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In the paper by Jalal et al. [1], the authors present phase portraits of a differential system exhibiting chaotic behavior with line equilibria. This commentary identifies inaccuracies in the provided figures and offers corrected versions. Specifically, discrepancies were found in Fig. 1 (phase portraits with parameters a = 15 and b = 1) and Fig. 2 (phase portraits with parameters a = 0 and b = 5). Corrected figures, along with MATLAB codes for verification, are provided. These corrections are discussed in the context of chaotic systems with line equilibria, referencing Jafari and Sprott's work [2], which explores similar systems. [ABSTRACT FROM AUTHOR]

Purpose: The purpose of this paper is to test the invulnerability of the guarantee network at the equilibrium point. Design/methodology/approach: This paper introduces a tractable guarantee network model that captures the invulnerability of the network in terms of cascade-based attack. Furthermore, the equilibrium points are introduced for banks to determine loan origination. Findings: The proposed approach not only develops equilibrium analysis as an extended perspective in the guarantee network, but also applies cascading failure method to construct the guarantee network. The equilibrium points are examined by simulating experiment. The invulnerability of the guarantee network is quantified by the survival of firms in the simulating progress. Research limitations/implications: There is less study in equilibrium analysis of the guarantee network. Additionally, cascading failure model is expressed in the presented approach. Moreover, agent-based model can be extended in generating the guarantee network in the future study. Originality/value: The approach of this paper presents a framework to analyze the equilibrium of the guarantee network. For this, the systemic risk of the whole guarantee network and each node's contribution are measured to predict the probability of default on cascading failure. Focusing on cascade failure process based on equilibrium point, the invulnerability of the guarantee network can be quantified. [ABSTRACT FROM AUTHOR]

In this paper, fuzzy SP-irresolute, fuzzy SP-irresolute open and fuzzy SP-irresolute closed functions between fuzzy topological spaces in Sˇostak sense are defined. Their properties and the relationships between these functions and other functions introduced previously are investigated. Next fuzzy SP-connectedness is introduced and studied with the help of r-fuzzy strongly preopen sets. [Copyright &y& Elsevier]

In this article, the author focuses on the importance of carrying momentum to succeed in daily life. She explains that while traveling with garbage-rescued cruiser bike with ape-hangers, a sissy bar and a 52-tooth chainring, a trademark balance was achieved. She also states that momentum is a most important part of life which is needed to move forward.

This paper analyzes a discrete-time Cournot competition model with time delays taking into consideration the dynamics of market interactions involving one public firm and multiple private firms. The model assumes that the production output of the public firm is influenced by the past output levels of the private firms. The output levels of the private firms are affected by the past production outputs of both the public firm and their private competitors. The study identifies two equilibrium states within the nonlinear system: a positive equilibrium and a boundary equilibrium. Through a stability analysis, it is established that the boundary equilibrium is a saddle point. In the absence of time delays, the stability region for the positive equilibrium is determined. The research explores various specific delay scenarios, finding conditions under which the positive equilibrium is asymptotic stability. Additionally, the occurrence of flip and Neimark–Sacker bifurcations are examined. In order to illustrate the complex dynamics, the paper provides a series of numerical examples. • We explore Cournot competition with delays, involving both public and private firms. • Two equilibria are found: positive and boundary, with boundary as a saddle point. • Stability of positive equilibrium is analyzed, with and without time delays. • Flip/Neimark–Sacker bifurcations are investigated supported by numerical examples. [ABSTRACT FROM AUTHOR]

The problem of existence and stability of equilibria of linear systems with constant power loads is addressed in this paper. First, we correct an unfortunate mistake in our recent paper refid="ref10"/ pertaining to the sufficiency of the condition for existence of equilibria in multiport systems given there. Second, we give two necessary conditions for existence of equilibria. The first one is a simple linear matrix inequality hence it can be easily verified with existing software. Third, we prove that the latter condition is also sufficient if a set defined by the problem data is convex, which is the case for single and two-port systems. Finally, sufficient conditions for stability and instability for a given equilibrium point are given. The results are illustrated with two benchmark examples. [ABSTRACT FROM PUBLISHER]

What happens when the only goal in a chess game is to capture at least one piece of the opposite side? Can both sides live peacefully in an equilibrium where neither can capture the other's pieces? In this short paper, we develop a new set of puzzles which we call chess equilibrium puzzles on this premise. We explain the rules of the game, analyze puzzles that have obvious and generalizable solutions, and provide several interesting puzzles for the reader to solve (solutions are provided at the end). Our puzzles provide an exciting twist to the realm of traditional chess puzzles. [ABSTRACT FROM AUTHOR]

This paper presents an HIV/AIDS model with ideological transfer in male susceptible individuals, and considers condoms and education as control measures. First, it gives the threshold R 0 of the model. Then it proves that the disease-free equilibrium is globally asymptotically stable when R 0 < 1. Under different threshold conditions, the local stability of the boundary equilibria is given. The existence of the endemic equilibrium is also given. We also perform numerical simulations for different parameter values. Furthermore, when considering control measures, the simulation results show that effective and long-term use of condoms significantly reduce the number of infected individuals and education should be vigorously promoted when the epidemic is controllable. This result provides theoretical guidance for effective control the spread of HIV. [ABSTRACT FROM AUTHOR]

