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In this article, exponential synchronization problem of complex networks with coupling delays using an impulsive observer and impulsive control is studied. Different from traditional network synchronization schemes, the state of the observer is fed back to the network nodes to achieve synchronization. The observer and controller, modeled by impulsive differential equations, are more secure and efficient. Both the cases of measurable and unmeasurable information of coupling time-varying delays are considered. Several criteria are obtained to investigate globally exponential synchronization of complex networks. The exponential convergence rate can be estimated effectively. By solving the linear matrix inequalities (LMIs) associated with the results, the gain matrices can be designed. Finally, theoretical results are verified through a numerical example. • This paper explores exponential synchronization in complex networks with coupling delays via impulsive observer and control. • Different from traditional schemes, the state of the observer is fed back to the network nodes to achieve synchronization. • It presents LMI conditions to solve cases of measurable and unmeasurable information of coupling time-varying delays. [ABSTRACT FROM AUTHOR]

This paper studies the secure synchronization problem for two-time-scale complex dynamical networks with unknown dynamics information and malicious attacks. The challenge is that under complex dynamic networks with unknown dynamic information, all system matrices are unknown. To ameliorate this conundrum, we design a reinforcement learning algorithm, conjoined with a full-order processing of singularly perturbed parameters, with the express objective of securing the stability of the synchronization error system within a context devoid of adversarial intrusions. Second, in order to reduce the impact of malicious attacks, a switching function is developed based on feedback gain. Besides, it is proved that the proposed distributed controllers can still guarantee the convergence of the synchronization error system under malicious attacks. Finally, two numerical examples are given to illustrate the applicability and effectiveness of the proposed algorithm. • A distributed model-free sliding mode controller is presented for SPCDNs. • An iterative algorithm based on RL is employed. • The controller can weaken the influence of the attacks and achieve secure synchronization. [ABSTRACT FROM AUTHOR]

In this paper we investigate explicit numerical approximations for stochastic differential delay equations (SDDEs) under a local Lipschitz condition by employing the adaptive Euler-Maruyama (EM) method. Working in both finite and infinite horizons, we achieve strong convergence results of the adaptive EM solution. We also obtain the order of convergence in finite horizon. In addition, we show almost sure exponential stability of the adaptive approximate solution for both SDEs and SDDEs. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the asymptotic tracking control for a class of strict-feedback nonlinear systems with unknown nonlinear functions and output constraint. The nonlinear transformation function-based system transformation approach is employed to convert the initial constrained system into an unconstrained one. To achieve the objective of asymptotic tracking, the tracking error is scaled by an exponential proportional function. Furthermore, to address the challenge of repetitive derivation of virtual control laws that occurs in the traditional design process of high-order backstepping methods, a first-order filter is applied. The designed control scheme guarantees that the system output will asymptotically track a predefined reference tracking trajectory while ensuring that the output constraint is not violated. Moreover, it is shown that the filter error will converge asymptotically to the origin. The effectiveness of the developed control strategy is verified through simulation results. • This study introduces a nonlinear system transformation function to convert the initial constrained control problem. • An exponential-type proportional function is constructed. • The asymptotic convergence performance can be guaranteed. [ABSTRACT FROM AUTHOR]

The topic of how to solve cooperation dilemmas has been intensively investigated. Numerous mechanisms have been presented to promote cooperation level by introducing various third-party strategies. This paper considers the more realistic strategy named punishing evil and promoting good (PEPG) in the public goods game, whose fundamental action is cooperation. Players selecting this strategy can not only reward cooperators but also penalize defectors, effectively amalgamating the advantages of both rewarders and punishers. Moreover, distinct strategic choices lead to dynamic changes in players' reputation, which in turn significantly influence their income distribution. Notably, individuals with elevated reputation are entitled to allocate more benefits rather than homogeneous distribution. Simulation experiments demonstrate that the PEPG strategy is pivotal in eliminating defectors and fostering cooperative behavior. Importantly, three intriguing phenomena are displayed in this research, including the resurgence of defectors, the polarization of reputation and the disparities of payoffs, all of which help us understand the emergence of cooperation in the real world. • The strategy of punishing evil and promoting goods is considered, which combines the advantages of rewarders and punishers. • Individuals with elevated reputations are entitled to allocate more benefits rather than homogeneous distribution. • The three intriguing phenomena can be observed in this research. • We explore the comprehensive influence on cooperation of various parameters. [ABSTRACT FROM AUTHOR]

We solve the so far open problem of constructing all spatial rational curves with rational arc length functions. More precisely, we present three different methods for this construction. The first method adapts a recent approach of (Kalkan et al. 2022) to rational PH curves and requires solving a modestly sized system of linear equations. The second constructs the curve by imposing zero-residue conditions, thus extending ideas of previous papers by (Farouki and Sakkalis 2019) and the authors themselves (Schröcker and Šír 2023). The third method generalizes the dual approach of (Pottmann 1995) from planar to spatial curves. The three methods share the same quaternion based representation in which not only the PH curve but also its arc length function are compactly expressed. We also present a new proof based on the quaternion polynomial factorization theory of the well known characterization of the Pythagorean quadruples. • Three methods to compute rational curves with rational arc length function. • Compact quaternionic formulations, extending formulas for rational PH curves. • Comparison and critical discussion of merits of each approach. • Extensive examples relating our findings to existing literature. [ABSTRACT FROM AUTHOR]

The secondary control of islanded microgrids holds paramount importance for maintaining power quality in smart grid regions, posing a significant challenge in recent research on multi-agent systems. This paper presents an improved secondary control approach designed in AC islanded microgrids (MGs) operating in a communication network environment. The proposed method first considers that the communication topology of distributed generation (DG) is switched randomly, and such switching is described by the semi-Markov process with an uncertain transition rate. Then, an asynchronous mechanism is used to correlate the evolution of system and controller modes in response to solve the difficulty of obtaining modal information. Based on the aforementioned foundation, an event-based distributed security secondary control algorithm is constructed, leveraging feedback linearization and the leader-follower consensus protocol. Specifically, a hybrid event-triggered mechanism, characterized by both memorability and dynamics, is introduced, effectively mitigating communication load. Furthermore, by taking the false-data-injection attack as a case study, the algorithm enhances the security of communication control. Besides, considering the limitation of model error and the influence of external noise, a sliding mode control is additionally applied to enhance the control robustness among the secondary control layer. The proposed method is verified and evaluated from theoretical and emulational perspectives. The research findings demonstrate that the suggested secondary control has robust performance for the environment constraint while satisfying the targeted frequency recovery and real power allocation. [ABSTRACT FROM AUTHOR]

