demonstrate that the dissipative property or conservative property coincides with theoretical results. [ABSTRACT FROM AUTHOR]

• Analytical and numerical results are obtained for a three-species food chain model that incorporates the cost and benefit of anti-predator behaviours. • A fear effect term is rigorously derived, the impact of the fear is fully expounded. • A comparison of dynamic behavior with the existing models, some new findings are presented. • The results may provide some useful biological insights into ecosystems containing predator-prey interactions. In this paper a three-species food chain model is formulated to investigate the impact of fear. First, we derive the predator's functional response by using the classical Holling's time budget argument and formulate a three-species food chain model where the cost and benefit of anti-predator behaviours are included. Then we study the dissipativity of the system and perform analysis on the existence and stability of equilibria. At last, we use numerical simulations to more visually explore the effects of fear on three species. The results show that the predator's fear effect can transform the system from chaotic dynamics to a stable state. Our results may provide some useful biological insights into ecosystems containing predator-prey interactions. [ABSTRACT FROM AUTHOR]