This paper studies the global dynamics of an SEIR (Susceptible–Exposed–Infectious–Recovered) model with nonlocal diffusion. We show the model's well-posedness, proving the solutions' existence, uniqueness, and positivity, along with a disease-free equilibrium. Next, we prove that the model admits the global threshold dynamics in terms of the basic reproduction number R 0 , defined as the spectral radius of the next-generation operator. We show that the solution map has a global compact attractor, offering insights into long-term dynamics. In particular, the analysis shows that for R 0 < 1 , the disease-free equilibrium is globally stable. Using the persistence theory, we show that there is an endemic equilibrium point for R 0 > 1. Moreover, by constructing an appropriate Lyapunov function, we establish the global stability of the unique endemic equilibrium in two distinct scenarios. [ABSTRACT FROM AUTHOR]

In this paper we adopt a rational expectations framework to formulate general equilibrium models with heterogeneous agents. The productivity dynamics are characterized by a jump–diffusion model, thus allowing to account for sudden and impactful events. The modelling approach utilizes Hamilton–Jacobi–Bellman (HJB) formulations to represent the endogenous decision-making of firms to remain or exit the industry. When firms decide to exit, they are instantaneously replaced by new entrants. This dynamic leads to the development of a probability density function for firms, which satisfies a Kolmogorov–Fokker–Planck (KFP) equation with a source term. Both HJB and KFP formulations involve partial-integro differential operators due to the presence of jumps. Equilibrium models are completed with the household problem and feasibility conditions. Since (semi-)analytical solutions are not available, a numerical methodology is considered. This approach involves a Crank–Nicolson scheme for the time discretization, an augmented Lagrangian active set method and a finite difference discretization for the HJB formulation, and an appropriate finite difference method for the KFP problem. Moreover, Adams–Bashforth schemes are employed to handle integral terms explicitly. For the global equilibrium problem, we introduce a Steffensen algorithm. Numerical examples are provided to showcase the performance of our proposed numerical methodologies and to illustrate the expected behaviour of computed economic variables. [ABSTRACT FROM AUTHOR]

This paper develops a neural network for solving the general nonsmooth convex optimization problems. The proposed neural network is modeled by a differential inclusion. Compared with the existing neural networks for solving nonsmooth convex optimization problems, this neural network has a wider domain for implementation. Under a suitable assumption on the constraint set, it is proved that for a given nonsmooth convex optimization problem and sufficiently large penalty parameters, any trajectory of the neural network can reach the feasible region in finite time and stays there thereafter. Moreover, we can prove that the trajectory of the neural network constructed by a differential inclusion and with arbitrarily given initial value, converges to the set consisting of the equilibrium points of the neural network, whose elements are all the optimal solutions of the primal constrained optimization problem. In particular, we give the condition that the equilibrium point set of the neural network coincides with the optimal solution set of the primal constrained optimization problem and the condition ensuring convergence to the optimal solution set in finite time. Furthermore, illustrative examples show the correctness of the results in this paper, and the good performance of the proposed neural network. [ABSTRACT FROM AUTHOR]

Multistable networks have attracted much interest in recent years, since multistability is of primary importance for some applications of recurrent neural networks where monostability exhibits some restrictions. This paper focuses on the analysis of dynamical property for a class of additive recurrent neural networks with nonsaturating linear threshold transfer functions. A milder condition is derived to guarantee the boundedness and global attractivity of the networks as compared to that presented in [6]. Dynamical properties of the equilibria of two-dimensional networks are analyzed theoretically, and the relationships between the equilibria features and network parameters (synaptic weights and external inputs) are revealed. In addition, the sufficient and necessary conditions for coexistence of multiple equilibria are obtained, which confirmed the observations in [14] with a cortex-inspired silicon circuit. The results obtained in this paper are applicable to both symmetric and nonsymmetric networks. Simulation examples are used to illustrate the theory developed in this paper. [ABSTRACT FROM AUTHOR]

Purpose: The purpose of this paper is to improve transient response and dynamic performance of automatic voltage regulator (AVR). Design/methodology/approach: This paper proposes a novel fractional order proportional–integral–derivative plus derivative (PIλDµDµ2) controller called FOPIDD for AVR system. The FOPIDD controller has seven optimization parameters and the equilibrium optimizer algorithm is used for tuning of controller parameters. The utilized objective function is widely preferred in AVR systems and consists of transient response characteristics. Findings: In this study, results of AVR system controlled by FOPIDD is compared with results of proportional–integral–derivative (PID), proportional–integral–derivative acceleration, PID plus second order derivative and fractional order PID controllers. FOPIDD outperforms compared controllers in terms of transient response criteria such as settling time, rise time and overshoot. Then, the frequency domain analysis is performed for the AVR system with FOPIDD controller, and the results are found satisfactory. In addition, robustness test is realized for evaluating performance of FOPIDD controller in perturbed system parameters. In robustness test, FOPIDD controller shows superior control performance. Originality/value: The FOPIDD controller is introduced for the first time to improve the control performance of the AVR system. The proposed FOPIDD controller has shown superior performance on AVR systems because of having seven optimization parameters and being fractional order based. [ABSTRACT FROM AUTHOR]