This paper is concerned with the linear quadratic decentralized control of forward and backward stochastic difference system (FBSDS) with multiplicative noises and nested asymmetric information. The main contribution is to give the equivalent condition for the solvability of the problem and the explicit decentralized optimal controllers in terms of Riccati equations. The key technology is solving the forward and backward stochastic difference equations with partial information derived from stochastic maximum principle. • LQ decentralized control of forward and backward stochastic difference system with nested asymmetric information. • Forward and backward stochastic difference equations with partial information. • Present solvability condition and explicitly decentralized optimal controllers of the problem. • Propose novel Riccati equations to solve forward and backward stochastic difference equations. [ABSTRACT FROM AUTHOR]

Recently, Li et al. [27] studied the hydrodynamic equations, proposed a relaxed block upper-lower triangular (RBULT) preconditioner. In this paper, we presented a modified relaxed block upper-lower triangular (MRBULT) preconditioner, which is an extension of the RBULT preconditioner. The advantage of this preconditioner is that it retains the computational advantage of RBULT preconditioner, and the choice of optimal parameters is simpler. We analyze the eigenvalue distribution and an upper bound of the degree of the minimal polynomial of the preconditioned matrix. Furthermore, the selection of the optimal parameter α and β are given. Finally, the theoretical analysis is proven by numerical examples, which has more advantages than previous existing preconditioners. • We present a modified RBULT preconditioner for generalized saddle point problems from the hydrodynamic equations. • This iterative algorithm is proved to be fast convergent. • Some numerical examples are tested to verify the efficiency and stability of the proposed algorithm. [ABSTRACT FROM AUTHOR]

• Use of two non-iterative explicit and implicit finite volume methods to calculate the transient flow of natural gas in distribution networks. • Solving the gas network as a whole at each time step. • Proposed finite volume approaches are more efficient than those of conventional methods. This paper describes two non-iterative explicit and implicit finite volume methods (FVM) to calculate the unsteady characteristics of natural gas in distribution systems. The proposed finite volume model could be used to solve the gas network as a whole for each time step. The main advantage of this approach is that it can analyze gas distribution networks with less computational cost. In addition, contrary to the conventional iterative algorithms, it does not consist of any divergence problems. To achieve this goal, the explicit FVM is derived from the governing equations. Then, the linearization process is performed, and the numerical solution of the linear equations is developed by using the implicit FVM. Three classic test cases are exemplified to check the performance of the proposed methods. The results confirm that both explicit and implicit methods, even in complicated distribution networks, can predict the elements of unsteady natural gas flows. Furthermore, when the governing equations are linearized and the implicit FVM is applied, satisfactory results with less computational cost, compared to the explicit one, are produced. [ABSTRACT FROM AUTHOR]

In graph theory, a packing within graph G is defined as a collection H = { H 1 , H 2 } , where H 1 and H 2 are both equivalent to G , yet they exist as edge-disjoint subgraphs within K n. Considering a positive integer k , we introduce D k , a set comprising graphs within G , each with a maximum degree not exceeding k. Extending the concept of a packing, a near packing that accommodates D k in graph G emerges, realized by overlapping two iterations of G. In this configuration, the intersecting subgraph formed by edges shared in both G copies is a member of D k. The notation m (n , D k) represents the highest possible value of m , ensuring each graph with order n and size at most m can incorporate a near packing that accommodates D k. A notable case is m (n , D 0) = n − 2 , highlighting that a near packing which admits D 0 equates to a standard packing. This paper focuses on establishing the value of m (n , D 1) = ⌊ 3 n − 5 2 ⌋. • We show that each graph of order n and size at most ⌊ 3 n − 5 2 ⌋ has a near packing. • Our proof is constructive. • The bound of our result is sharp in general. [ABSTRACT FROM AUTHOR]

As a social welfare, pension insurance has provided a great assistance to retiree in maintaining their livelihoods. In order to reward the contributions made by cooperators, an additional pension income is provided to them. In this paper, we study a public goods game based on the pension system on population structure, in which only cooperators can purchase pension insurance and receive pension benefits upon meeting specified conditions. In the simulation, we demonstrate the evolution of cooperation frequency over time and investigate the influence of insurance premium, pension baseline, the number of defected neighbors of cooperators, and the enhancement factor on the cooperation frequency. Additionally, we examine the impact of the pension system on cooperative behavior in triangular and hexagonal networks. Our results suggest that in the public goods game on networked structure, the pension system not only fails to promote cooperation but actually inhibits it, even when cooperators do not have to pay any costs and can receive pensions when conditions are met. [ABSTRACT FROM AUTHOR]

This paper considers the time-varying formation (TVF) tracking issue of a class of multiagent systems (MASs) with input saturation. Meanwhile, the prescribed transient and steady-state performance of MASs is guaranteed. Compared to the existing prescribed performance control (PPC) approaches, the maximum convergence time of the designed algorithm is preassigned. To save network resources, an event-based protocol is introduced to achieve lower information communication frequency, where Zeno behavior is excluded. Furthermore, an improved control strategy is employed where an auxiliary system is designed to improve the performance under actuator saturation. According to Lyapunov stability theory, the expected practical TVF tracking control problem is solved for the MASs. Finally, the effectiveness of the designed schemes is demonstrated by the simulation examples. • A low-complexity, static, state-feedback control algorithm is proposed. • The prescribed performance control law achieves the appointed-time convergence. • The tracking precision of the control law is guaranteed under actuator saturation. [ABSTRACT FROM AUTHOR]

Let C m be a cycle with length m. The k -uniform hypercycle with length m is obtained by adding k − 2 new vertices in every edge of C m , denoted by C m , k. In this paper, we obtain some trace formulas of uniform hypercycles with length four. Moreover, we give the characteristic polynomials of uniform hypercycles with length four. • By the Matrix-Tree Theorem and BEST Theorem, etc., we present some trace formulas of uniform hypercycles with length four. • By applying the trace formulas, we study the characteristic polynomials of uniform hypercycles with length four. • We study the traces and obtain an explicit characteristic polynomial expression of uniform hypercycles with length four. [ABSTRACT FROM AUTHOR]