In this paper, we study the split common fixed point problem for two nonlinear mappings in p -uniformly convex and uniformly smooth Banach spaces. We propose an algorithm which uses dynamic stepsize, it allows to be easily implemented without prior information about operator norm. We further apply our result to solve the split variational inclusion problem, equilibrium problem and convexly constrained linear inverse problem. Moreover, we provide numerical examples to verify efficiency of our algorithm. • Presents an iteration process for splitting problems, with dynamic choosing step size. • Implement the result for solving wide classes of mathematical engineering problems. • Includes nontrivial example with computer simulation to compare our findings with existing ones. [ABSTRACT FROM AUTHOR]

The slowly forced Duffing system has been found to exhibit unique fast–slow behaviors in descending frequency switching scheme, the typical of which is the sliding bursting, which is instructive for understanding the dynamics of ubiquitous frequency switching systems in scientific research and practical engineering. This paper is devoted to further refining the fast–slow dynamics of the slowly forced Duffing system with another commonly encountered frequency switching scheme, i.e., the ascending frequency switching, characterized by switching the frequency synchronously according to the increase or decrease of state variable values. Taking the forcing amplitude as an example, this paper provides a theoretical way to fully summarize the fast–slow behaviors in frequency switching systems with the variation of parameter values based on the proposed superposition analysis of one- and two-parameter bifurcations of subsystems. As a result, two typical bifurcation structures and eight threshold windows with distinct switching vector fields therein are identified, inducing up to ten different bursting patterns. Among them, several novel fast–slow dynamics, such as multiple jumps hysteresis loop formed by boundary equilibrium bifurcations and the switching failure phenomena, are presented and investigated. In particular, the underlying mechanism of threshold modulation, i.e., the evolution of unconventional bifurcations, is also found. These findings contribute to complementing and contrasting the existing studies on the fast–slow dynamics of frequency switching systems. • Theoretical way for identifying threshold windows under different parameter values. • Novel bursting patterns induced by several novel multi-jump hysteresis loops. • Underlying mechanisms of the threshold modulating the dynamical behaviors. • Effect of frequency switching scheme and system parameters on the fast–slow dynamics. [ABSTRACT FROM AUTHOR]

Dichotomy, or monostability, is one of the most important properties of nonlinear dynamic systems. For a dichotomous system, the solution of the system is either unbounded or convergent to a certain equilibrium, thus periodic or chaotic states cannot exist in the system. In this paper, a new methodology for the analysis of dichotomy of a class of nonlinear systems is proposed, and a linear matrix inequality (LMI)-based criterion is derived. The results are then extended to uncertain systems with real convex poly- topic uncertainties in the linear part, and the LMI representation for robust dichotomy allows the use of parameter-dependent Lyapunov function. Based on the results, a dynamic output feedback controller guaranteeing robust dichotomy is designed, and the controller parameters are explicitly expressed by a set of feasible solutions of corresponding linear matrix inequalities. An extended Chua's circuit with two nonlinear resistors is given at the end of the paper to demonstrate the validity and applicability of the proposed approach. It is shown that by investigating the convergence of the bounded oscillating solutions of the system, our results suggests a viable and effective way for chaos control in nonlinear circuits. [ABSTRACT FROM AUTHOR]

Suggests activities and crafts designed to develop balance in children. Instructions for making mobiles; Exercise with large beach or vestibular balls; Art works with paper cone pendulums.

We consider an environmentwhere players need to decide whether to buy a certain product (or adopt a technology) or not. The product is either good or bad, but its true value is unknown to the players. Instead, each player has her own private information on its quality. Each player can observe the previous actions of other players and estimate the quality of the product. A classic result in the literature shows that in similar settings, informational cascades occur, where learning stops for the whole network and players repeat the actions of their predecessors. In contrast to this literature, in this paper, players get more than one opportunity to act. In each turn, a player is chosen uniformly at random from all the players and can decide to buy the product and leave the market or wait. Her utility is the total expected discounted reward, and thus, myopic strategies may not constitute equilibria. We provide a characterization of perfect Bayesian equilibria (PBEs) with forward-looking strategies through a fixed-point equation of dimensionality that grows only quadratically with the number of players. Using this tractable fixed-point equation, we show the existence of a PBE and characterize PBEs with threshold strategies. Based on this characterization, we study informational cascades in two regimes. First, we show that for a discount factor $\delta$ strictly smaller than 1, informational cascades happen with high probability as the number of players $N$ increases. Furthermore, only a small portion of the total information in the system is revealed before a cascade occurs. Second, and more surprisingly, we show that for a fixed $N$ , and for a sufficiently large $\delta < 1$ , when the product is bad, there exists an equilibrium where an informational cascade can happen only after at least half of the players revealed their private information, and consequently, the probability for a “bad cascade” where all the players buy the product vanishes exponentially with $N$. Finally, when $\delta =1$ and the product is bad, there exists an equilibrium where informational cascades do not happen at all. [ABSTRACT FROM AUTHOR]