This paper focuses on non-instantaneous impulsive systems with double state-dependent delays. The finite-time stability criteria of such systems are obtained via impulsive control theory and Lyapunov function method, and the effects of destabilizing impulse and stabilizing impulse on the finite-time stability are considered, respectively. Meanwhile, the obtained results are applied to finite-time stability of genetic regulatory networks with state-dependent delays and non-instantaneous impulses as an example. • The relationship between state-dependent delay and non-instantaneous impulse is revealed. • The restrictions of state-dependent delays are reduced in order to make the conditions more applicable. • The effects of destabilizing and stabilizing impulse on the finite-time stability are considered. • The obtained results are applied to genetic regulatory networks. [ABSTRACT FROM AUTHOR]

This study proposes a search algorithm for the compact uncertainty region of system elements with affine constraints and interval information. While existing studies have considered only the interval information of uncertainty, the proposed algorithm reflects both the interval and affine constraints of uncertain elements to reduce conservatism due to uncertainties. First, the affine constraints of uncertainty are plotted in an N -dimensional space, where N is the number of elements that should be considered. Due to the properties of the affine constraints, the affine constraints are expressed as an (N -1) dimensional structure. The interval information is then expressed as an N -dimensional structure. The algorithm iteratively checks the edges of the N -dimensional structure between two geometrical structures to determine the intersection region. The algorithm then collects the vertices of the convex polytope that overlap the two structures. To show the effectiveness of proposed algorithm, this study proposes a consensus criterion for a multi-agent time-delayed system with uncertain switching topologies as an application of a system with uncertainties in system elements. • We clarify the motivation and contribution of this paper. • We update figures in Example 1. • We add the Example 2 to better show the efficiency of proposed algorithm. • To help us understand, we added Remarks 1, 8. [ABSTRACT FROM AUTHOR]

This paper specializes in exponential output synchronization of reaction-diffusion neural networks (RDNNs) for two different cases of output couplings. The proposed model is novel for incorporating multiple output couplings related to their own output states as well as output spatial diffusion couplings in the absence of ordinary differential equations (ODE) systems. Generalized intermittent control based on spatial sampled-data is adopted for the first time to handle error systems consisting of the states of RDNNs with different output couplings scenarios and the average of the sum of their output states to achieve exponential stability. Some sufficient conditions for determining the proposed network models are established by control protocols based on the Lyapunov functional method. Finally, the obtained theoretical results are demonstrated to be valid by numerical simulations. • A novel RDNNs with multiple output spatial diffusion coupling is proposed. • The method of intermittent spatial sampled-data control is first employed in the output-coupled model. • The output synchronization criterion is established by means of Lyapunov functions and intermittent control. [ABSTRACT FROM AUTHOR]

Given a non-negative integer g , the g - e x t r a c o n n e c t i v i t y in a connected graph G is defined as the minimum number of vertices that, when removed, results in G becoming disconnected and each remaining component containing more than g vertices. As an extension of the connectivity, the g - e x t r a c o n n e c t i v i t y provides a better measurement for the fault tolerance of an interconnection network. The strong product G 1 ⊠ G 2 of graphs G 1 and G 2 is the graph with vertex set V (G 1) × V (G 2) , where two distinct vertices (a 1 , b 1) and (a 2 , b 2) are adjacent if and only if a 1 = a 2 and b 1 b 2 ∈ E (G 2) , or b 1 = b 2 and a 1 a 2 ∈ E (G 1) , or a 1 a 2 ∈ E (G 1) and b 1 b 2 ∈ E (G 2). In this paper, we focus on networks modeled by the s t r o n g p r o d u c t G 1 ⊠ G 2. We determine the g (≤ 3) - e x t r a c o n n e c t i v i t y of G 1 ⊠ G 2 , where G 1 and G 2 are regular and maximally connected graphs with girth at least g + 4. Additionally, we give the g (≤ 3) - e x t r a conditional diagnosability of G 1 ⊠ G 2 under PMC model. • Determined the g (≤3)- extra connectivity of the strong product of regular and maximally connected graphs. • Gave the g (≤ 3) - extra conditional diagnosability of a class of strong product graphs under PMC model. • Provided two reliability-measurements for the interconnection networks modeled by a class of strong product graphs. [ABSTRACT FROM AUTHOR]

The problem of stochastic input-to-state stability in probability (ISSiP) is studied in this paper for a class of constrained impulsive switched systems with unstable subsystems, where the switching signal and impulsive signal are confined to the subsets of all modes. Especially, the impulsive intensity and impulsive density are both stochastic, which triggered by a renewal process. Taking into account asynchrony of switching and impulsive signal, a new comprehensive admissible edge-dependent average impulsive interval (CAED-AII) concept is proposed to characterize the impulsive signal. Under constrained switching and stochastic impulses, we investigate ISSiP of impulsive switched systems in two cases: the impulsive intensity is supposed to be stochastic and impulsive density is fixed, and the impulsive intensity and density are both stochastic. In these cases, the situation of the subsystem is divided into two scenarios for discussion as well. The first scenario is that all subsystems are unstable. The second scenario is that some subsystems are unstable and some subsystems are stable. With the help of the admissible edge-dependent average dwell time (AED-ADT), CAED-AII, and Lyapunov-like function, sufficient conditions are established for ISSiP of the whole system. Finally, two numerical examples are given to illustrate the developed results. • A more general framework of nonlinear systems is constructed in the presence of switching signals and impulsive signals, which contains constrained switchings and impulses, and stabilizing or destabilizing impulses with stochastic intensity and density. Due to the asynchronous switchings and impulses, two kinds of time sequence need to be tackled separately. • A novel CAED-AII concept is proposed for impulsive signals. Since the impulsive signals occur independently of the switching signals, it is less conservative. For impulsive intensity and density of stochastic impulses, the AII of impulse mode with different intensity and density is related to the current and historical impulse mode. • The ISSiP problem is addressed for two cases. When all subsystems are unstable but the impulses are stabilizing, the system is ISSiP while the impulses only have stochastic intensity or the impulses have stochastic intensity and density. When some subsystems are stable and the impulses are destabilizing, the system is ISSiP with the impulses are as before. • A comprehensive modeling paradigm that can describe stochastic impulses from the method of CAED-AII is constructed through the impulsive effects of stochastic impulsive intensity and density. [ABSTRACT FROM AUTHOR]