In this paper we combine partial stability and set invariance methods, which is a necessary development for applications such as synchronization. We use set invariance methods to ensure that the ‘auxiliary’ variables remain on a restricted domain, and then use this framework to develop new results for both local and global partial stability theory. We apply the methodology to identical synchronization of oscillator networks, which gives rigorous conditions for existing local methodology as well as novel global methodology. The work also allows a common framework for synchronization analysis of both oscillator networks and power systems. We show the applicability by finding novel synchronization conditions for an example oscillator network. [ABSTRACT FROM PUBLISHER]

This paper addresses robustness to external disturbances of switched discrete and continuous systems with multiple equilibria. First, we prove that if each subsystem of the switched system is input-to-state stable (ISS), then under switching signals that satisfy an average dwell-time bound, the solutions are ultimately bounded within a compact set. The size of this set varies monotonically with the supremum norm of the disturbance signal. These results generalize existing ones in the common equilibrium case to accommodate multiple equilibria. Then, we relax the (global) ISS conditions to consider equilibria that are locally exponentially stable (LES), and we establish practical stability for such switched systems under disturbances. Our motivation for studying this class of switched systems arises from certain motion planning problems in robotics, where primitive movements, each corresponding to an equilibrium point of a dynamical system, must be composed to obtain more complex motions. As a concrete example, we consider the problem of realizing safe adaptive locomotion of a three-dimensional biped under persistent external force by switching among motion primitives characterized by LES limit cycles. The results of this paper, however, are relevant to a much broader class of applications, in which composition of different modes of behavior is required. [ABSTRACT FROM AUTHOR]

Multiagent systems (MASs) are usually applied with agents classified into leaders and followers, where selecting appropriate leaders is an important issue for formation control applications. In this paper, we investigate two leader selection problems in second-order MAS, namely, the problem of choosing up to a given number of leaders to minimize the formation error and the problem of choosing the minimum number of leaders to achieve a tolerated level of error. We propose a game theoretical method to address them. Specifically, we design a supermodular game for the leader selection problems and theoretically prove its supermodularity. In order to reach Nash equilibrium of the game, we propose strategies for the agents to learn to select leaders based on stochastic fictitious play. Extensive simulation results demonstrate that our method outperforms existing ones. [ABSTRACT FROM AUTHOR]

In this paper, an age-structured infectious disease dynamical model that considers two diseases simultaneously but with limited medical resources is proposed and analyzed. The asymptotic smoothness and persistence of the solution semi-flow are investigated. Then conditions for the existence of a global attractor are derived, which means that disease persists when ℜ 0 > 1. By using a Lyapunov function, it is shown that the infection-free equilibrium is globally asymptotically stable if ℜ 0 < 1 and the infection equilibrium is globally asymptotically stable if ℜ 0 > 1. In the presence of limited medical resources, the results suggest that equitable distribution for the limited medical resources is significant when treating low-risk and high-risk diseases and that keeping a resource sharing coefficient at a moderate level helps to eliminate the disease. • We proposed an age-structured infectious disease model with limited medical resources. • some important parameters were included in ℜ 0. • ℜ 0 determines the global stability of the equilibria. • Discussions of the effects of the parameters and biological significance were provided. [ABSTRACT FROM AUTHOR]

In this paper, we study the static equilibrium of a double catenary formed when a closed, ideal chain of length L * is draped over two frictionless pins at the same vertical height and separated by horizontal distance D * . Each of the two segments forms a catenary at equilibrium. At issue is whether or not a given equilibrium solution is stable. First, we show that the trivial solution of two identical catenaries is always an equilibrium configuration. However, it is not always stable. There exists a critical length L c * such that the trivial solution becomes unstable for L * > L c * . in such cases, the system is stable only for catenaries of differing lengths. [ABSTRACT FROM AUTHOR]

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

In this paper, we consider the following problem: what passivity properties can be inferred for a system by studying only an approximate mathematical model for it. Our results show that an excess of passivity (whether in the form of input strictly passive, output strictly passive or very strictly passive) in the approximate model guarantees a certain passivity index for the system, provided that the norm of the error between the approximate and the true models is sufficiently small in a suitably defined sense. Further, we consider $(Q,S,R)$-dissipative systems and show that $(Q,S,R)$- dissipativity has a similar robustness property, even though the supply rates for the system and its approximation may be different. These results may be particularly useful if either the approximate model is much easier to analyze, or if the precise system model is unknown. We illustrate the results by considering particular approximation methods, e.g., model reduction, discretization, quantization, and linearization around an equilibrium point. [ABSTRACT FROM PUBLISHER]