Let G ˙ be an unbalanced signed graph. In this paper, we establish that e (G ˙) ⩽ 1 2 (n 2 − n) − (n − 3) and ρ (G ˙) ⩽ n − 2 if G ˙ does not contain unbalanced K 4 as a signed subgraph. Moreover, comprehensive characterizations of the extremal signed graphs have been obtained. • The situation that the index of signed graph equals to the spectral radius of underlying graph has been characterized. • We determine the Turán number of K 4 − in unbalanced signed graph, and we characterize all the extremal signed graphs. • We solve the spectral Turán problem among all K 4 − -free unbalanced signed graphs. [ABSTRACT FROM AUTHOR]

In this paper, we explore proper vertex colorings of graphs with the additional constraint that the colors used on the closed neighborhood of every vertex form an integer interval; such colorings are called proper closed interval vertex colorings (PCIV colorings for short). As not every graph admits such a coloring, we are concerned with the problem of deciding which graphs admit PCIV colorings, describing several sufficient conditions (among them unique and 3-colorability, partial decomposability into two bipartite graphs, and density constraints) as well as some constructions of PCIV uncolorable graphs. We also present an algorithmic support for PCIV colorability based on a modification of the standard greedy algorithm for standard proper coloring. • Propose a new proper coloring (PCIV coloring) where colors on closed neighborhoods of vertices form integer intervals. • Particular sufficient conditions for PCIV colorability are proved. • Several constructions of PCIV uncolorable graphs are presented. • Greedy algorithm for finding a PCIV coloring is developed. [ABSTRACT FROM AUTHOR]

In this paper, the Lagrange stability and passivity of a class of discrete-time stochastic Markovian switching neural networks with mixed delays are studied. The mixed delays include time-varying transmission and distribution delays. Global exponential Lagrange stability criterion and passivity criterion in the mean square sense are derived. A numerical example illustrates the usefulness of these criteria. The originality of the proposed method lies in: (i) the criteria are derived directly from the definitions rather than constructing Lyapunov–Krasovskii functional; (ii) the inequalities in the criteria contain only a few decision variables, which is easy to solve and leads to low computational complexity. • Lagrange stability and passivity of discrete-time SM-SNNs with mixed delays are studied. • The time-varying transmission and distribution delays are involved. • Global exponential Lagrange stability criterion and passivity criterion in MMS are derived. • The inequalities in the criteria are easy to solve and lead to low computational complexity. [ABSTRACT FROM AUTHOR]

In this paper, we examine the solvability of a functional equation in a Lipschitz space. As an application, we use our result to determine the existence and uniqueness of solutions to an equation describing a specific type of choice behavior model for the learning process of the paradise fish. Finally, we present some concrete examples where, using numerical techniques, we obtain approximations to the solution of the functional equation. As the straightforward Picard's iteration can be very expensive, we show that an analytical suboptimal least-squares approximation can be chosen in practice, resulting in very good accuracy. • Functional equation from behavioral science is considered. • A unique solution is found in the Lipschitz space. • An analytical accurate approximate solution is devised. [ABSTRACT FROM AUTHOR]

The stabilized collocation method (SCM) is a promising meshless collocation method that can overcome the instability defects in the classical direct collocation method. To improve the performance of the SCM, a superconvergent stabilized collocation method (SSCM) is developed in this paper for linear and nonlinear elliptic problems through the use of the moving least squares (MLS) approximation and its smoothed derivatives. Accuracy of the SSCM and the SCM is analyzed with an emphasis on the influence of boundary conditions, and precise error measures are presented for different types of boundary conditions. Numerical results validate the superconvergence of the SSCM and confirm the theoretical analysis. • A superconvergent stabilized collocation method is developed for linear and nonlinear elliptic problems. • Theoretical accuracy of the method is analyzed detailedly. • Theoretical and numerical results reveal that the method possesses superconvergence property. [ABSTRACT FROM AUTHOR]

This paper focuses on solving stochastic linear quadratic nonzero-sum differential games with completely unknown system matrices and long-time average costs. Firstly, for the case that the system matrices are known, we design a model-based value iteration (VI) algorithm and a model-based robust VI algorithm to solve the coupled algebraic Riccati equations (AREs), which are directly related to the construction of the feedback Nash equilibrium solution. Then, we present a model-free VI algorithm, based on which feedback control strategies for players are designed using only the data of states and inputs, without requiring the prior knowledge of the system matrices. Moreover, we show that the designed feedback control strategies make the closed-loop game system stable and reach the Nash equilibrium. Finally, a numerical simulation is given to demonstrate the effectiveness of the proposed method. • A model-based robust VI algorithm is proposed to solve stochastic linear quadratic nonzero-sum games. • A novel model-free VI algorithm is proposed for unknown stochastic linear quadratic nonzero-sum differential games. • Feedback control strategies for players are designed by using only the data of states and inputs. • The designed feedback control strategies can make the closed-loop game system stable and reach the Nash equilibrium. [ABSTRACT FROM AUTHOR]

As a fundamental task, medical image reconstruction has attracted growing attention in clinical diagnosis. Aiming at promising performance, it is critical to deeply understand and effectively design advanced model for image reconstruction. Indeed, one possible solution is to integrate the deep learning methods with the variational approaches to absorb benefits from both parts. In this paper, to protect more details and a better balance between the computational burden and the numerical performance, we carefully choose the multi-level wavelet convolutional neural network (MWCNN) for this issue. As the tight frame regularizer has the capability of maintaining edge information in image, we combine the MWCNN with the tight frame regularizer to reconstruct images. The proposed model can be solved by the celebrated proximal alternating minimization (PAM) algorithm. Furthermore, our method is a noise-adaptive framework as it can also handle real-world images. To prove the robustness of our strategy, we address two important medical image reconstruction tasks: Magnetic Resonance Imaging (MRI) and Positron Emission Tomography (PET). Extensive numerical experiments show clearly that our approach achieves better performance over several state-of-the-art methods. • We explore a medical image reconstruction framework with deep learning-based method. • To solve the hybrid model, we introduce the PAM algorithm. • We analyze the convergence of the proposed method with numerical results. • The robustness of our method can be illustrated by the PET and MRI reconstruction. [ABSTRACT FROM AUTHOR]

As an important part of the evolution of cooperation, interaction diversity has attracted extensive attention. Traditional studies mainly focus exclusively on either time or space, and the comprehensive impact of these two dimensions on the evolution of cooperation deserves further exploration. In this paper, the interaction willingness is abstracted as the network weight, and an adaptive mechanism for interaction willingness based on the delay effect is proposed by combining the network weight and accumulated payoff. Specifically, individuals adjust their willingness to participate in the game with a delay, considering the change in accumulated payoffs within a certain period, rather than the change in immediate payoffs. The degree of delay is reflected in the length of individuals' memory of historical payoffs. Simulation results deviate from the intuition that more information facilitates cooperation, and in certain modes of interaction willingness adjustment, the correlation between cooperation level and memory–length is non–monotonic. Long–memory is conducive to promoting cooperation under the condition of a relatively large range adjustment of interaction willingness. Conversely, with a narrow adjustment range of interaction willingness, cooperation can be effectively established with short–memory. These results hold substantial implications for both theory and practice, as slight adjustment often corresponds to low loss and high operability. In addition, the underlying mechanism is furthermore explored, revealing that the 'isolation zones' composed of individuals with low interaction willingness play an important role in promoting cooperation. • The self-adaptive mechanism based on memory influences cooperation evolution. • The relationship between cooperation level and memory-length is non-monotonic. • Memory-length affect cooperation differently based on willingness adjustment mode. [ABSTRACT FROM AUTHOR]

Hedetniemi's conjecture, a significant problem in graph theory, was disproved by Shitov in 2019. This breakthrough stimulates us to consider the signed graphs version of Hedetniemi's conjecture, where signed graphs are graphs with an assignment of { ± 1 } to the edges. In this paper, we provide counterexamples to the signed analog of Hedetniemi's conjecture. Nevertheless, we show that the chromatic number of the product is determined by its factors in some instances, i.e., if Γ → Θ and (Θ , π) is balanced, then χ ((Γ , σ) × (Θ , π)) = χ ((Γ , σ)). As a byproduct, we investigate the homomorphisms from a factor of (Γ , σ) × (Θ , π) to the other, say from (Γ , σ) to (Θ , π). We find that if such homomorphisms exist, then the chromatic number of (Γ , +) provides a lower bound for χ ((Γ , σ) × (Θ , π)). Moreover, if Γ is a core, then the number of balanced subgraphs isomorphic to (Γ , +) in (Γ , σ) × (Θ , π) is equal to the number of such homomorphisms. [ABSTRACT FROM AUTHOR]

This paper presents a new interface sampling and reduction method for varying parameters within the framework of the component mode synthesis (CMS) method. In the conventional interpolation-based parametric reduced-order model (IB-PROM) based on CMS, reduced subsystems are synthesized, and interface degrees-of-freedom are reduced via the secondary eigenvalue analysis in the online stage, which is essential to obtain highly reduced system matrices. However, it inhibits rapid parameter updating and requires extra computational resources in the online stage. In the present study, the pre-reduced interface sampling method is proposed by synthesizing substructures and reducing the interface before beginning the online stage. The interface solution is obtained through offline and online adaptation, likewise the procedure of computing the interior solution of previous studies. Additionally, congruence transformations are applied to each reduced interface, which is mandatory for interpolating symmetric positive definite matrices with respect to a new input parameter. As a result, the online performance of IB-PROM is greatly enhanced, showing rapid and near real-time adaptations to the variations in system parameters. • A new interface sampling and reduction method for varying parameters. • Efficient online reduced-order model with pre-reduced interface degrees of freedom. • System-level interface sampling and reduction in the offline stage. • Online interpolation of pre-reduced interfaces via congruence transformation. [ABSTRACT FROM AUTHOR]

• A self-propelled particle model considering leadership and experience is proposed. • Group queuing and stable movement are achieved under the guidance of leaders. • The optimal leadership and vision intervals are obtained. The coordinated and ordered collective behavior arising from dispersed and local self-organizing interactions among individuals is widely observed in many biological communities. For populations engaging in group foraging or migration, complex societies with multiple hierarchical levels are common. A prevalent scenario involves two dominant levels, one corresponding to leaders and the other composed of followers. In this paper, we explore how a group can be influenced by a set of leaders to organize and move in a coordinated manner towards a preferred direction. Specifically, we balance social interactions among group members through two weighted factors, corresponding to the leadership and experience of leaders. Through computer simulations, we observe a linear relationship between the proportion of leaders guiding group movement and the group size, with larger groups requiring a higher proportion of leaders while maintaining initial density. Additionally, we identify an optimal leadership interval where group movement performance enhances with increasing leadership up to a certain point, beyond which it starts to decline. Similar patterns are observed concerning individual visual field, indicating the existence of an optimal interval for visual angle. Our findings not only contribute to understanding collective motion in natural biological systems but also provide new insights into effective leadership mechanisms in artificial swarm systems. [ABSTRACT FROM AUTHOR]

In this paper, we explore and analyze a stochastic epidemiological model with vaccination and two mean-reverting Ornstein-Uhlenbeck processes to describe the dynamics of HBV transmission. Our study begins by deriving a sufficient condition for the extinction of hepatitis B. Additionally, we determine the presence of a stationary distribution using the Markov process stability theory. Next, we utilize algebraic equation theory and the associated Fokker-Planck equation to derive an explicit expression for the unique probability density function. To validate our theoretical findings, we conduct several numerical simulations using hepatitis B data provided by the World Health Organization (WHO). The results of these simulations support and complement our analytical approach. • A stochastic hepatitis B epidemic model with vaccination and OU process is proposed. • The dynamics of the stochastic model and its corresponding deterministic version are consistent. • The existence of ergodic stationary distribution for the proposed model is proved. • An explicit expression for the unique probability density function is derived. • Numerical simulations are taken to support and complement the analytical approach. [ABSTRACT FROM AUTHOR]

The motivation of this paper is inspired by the research of Zanette et al. (2011) [4] , which reveals that the perception of predation risk can reduce the number of juvenile prey. To investigate how fear affects the behavior of prey population, we formulate a predator-prey model with a stage structure for prey in which adult prey is undergoing the interference of fear. First, the dynamic properties of the model are analyzed, including the existence and stability of equilibria, bifurcations dynamics, such as transcritical, saddle-node, and Hopf bifurcations. Our results reveal that fear can promote the formation of the stability of the model by transforming it from an unstable state to a stable state. Numerical simulations and theoretical analysis indicated that when the level of fear reaches 25.4, the number of juvenile prey affected by fear decreases by 40% compared to those not affected by fear. Terrifyingly, the predators are on the brink of extinction in high levels of fear. Moreover, we examine the impact of time delay on fear in the previous model as prey needs some time to assess the predation risk. Some dynamic characteristics of this delayed model are also analyzed. Results indicate that the fear effect is conducive to generating a stable state, while delay is detrimental to generating a stable state, leading to limit cycles or even chaos. • A stage-structure predator-prey model with fear and delay is proposed. • The feasibility equilibria and bifurcation dynamics of the model are studied. • Fear can transform a model from an unstable state to a stable state. • Delay is detrimental to generating a stable state, leading to limit cycles or chaos. [ABSTRACT FROM AUTHOR]