This paper is concerned with the winner-take-all (WTA) problem on networks. We propose the first distributed protocol to address this problem dynamically. This protocol features strong nonlinearity. Theoretical analysis reveals that it contains invariant quantities, symmetric solutions, and multiple equilibrium points. By leveraging these properties, this work proves the instability of its non-WTA solutions, and global convergence to the WTA solution via Lyapunov theory. Two simulations over networks with 10 and 200 nodes, respectively, are conducted. Simulation results have well verified the theoretical conclusions drawn in this paper. [ABSTRACT FROM PUBLISHER]

The DeGroot–Friedkin (DF) model is a recently proposed dynamical description of the evolution of individuals’ self-appraisal and social power in a social influence network. Most studies of this system and its variations have so far focused on models with a time-invariant influence network. This paper proposes novel models and analysis results for DF models over switching influence networks, and with or without environment noise. First, for a DF model over switching influence networks, we show that the trajectory of the social power converges to a ball centered at the equilibrium reached by the original DF model. For the DF model with memory on random interactions, we show that the social power converges to the equilibrium of the original DF model almost surely. Additionally, this paper studies a DF model that contains random interactions and environment noise, and has memory on the self-appraisal. We show that such a system converges to an equilibrium or a set almost surely. Finally, as a by-product, we provide novel results on the convergence rates of the original DF model and convergence results for a continuous-time DF model. [ABSTRACT FROM AUTHOR]

Purpose – The purpose of this paper is to examine the criteria of uniqueness of the equilibrium point and the new stability criteria for stability of the equilibrium point. The new stability condition is dependent on the size of delays. Design/methodology/approach – The global asymptotic stability of a class of delayed bi-directional associative memory (BAM) neural networks is studied. Some new sufficient conditions are presented for the unique equilibrium point and the global stability of BAM neural networks with time delays by constructing Lyapunov functions and using the linear matrix inequality. A numerical example is presented to illustrate the effectiveness of the theoretical results. Findings – Based on the mathematical method and matrixes inequality skill, some criteria are obtained which contain the unique equilibrium point and the global stability of BAM neural networks. Research limitations/implications – The paper proposes the new Lyapunov function and new skill to compose matrixes inequality. Practical implications – A very useful method for BAM neural network to judge the uniqueness of the equilibrium point and stability. Originality/value – The new mathematical model is proposed about the production process, and the new control method is used in the temperature system for a double layers welded pipe in welding process. [ABSTRACT FROM AUTHOR]

This paper proposes a design methodology to stabilize isolated relative equilibria in a model of all-to-all coupled identical particles moving in the plane at unit speed. Isolated relative equilibria correspond to either parallel motion of all particles with fixed relative spacing or to circular motion of all particles with fixed relative phases. The stabilizing feedbacks derive from Lyapunov functions that prove exponential stability and suggest almost global convergence properties. The results of the paper provide a low-order parametric family of stabilizable collectives that offer a set of primitives for the design of higher-level tasks at the group level. [ABSTRACT FROM AUTHOR]

In this paper, we explore a variety of new multistability scenarios in the general delayed neural network system. Geometric structure embedded in equations is exploited and incorporated into the analysis to elucidate the underlying dynamics. Criteria derived from different geometric configurations lead to disparate numbers of equilibria. A new approach named sequential contracting is applied to conclude the global convergence to multiple equilibrium points of the system. The formulation accommodates both smooth sigmoidal and piecewise-linear activation functions. Several numerical examples illustrate the present analytic theory. [ABSTRACT FROM PUBLISHER]

In this paper, we discuss the coexistence and dynamical behaviors of multiple equilibrium points for recurrent neural networks with a class of discontinuous nonmonotonic piecewise linear activation functions. It is proved that under some conditions, such n -neuron neural networks can have at least 5^{n} equilibrium points, 3^{n}$ of which are locally stable and the others are unstable, based on the contraction mapping theorem and the theory of strict diagonal dominance matrix. The investigation shows that the neural networks with the discontinuous activation functions introduced in this paper can have both more total equilibrium points and more locally stable equilibrium points than the ones with continuous Mexican-hat-type activation function or discontinuous two-level activation functions. An illustrative example with computer simulations is presented to verify the theoretical analysis. [ABSTRACT FROM PUBLISHER]

This paper introduces a deterministic fractional-order epidemic model (FOEM) for studying the transmission dynamics of the human immunodeficiency virus (HIV) and acquired immunodeficiency syndrome (AIDS). The model highlights the substantial role of unaware and undetected HIV-infected individuals in spreading the disease. Control strategies, such as wielding condoms, level of preventive measures to avoid infection, and self-strictness of susceptibles in sexual contact, have been incorporated into the study. The basic reproduction number ℛ 0 α has been derived, which suggests the conditions for ensuring the persistence and elimination of the disease. Further, to validate the model, actual HIV data taken from Mexico and India separately have been used. The disease dynamics and its control in both countries are analyzed broadly. The values of biological parameters are estimated at which numerical solutions better match the actual data of HIV patients in the case of fractional-order (FO) instead of integer-order (IO). Moreover, in the light of ℛ 0 α , our findings forecast that the disease will abide in the population in Mexico, and at the same time, it will die out from India after a long time. [ABSTRACT FROM AUTHOR]