This paper investigates the fixed-time cooperative output regulation (FTCOR) problem with unknown exosystem of second-order nonlinear multiagent systems. To solve this problem, it is necessary to ensure that each agent can obtain the reference trajectory originating from an unknown exosystem and reconstruct the solution of the regulator equation containing the exogenous signal and unknown parameter vectors within a fixed time, and the unknown exosystem makes it difficult to solve these two problems with existing tools. By using auxiliary filtering variables, we propose a fixed-time adaptive distributed observer (FTADO) and design a fixed-time distributed internal model (FTDIM) based on this FTADO. These two new tools ensure the solvability of the two problems just mentioned. Then, based on these two tools, we transform the problem into a fixed-time stabilization problem of a so-called augmented system. Through the backstepping technique, we propose a distributed controller that solves this fixed-time stabilization problem while relaxing the assumptions of similar references. • More general and difficult controlled objects and research problem are considered. • A fixed-time adaptive distributed observer for an unknown leader system is designed. • A compensator is designed to reconstruct the ideal value of the system in fixed time. • Additional assumption for similar work is avoided. [ABSTRACT FROM AUTHOR]

This paper investigates practical stabilization of multi-links highly nonlinear Takagi-Sugeno fuzzy complex networks with Lévy noise (MHFNLs) via aperiodically intermittent discrete-time observation control (AIDOC). It is worth pointing out that AIDOC is considered into MHFNLs for the first time. Due to the difficulty of satisfying linear conditions in real life, polynomial growth conditions are used instead of linear growth conditions. Meanwhile, a global Lyapunov function is constructed and sufficient conditions which can remain MHFNLs practically stable are gained. Finally, an application and its numerical simulations are presented to verify the effectiveness and availability of the theoretical results. • Polynomial growth conditions are used instead of linear growth conditions. • We consider Lévy noise and construct a global Lyapunov function. • Aperiodically intermittent discrete-time observation control is designed. • The theoretical result is applied to Van der Pol-Duffing oscillator. [ABSTRACT FROM AUTHOR]

Let G be a graph with vertex set V and edge set E. A topological index has the form T I (G) = ∑ u v ∈ E f (d u , d v) , where f = f (x , y) is a pertinently chosen function which must be symmetric and real-valued for all x , y pertaining to vertex degrees of the graph G. Particularly interesting are the Sombor index SO and the elliptic Sombor index ESO , induced by the functions f (x , y) = x 2 + y 2 and f (x , y) = (x + y) x 2 + y 2 , respectively. In this paper we analyze the ordering relations in benzenoid systems with respect to these two important topological indices. Also, we extend the results to general Sombor index SO α , β and general elliptic Sombor index ESO α. [ABSTRACT FROM AUTHOR]

A signed graph G ˙ is defined to be harmonic if its net-degree vector d ± (G ˙) is a zero vector or an eigenvector of adjacency matrix A (G ˙) corresponding to an eigenvalue λ. In this paper, we demonstrate the existence of λ -harmonic signed trees with arbitrary diameter and the nonexistence of finite 0-harmonic signed trees. And we determine all harmonic signed trees with diameter at most 5. Moreover, we give some general properties of finite harmonic signed trees. In particular, a 1-harmonic signed tree must be 1 net-regular. • The existence of harmonic signed tree with arbitrary diameter is proved. • The nonexistence of finite 0-harmonic signed trees is proved. • All harmonic signed trees with diameter less than 6 are determined. • Some general properties of finite harmonic signed trees are given. • 1-harmonic signed tree must be 1 net-regular. [ABSTRACT FROM AUTHOR]

Due to their dynamic properties such as irregular sampling rate and high-frequency sampling, Continuous Time Series (CTS) are found in many applications. Since CTS with irregular sampling rate are difficult to model with standard Recurrent Neural Networks (RNNs), RNNs have been generalised to have continuous-time hidden dynamics defined by a Neural Ordinary Differential Equation (Neural ODE), leading to the ODE-RNN model. Another approach that provides a better modelling is that of the Latent ODE model, which constructs a continuous-time model where a latent state is defined at all times. The Latent ODE model uses a standard RNN as the encoder and a Neural ODE as the decoder. However, since the RNN encoder leads to difficulties with missing data and ill-defined latent variables, a Latent ODE-RNN model has recently been proposed that uses a ODE-RNN model as the encoder instead. Both the Latent ODE and Latent ODE-RNN models are difficult to train due to the vanishing and exploding gradients problem. To overcome this problem, the main contribution of this paper is to propose and illustrate a new model based on a new Latent ODE using an ODE-LSTM (Long Short-Term Memory) network as an encoder - the Latent ODE-LSTM model. To limit the growth of the gradients, the Norm Gradient Clipping strategy was embedded on the Latent ODE-LSTM model. The performance evaluation of the new Latent ODE-LSTM (with and without Norm Gradient Clipping) for modelling CTS with regular and irregular sampling rates is then demonstrated. Numerical experiments show that the new Latent ODE-LSTM performs better than Latent ODE-RNNs and can avoid the vanishing and exploding gradients during training. Code implementations developed in this work are available at github.com/CeciliaCoelho/LatentODELSTM. [ABSTRACT FROM AUTHOR]

The issue of composite adaptive exponential tracking control for a class of large-scale nonlinear systems under model uncertainties, external disturbances, along with multiplicative and additive time-varying sensor faults is considered in this paper. The command filtered backstepping approach is employed to address the "explosion of terms" issue inherent in standard backstepping method. A novel compensating system is incorporated to mitigate the effects of filtering errors and enhance the convergence of tracking errors. The composite estimation laws are designed by integrating the compensated tracking errors, prediction errors stemming from output estimators, along with the proportional and integral estimation errors of faulty terms. This on-line estimation framework enables achieving fast, robust and accurate estimation, even when employing low learning and modification gains. By incorporating modification terms with appropriate time-varying gains, it is demonstrated that the resulting system is globally exponentially stable. Finally, the effectiveness of the presented FTC approach is illustrated through two simulation examples. • A composite adaptive FTC strategy is developed under time-varying sensor faults. • The proposed method incorporates composite estimation laws into the CFB framework. • A novel compensating system is utilized to mitigate the effects of filtering errors. • The on-line estimation architecture enables fast, robust and accurate estimation. • The closed-loop system is rigorously proven to be globally exponentially stable. [ABSTRACT FROM AUTHOR]