This paper develops the fundamentals of optimal-tuning periodic-gain fractional delayed state feedback control for a class of linear fractional-order periodic time-delayed systems. Although there exist techniques for the state feedback control of linear periodic time-delayed systems by discretization of the monodromy operator, there is no systematic method to design state feedback control for linear fractional periodic time-delayed (FPTD) systems. This paper is devoted to defining and approximating the monodromy operator for a steady-state solution of FPTD systems. It is shown that the monodromy operator cannot be achieved in a closed form for FPTD systems, and hence, the short-memory principle along with the fractional Chebyshev collocation method is used to approximate the monodromy operator. The proposed method guarantees a near-optimal solution for FPTD systems with fractional orders close to unity. The proposed technique is illustrated in examples, specifically in finding optimal linear periodic-gain fractional delayed state feedback control laws for the fractional damped Mathieu equation and a double inverted pendulum subjected to a periodic retarded follower force with fractional dampers, in which it is demonstrated that the use of time-periodic control gains in the fractional feedback control generally leads to a faster response. [ABSTRACT FROM PUBLISHER]

In this paper, an SIR epidemic model considering hospital resources and vaccination is established and the rich dynamics and complex bifurcations are investigated. Firstly, the existence of disease-free equilibrium and endemic equilibria is explored. It is founded that when the vaccination rate is not high, the number of endemic equilibrium points changed easily with the number of hospital resources and vaccination, resulting in transcritical bifurcation and saddle–node bifurcations. Secondly, different types singularities such as degenerate saddle–node of codimension 1 at the disease-free equilibrium, and cusp or focus type Bogdanov–Takens singularities of codimension 3 at endemic equilibria are presented. Thirdly, bifurcation analysis at these equilibria is investigated, and it is found that the system undergoes a sequence of bifurcations, including Hopf, degenerate Hopf bifurcation, homoclinic bifurcation, the cusp type Bogdanov–Takens bifurcation of codimension 2, and the focus type Bogdanov–Takens bifurcation of codimension 3 which are the organizing centers for a series of bifurcations with lower codimension. And the system shows very rich dynamics such as the existence of multiple coexistent periodic orbits, homoclinic loops. Finally, numerical simulations are presented to verify the theoretical results. [ABSTRACT FROM AUTHOR]

The article presents abstracts related to electric engineering which include the identifiability of bilinear systems, impact of the theory of learning in games on equilibria emerging in non-cooperative games and data rate theorem for the stabilization of a linear, discrete-time, dynamical system with arbitrarily large disturbances.

Game theory is emerging as a popular tool for distributed control of multiagent systems. To take advantage of these game theoretic tools, the interactions of the autonomous agents must be designed within a game-theoretic environment. A central component of this game-theoretic design is the assignment of a local utility function to each agent. One promising approach to utility design is assigning each agent a utility function according to the agent's Shapley value. This method frequently results in games that possess many desirable features, such as the existence of pure Nash equilibria with near-optimal efficiency. In this paper, we explore the relationship between the Shapley value utility design and the resulting efficiency of both pure Nash equilibria and coarse correlated equilibria. To study this relationship, we introduce a simple class of resource allocation problems. Within this class, we derive an explicit relationship between the structure of the resource allocation problem and the efficiency of the resulting equilibria. Lastly, we derive a bicriteria bound for this class of resource allocation problems—a bound on the value of the optimal allocation relative to the value of an equilibrium allocation with additional agents. [ABSTRACT FROM AUTHOR]

Automatic clustering and dimension reduction are two of the most intriguing topics in the field of clustering. Affinity propagation (AP) is a representative graph-based clustering algorithm in unsupervised learning. However, extracting features from high-dimensional data and providing satisfactory clustering results is a serious challenge for the AP algorithm. Besides, the clustering performance of the AP algorithm is sensitive to preference. In this paper, an improved affinity propagation based on optimization of preference (APBOP) is proposed for automatic clustering on high-dimensional data. This method is optimized to solve the difficult problem of determining the preference of affinity propagation and the poor clustering effect for non-convex data distribution. First, t-distributed stochastic neighbor embedding is introduced to reduce the dimensionality of the original data to solve the redundancy problem caused by excessively high dimensionality. Second, an improved hybrid equilibrium optimizer based on the crisscross strategy (HEOC) is proposed to optimize preference selection. HEOC introduces the crisscross strategy to enhance local search and convergence efficiency. The benchmark function experiments indicate that the HEOC algorithm has better accuracy and convergence rate than other swarm intelligence algorithms. Simulation experiments on high-dimensional and real-world datasets show that APBOP has better effectiveness. [ABSTRACT FROM AUTHOR]

Purpose – This paper aims to introduce a new class of entire functions whose zeros (zk)k蠅1 satisfy ∑k=1∞Im zk=O(1). Design/methodology/approach – This is done by means of a Ritt's formula which is used to prove that every partial sum of the Riemann Zeta function, ζn(z):=∑k=1n1/kz, n蠅2, has zeros (snk)k蠅1 verifying ∑k=1∞Re snk=O(1) and extending this property to a large class of entire functions denoted by AO. Findings – It is found that this new class AO has a part in common with the class A introduced by Levin but is distinct from it. It is shown that, in particular, AO contains every partial sum of the Riemann Zeta function ζn(iz) and every finite truncation of the alternating Dirichlet series expansion of the Riemann zeta function, Tn(iz):=∑k=1n(-1)k-1/kiz, for all n蠅2. Practical implications – With the exception of the n=2 case, numerical experiences show that all zeros of ζn(z) and Tn(z) are not symmetrically distributed around the imaginary axis. However, the fact consisting of every function ζn(iz) and Tn(iz) to be in the class AO implies the existence of a very precise physical equilibrium between the zeros situated on the left half-plane and the zeros situated on the right half-plane of each function. This is a relevant fact and it points out that there is certain internal rule that distributes the zeros of ζn(z) and Tn(z) in such a way that few zeros on the left of the imaginary axis and far away from it, must be compensated with a lot of zeros on the right of the imaginary axis and close to it, and vice versa. Originality/value – The paper presents an original class of entire functions that provides a new point of view to study the approximants and the alternating Dirichlet truncations of the Riemann zeta function. [ABSTRACT FROM AUTHOR]

In this paper, we first introduce the model of single-driving double-response system (SDDRS), which consists of a driving system (subsystem) and two response systems (sub- systems). By applying the theory of Lyapunov stability, we study the projective synchronization of SDDRS between the driving and response systems. The sufficient conditions for achieving projec- tive synchronization are obtained when the driving system has either a globally stable equilibrium point or a chaotic attractor. Furthermore, we use the SDDRS for cryptography in secure communication and present a novel scheme for encryption and decryption based on its projection synchronization. The results of numerical simulations verify the efficiency of the presented control schemes and the excellence of cryptography. [ABSTRACT FROM AUTHOR]

This paper deals with the analysis of the nonisothermal axial dispersion tubular reactor. The existence of equilibrium profiles is investigated. In particular, for equal Peclet numbers, it is shown that one or three equilibria can be exhibited, depending on the parameters of the system, especially on the diffusion coefficient. In addition, different and close Peclet numbers are also considered. In these cases, it is also shown that the reactor has one or three equilibrium profiles. Some numerical simulations support the theoretical results. [ABSTRACT FROM AUTHOR]

This paper proposes a design methodology to stabilize relative equilibria in a model of identical, steered particles moving in the plane at unit speed. Relative equilibria either correspond to parallel motion of all particles with fixed relative spacing or to circular motion of all particles around the same circle. Particles exchange relative information according to a communication graph that can be undirected or directed and time-invariant or time-varying. The emphasis of this paper is to show how previous results assuming all-to-all communication can be extended to a general communication framework. [ABSTRACT FROM AUTHOR]

In this paper, we study the Cohen-Grossberg neural networks with discrete and distributed delays. For a general class of internal decay functions, without assuming the boundedness, differentiability, and monotonicity of the activation functions, we establish some sufficient conditions for the existence of a unique equilibrium and its global asymptotic stability. Theory of M-matrices and Lyapunov functional technique are employed. The criteria are independent of delays and hence delays are harmless in our case. Our results improve and generalize some existing ones. [ABSTRACT FROM AUTHOR]

Inspired by the so-called `tugs" problem front mathematics, we study the geometric formations of multivehicle systems under cyclic pursuit. First, we introduce the notion of cyclic pursuit by examining a system of identical linear agents in the plane. This idea is then extends to a system of wheeled vehicles, each subject to a single nonholonomic constraint (i.e., unicycles), which is the principal focus of this paper. The pursuit framework is particularly simple in that the n identical vehicles are ordered such that vehicle i pursues vehicle i &plus; 1 modulo n. In this paper, we as- suine each vehicle has the same constant forward speed. We show that the system's equilibrium formations are generalized regular polygons and it is exposed how the multiivehicle system's global behavior can be shaped through appropriate controller gain assignments. We then study the local stability of these equilibrium polygons, revealing which formations are stable and which are not. [ABSTRACT FROM AUTHOR]

This paper addresses the problem of determining switching rules for affine switched systems such that the system state is driven to a desired point and a guaranteed cost is minimized. The switching rule is determined by solving an LMI problem and global asymptotic stability of the tracking error dynamics is guaranteed even if sliding motions occur on any switching surface of the system. The potential of the results is illustrated on a true refrigeration system, namely a domestic refrigerator, where the purpose is to control the temperature in the fresh food and the freezer compartments. [ABSTRACT FROM PUBLISHER]