The Collatz conjecture is a famous conjecture aiming to estimate the behaviors of positive integers after a series of iterations of the Collatz function f (n) , where f (n) = { 3 n + 1 2 if n is odd , n 2 if n is even. Collatz conjecture has attracted a great deal of attention since it was put forward in 1930s due to the surprising nature. However, after strenuous efforts, many mathematicians think the current knowledge seems insufficient to solve Collatz conjecture, although it appears in a very simple form. As a replacement, a "periodic" (weaker) version of Collatz conjecture (called Collatz periodic conjecture) was proposed and discussed. In this paper, we would like to explore what properties Collatz (periodic) conjecture could imply. Maybe it is another possible direction to confirm or disprove Collatz (periodic) conjecture. • Provide some equivalent forms of Collatz (periodic) conjecture. • Consider the Collatz (periodic) conjecture in the aspect of digraphs and matrices. • Reveal some further properties what Collatz (periodic) conjecture could imply. [ABSTRACT FROM AUTHOR]

In this paper, the cost function of a single-server queueing system with Poisson input stream and Erlang- k service time will be analyzed. Treated as a function of the traffic intensity ρ , with respect to some known constants, we will show that its stationary points are solutions of a fourth-degree polynomial equation with real coefficients. Moreover, an explicit form of these solutions is given and it is shown the function reaches a minimum value at some of these points. For illustration, a numerical analysis of the cost function is carried out by changing the values of the costs, which are changed according to the principle of arithmetic progression. Also, a statistical analysis of the relationship between optimal solutions ρ ⁎ and Φ (ρ ⁎) is done. • A cost function is formed for single-server queueing system with Poisson input stream and Erlang-k service time. • Stationary points of the proposed cost function are solutions of fourth-degree polynomial equation with real coefficients. • The sensitivity of the parameters is tested according to the principle of arithmetic progression. • A statistical analysis of the optimal solutions is made and an economic interpretation of the results is given. [ABSTRACT FROM AUTHOR]

In the paper, an unconditional energy-stable scheme is developed with the help of the scalar auxiliary variable (SAV) approach and the leapfrog time-discretization. The fully discrete scheme gives a linear and decoupled system, whereas previous SAV-type methods usually gave a coupled linear system. Therefore, the proposed scheme is decoupled and thus easier to be implemented. Moreover, it is proved that the proposed scheme has second-order time accuracy in the temporal direction. Several numerical examples are provided to support the proposed theoretical results. • An unconditional energy-stable scheme is developed to numerically solve the Swift-Hohenberg equations. • The proposed scheme is linearly implicit and decoupled and thus easier to be implemented. • Convergence of the semi-discrete scheme is obtained. [ABSTRACT FROM AUTHOR]

In this paper, we propose new sufficient conditions for stability of solutions of systems of Volterra linear integral equations and systems of linear integro-differential Volterra equations. Solution stability conditions for systems of Volterra linear integral equations are studied with perturbed right hand sides of the equations. These new sufficient conditions are expressed in terms of logarithmic norms of matrices consisting of coefficients of equations and derivatives of their kernels. For Volterra integro-differential equations we also provide sufficient conditions for stability of solutions when the initial conditions are perturbed. • Extending the Lyapunov first method to systems of Volterra integral equations and Volterra integro-differential equations. • Using the logarithmic norm to obtain sufficient stability conditions for integral and integro-differential Volterra equations. • Common approach to constructing the sufficient stability conditions for integral and integro-differential Volterra equations. • Explicit sufficient conditions via kernels and coefficients of the equations. [ABSTRACT FROM AUTHOR]

In this article, the null controllability of a coupled ODE-heat system is considered by backstepping method. First, according to a backstepping transformation, the system is transferred into an intermediate system which has an equivalence of null controllability with the original system. Second, we divide the finite time interval into a sequence of time intervals, and on each time interval, transform the intermediate system into a target system through another backstepping transformation. Consequently, we obtain the crucial norm estimates of states and control laws. Finally, based on the invertibility of the target system, we prove null controllability of the intermediate system and ultimately, show the null controllability of the original coupled ODE-heat system. In addition, numerical experiment results are presented to examine the validity for the presented controller. • We construct a piecewise constant feedback controller for the null controllability of coupled ODE-heat system coupled at boundary and internal term. • We obtain the crucial norm estimates for the kernels, states and control. Especially, we obtain the kernels estimate by using suitable induction hypothesis expression in successive approximation. • We obtain the uniform estimate of the decay rate β for the target system, which is independent of the time intervals. • The procedure of this paper avoiding some technical skills for constructing continuous feedback control in references, which can be extended to similar problems for some other PDE-ODE system. [ABSTRACT FROM AUTHOR]

This paper designs the asynchronous finite-time extended dissipative filter on neutral semi-Markov jump systems subject to sensor saturation and dual-channel denial-of-service attacks. Based on event-triggering scheme and hidden Markov model, the asynchronous filter is designed. Sufficient conditions which filter error system is extended dissipative stochastic finite-time bound are derived. In the meantime, we present novel results on extended dissipative stochastic finite-time contractive bound. Finally, the partial element equivalent circuits (PEECs) are employed to evidence the validness of results. • Because system modes might be unavailable, hidden Markov model is used to explain non-synchronous phenomenon between system and filter, which subsystems could be detected with the conditional probability matrix. • As for dual-channel DoS attacks, two mutually independent Bernoulli variables are utilized to depict whether two channels are attacked or not. • Both finite-time bound and finite-time contractive bound are considered to reflect the transient performance of filter error system. By adjusting parameter matrices of extended dissipative, different performances could be obtained, such as passive, (Q , S , R)-dissipative, etc. [ABSTRACT FROM AUTHOR]