The delayed logistic equation, also known as Hutchinson's equation, is a simple and elegant model commonly used to capture critical features of complex phenomena in biology, medicine, and economics. This paper studies the stability of a single-species logistic model with a general delay distribution and a constant inflow of nutritional resources. We provide conditions for the linear stability of the positive equilibrium and the occurrence of Hopf bifurcation. The findings complement existing literature and are applied to specific delay distributions: Uniform, Dirac-delta, and gamma distributions. Without resource inflow, we find that the positive equilibrium is stable for short delays but loses stability through Hopf bifurcation as the mean delay increases. The model's dynamics vary with resource inflow based on the delay distribution: in uniform and Dirac-delta distributions, the dynamics are similar to the no-inflow case, whereas for the gamma distribution, stability depends on the delay order p = 1 , 2 , 3. • Stability of a logistic model with distributed delay and nutrient inflow is studied. • Sufficient criteria for Hopf bifurcation with a generic delay kernel are established. • The results are applied to uniform, Dirac-delta, and gamma distributions. • The results can be applied to othe distributions. [ABSTRACT FROM AUTHOR]

This paper proposed a new predefined time nonsingular sliding mode control (SMC) method for nonlinear systems. Firstly, based on the definition of predefined time stability (PTS), a new sufficient condition is designed to ensure that the system states converge to the origin within a predefined time. The design of a simple variable exponent not only guarantees PTS, but also enables adaptive adjustment when the system states are far away from and near the equilibrium point. And compared with traditional methods, the proposed Lemma 2 enhances the control effect and achieves faster convergence whether the system states are far from or near to the equilibrium point. Secondly, based on the proposed stability condition, a new nonsingular SMC method is designed to ensure that the tracking error converges to an arbitrarily small region within a predefined time. Finally, the proposed method is verified to have good performance through simulation and physical experiments. [ABSTRACT FROM AUTHOR]

The diversity of firing patterns and their bifurcation mechanisms of a neuron circuit are crucial for exploiting bionic applications. This paper constructs an N-type locally active memristor (LAM)-based FitzHugh–Nagumo (FHN) circuit by replacing the tunnel-diode in the original FHN circuit. Numerical simulations and hardware measurements reveal that the N-type LAM-based FHN circuit can reproduce rich neuromorphic firing patterns of quasi-periodic/periodic bursting behaviors and chaotic/periodic spiking behaviors. The bursting and spiking behaviors are triggered by a low-frequency stimulus and a high-frequency one, respectively. Besides, the fold and Hopf bifurcation sets are depicted. Then, the bifurcation mechanisms for the quasi-periodic and periodic bursting behaviors via Hopf/Hopf and Hopf/fold bifurcations are theoretically deduced by time-domain waveform and equilibrium trajectory. The numerical results and hardware experiments exhibit the effectiveness of the proposed memristive FHN circuit in reproducing abundant firing patterns of bursting and spiking behaviors. • An N-type locally active memristor-based FHN circuit is first built by replace the tunnel-diode. • Fast–slow dynamics versus to high-frequency and low-frequency stimuli are explored. • Bifurcation mechanisms for quasi-periodic and periodic bursting behaviors are deduced. • Hardware experiments are executed to validate the generation of spiking and bursting behaviors. [ABSTRACT FROM AUTHOR]

This paper aims to explore the dynamical mechanism of compound relaxation oscillations generated in the Filippov system, focusing on the influence of bifurcations induced by discontinuities on compound relaxation oscillations. A non-smooth Filippov system with two scales is built by introducing a non-smooth term and a external excitation into the Langford system based on the unfolding cusp-Hopf normal form. By specifying the slow variable as the bifurcation parameter, regular equilibrium bifurcation, boundary equilibrium bifurcation, and a variety of sliding bifurcations in the fast subsystem are obtained. Three types of oscillation modes under different parameter conditions are given through the numerical simulation. The dynamical mechanisms of each compound relaxation oscillation are also explained by combining the bifurcation analysis and slow–fast analysis. It is observed that persistence bifurcation realizes the stable transition between the pseudo-equilibrium branch and equilibrium branch. The grazing–sliding and switching–sliding bifurcations only change the topology of the limit cycle. None of these non-smooth bifurcations results in the transition between the quiescent state and spiking state, whereas the jumping behavior between different equilibrium branches may lead to the restructuring of the relaxation oscillation mode. In addition, a distinctive silent mode is observed, manifested by the complete disappearance of the compound relaxation oscillation. • A Filippov system with two scales is constructed based on the unfolding cusp-Hopf normal form. • Several non-smooth bifurcations are found in the fast subsystem, including boundary equilibrium bifurcations and two types of the sliding bifurcations. • Three different oscillation modes and their respective mechanisms are investigated. [ABSTRACT FROM AUTHOR]

This paper presents a new 4D chaotic system having both single-scroll and double-scroll, self-excited and hidden attractors with the variation of system parameters. The proposed system exhibits several equilibrium points and their associated attractors. There are four interseting sets of equilibria with self-excited and hidden attractors. The system characteristics are demonstrated using phase plots, bifurcation diagrams and Lyapunov exponents. Offset boosting, amplitude control, the coexistence of attractors and antimonotonicity are demonstrated by the new system. Further, a predefined-time sliding mode controller with three control inputs for a 4D system is designed for successful synchronization between the two identical master and slave systems. Such an underactuated controller for synchronization is seldom seen in the literature. The developed 4D chaotic system is realized first using the NI Multisim circuit simulator and then in an NI Academic RIO hardware platform. The outcomes validate the numerical results. [ABSTRACT FROM AUTHOR]