This paper focuses on investigating the stochastic Volterra integral equations (SVIEs) with double weakly singular kernels. Our primary objective is to examine the well-posedness of the proposed equation. Specifically, we explore the presence of existence, uniqueness, boundedness, and the continuous dependence of the exact solution on the initial data. Additionally, we develop a stochastic θ -scheme as a numerical solution for the equation and demonstrate that the convergence rate of the scheme is influenced by the kernel parameters. To validate the accuracy and reliability of our theoretical findings, we present two numerical examples. Notably, one of these examples concentrates on estimating the price of a European call option using the Heston stochastic volatility model with a singular kernel. Our results, when compared to the corresponding findings by Li et al. [3] , not only relax the integrable limitations of singular kernels but also establish a precise convergence order. In addition, we propose an improved scheme, based on the efficient sum-of-exponentials (SOE) approximation, to address the low computational efficiency of the stochastic θ -scheme. The results confirm that our approach aligns significantly with the expected physical interpretations. • We introduce the novel double singular stochastic Volterra integral equation (SVIE). • We investigate the well-posedness of SVIEs with double weakly singular kernels. • Additionally, the boundedness of the p th moment of the proposed equation is achieved. • The proposed scheme, which is built upon the SOE approximation, effectively reduces the computational cost and storage requirements to O (N log ⁡ N) and O (log ⁡ N) when T ≫ 1 , respectively. When T ≈ 1 , the computational cost and storage requirements are further reduced to O (N log 2 ⁡ N) and O (log 2 ⁡ N) , respectively. • In addition, through our numerical experiments, we have substantiated that the computational efficiency of the proposed scheme, utilizing the SOE approximation, outperforms conventional methods such as the Euler method and the Galerkin approximation. [ABSTRACT FROM AUTHOR]

This paper is concerned with the protocol-based fault detection (FD) issue for state-saturated delayed nonlinear systems with both redundant channels and sensor nonlinearities. Specifically, in pursuit of relieving the communication pressure, the stochastic communication protocol is adopted, which can regulate the transmission of output signal during the communication process. Moreover, for the purpose of decreasing the negative impact of one channel failure on network communication and enhancing data reliability, the redundant channels transmission is introduced. Then, the protocol-based FD filter is constructed such that the resultant FD dynamics system attains the global asymptotic stability in mean-square sense and satisfies the H ∞ performance index. Moreover, an implement-to-easy algorithm is developed to facilitate the solvability of the desirable FD filtering method. In the end, two illustrative examples are utilized to demonstrate the validity of the developed FD filtering algorithm. • The stochastic communication protocol is adopted to regulate the network communication. • The redundant channels transmission mechanism is introduced into the signal communication process. • New sufficient criteria are derived to assure the existence of desirable FD filtering scheme for state-saturated systems. [ABSTRACT FROM AUTHOR]

Hexagonal chains are graph representations of benzenoid polymers which are common but important molecular structures in chemistry. A spanning tree of a graph is a subgraph which is a tree and contains all the vertices of the graph. A perfect matching of a graph is a set of disjoint edges covering all vertices of the graph. In this paper, we give exact expression for the number of spanning trees containing perfect matchings in hexagonal chains with a unique kink. • The number of spanning trees containing perfect matchings in hexagonal chains with a unique kink is derived. • The number of spanning trees in the graph obtained from hexagonal chain by contracting a perfecting matching is obtained. • Graphs obtained from the hexagonal chain with a unique kink by contracting a perfecting matching are characterized. [ABSTRACT FROM AUTHOR]

This paper presents an efficient and concise double fast algorithm to solve high dimensional time-space fractional diffusion problems with spectral fractional Laplacian. We first establish semi-discrete scheme of time-space fractional diffusion equation, which uses linear finite element or fourth-order compact difference method combining with matrix transfer technique to approximate spectral fractional Laplacian. Then we introduce a fast time-stepping L1 scheme for time discretization. The proposed scheme can exactly evaluate fractional power of matrix and perform matrix-vector multiplication at per time level by using discrete sine transform, which doesn't need to resort to any iteration method and can significantly reduce computation cost and memory requirement. Further, we address stability and convergence analyses of full discrete scheme based on fast time-stepping L1 scheme on graded time mesh. Our error analysis shows that the choice of graded mesh factor ω = (2 − α) / α shall give an optimal temporal convergence O (N − (2 − α)) with N denoting the number of time mesh. Finally, ample numerical examples are delivered to illustrate our theoretical analysis and the efficiency of the suggested scheme. • Efficient spatial semi-discretization using matrix transfer technique for time-space fractional diffusion problem is established. • Stability and error bounds of full discrete scheme based on fast time-stepping L1 method on graded time meshes are discussed. • Fast and accurate solver for system of equations is designed by discrete sine transform at per time level. • Commendably concise and implementation-friendly double fast algorithm is proposed. [ABSTRACT FROM AUTHOR]

In this paper, the robust security protection control for switched systems is studied. A robust anti-disturbance mechanism using the Radial Basis Function Neural Network (RBFNN) is proposed for switched systems, which can approximate and compensate for the impact of unknown disturbance on the system state. Then, a network security protection mechanism based on encoder and decoder is presented, which has the ability to resist the dual impact on the feedback information caused by the network privacy snooping and data injection attacks. Accordingly, stability analysis and state-feedback controller design are given for the switched systems under unknown disturbance, network privacy snooping and injection attacks. Finally, simulation results illustrate the effectiveness of the proposed method. • A data-based robust anti-disturbance mechanism is adopted to approximate any unknown disturbance. • A novel network security protection mechanism is proposed to resist data injection attacks and prevent privacy snooping. • For the first time in the control field, the privacy protection level of encoder and decoder is analyzed. [ABSTRACT FROM AUTHOR]

In this paper, a type of delayed Clifford-valued memristor-based neural networks (CLVMNNs) with impulsive disturbances is established, and the global exponential stability is investigated by using generalized norm. Firstly, the n-dimensional Clifford-valued systems are decomposed into 2 m n -dimensional real-valued systems to address the non-commutativity problem of the multiplication of Clifford numbers. Secondly, the generalized ∞-norm and 1-norm are introduced to induce the global exponential stability for CLVMNNs, and two special Lyapunov functionals are established to prove the stability. Thirdly, the strict assumption of the boundedness of activation function in previous research is loosened, and some less conservative conditions of stability are obtained based on the constructed Lyapunov functionals. Finally, the theoretical results are verified by two numerical simulations, and an image encryption scheme is proposed to show the application in real world situation based on the delayed CLVMNNs. • Impulsive disturbances and memristor are taken into account for Clifford-valued NNs. • The generalized { ξ , 1 } -norm and { ξ , ∞ } -norm are introduced to investigate the global exponential stability of CLVMNNs. • The global exponential stability of CLVMNNs is discussed and some new criteria are derived. [ABSTRACT FROM AUTHOR]