Equilibrium Point Research Articles

The influence of prevention and isolation measures to control the infections of the fractional Chickenpox disease model

In this paper, we propose a mathematical model using the Caputo fractional derivative (CFD) and two control signals to study the transmission dynamics and control of Chickenpox (Varicella) outbreak. The model consists of six compartments representing susceptible, vaccinated, exposed, infected with complications, infected without complications, and recovered individuals. We analyze the theoretical properties of the model, including existence, uniqueness, and boundedness of solutions, and calculate the basic reproduction number (R0). We identify equilibrium points and establish conditions for their stability. Sensitivity analysis helps identify the most influential parameters on R0. We formulate a fractional optimal control problem (FOCP) by incorporating time-dependent prevention and isolation measures. The necessary optimality conditions are derived using Pontryagin’s maximum principle. Numerical simulations based on the Adams–Bashforth-Moulton (ABM) method illustrate the impact of control measures and fractional order on disease propagation. The results highlight the effectiveness of optimal controls and fractional order in understanding and managing epidemics, enhancing stability conditions. The study contributes to a better understanding of Chickenpox transmission dynamics and provides insights for disease control and management, aiding decision-makers and governments in taking preventive measures.

Dynamics, periodic orbits of a novel four-dimensional hyperchaotic system with hidden attractors

In recent years, the investigation of systems featuring hidden attractors and coexisting attractors has garnered significant attention. This paper presents a novel four-dimensional (4D) hyperchaotic system devoid of equilibrium points, achieved by formulating an equation without a solution or constructing a system without fixed points. Due to the complex shape of this attractor, a novel coding method is utilized to establish symbol dynamics using eight letters. The proposed system exhibits highly intricate dynamics, including variations in topological structure with alterations in system parameters, as well as an exploration and discussion of four types of coexisting attractors. Our extensive practice has led us to propose a new conjecture: hyperchaotic systems with parameters close to the bifurcation point frequently display multistable states. Furthermore, the unstable periodic orbits with different topological lengths in the hidden hyperchaotic attractor are calculated systematically using the variational method. Additionally, the DSP circuit implementation is employed to validate the numerical simulation results for this new 4D system. Finally, adaptive synchronization is successfully realized within the system, thereby confirming its feasibility.

Convex equilibrium‐free stability and performance analysis of discrete‐time nonlinear systems

AbstractThis paper considers the equilibrium‐free stability and performance analysis of discrete‐time nonlinear systems. Two types of equilibrium‐free notions are considered. Namely, the universal shifted concept, which considers stability and performance w.r.t. all equilibrium points of the system, and the incremental concept, which considers stability and performance between trajectories of the system. This paper shows how universal shifted stability and performance of discrete‐time systems can be analysed by making use of the time‐difference dynamics. Moreover, the existing results are extended for incremental dissipativity for discrete‐time systems based on dissipativity analysis of the differential dynamics to more general state‐dependent storage functions for less conservative results. Finally, it is shown how both these equilibrium‐free notions can be cast as a convex analysis problem by making use of the linear parameter‐varying framework, which is also demonstrated by means of an example.

EFFECTS OF SYMBIOTIC BACTERIA IN PATHOGENIC INTERACTIONS: THE CASE OF BATRACHOCHYTRIUM DENDROBATIDIS AND PSEUDOMONAS SP. IN AMPHIBIAN POPULATIONS

The decline in amphibian populations in recent decades may be linked to the occurrence of infectious diseases such as chytridiomycosis, which is caused by the chytrid fungus Batrachochytrium dendrobatidis (Bd). It is known that symbiotic bacteria protect the host due to their inhibitory nature. However, how the population dynamics of amphibians is affected by additional effects provided by symbiotic bacteria has not been analyzed in depth. In this paper, a model is proposed to describe the interaction among susceptible amphibians, susceptible amphibians with symbiotic bacteria and amphibians with chytrid fungus. When the modeling takes into account the additional reproductive benefits that the symbiont Pseudomonas sp. grants to the host, multiple endemic equilibrium points can exist if [Formula: see text] ([Formula: see text] is the basic reproduction number for Bd). In this scenario, the existence of a subcritical bifurcation at [Formula: see text], which can occur in two different disease-free equilibrium points, gives rise to complex dynamics and stability scenarios. Particularly, the analysis of the model shows that a sudden increase of fungus-infected amphibians can occur even when [Formula: see text] due to bistability phenomena. In this scenario, the existence of a subcritical bifurcation, which translates for the fungus into colonization even for values of [Formula: see text] less than one, represents an advantage for the chytrid fungus Batrachochytrium dendrobatidis since the pathogen should benefit from remaining as close as possible to an endemic equilibrium. To control the fungal infection, [Formula: see text] must be reduced to a value below one until the endemic equilibrium points disappear. Finally, we show that the amphibian population can reach a critical population level close to an extinction scenario when [Formula: see text] increases.

Fractional-Order Tabu Learning Neuron Models and Their Dynamics

In this paper, by replacing the exponential memory kernel function of a tabu learning single-neuron model with the power-law memory kernel function, a novel Caputo’s fractional-order tabu learning single-neuron model and a network of two interacting fractional-order tabu learning neurons are constructed firstly. Different from the integer-order tabu learning model, the order of the fractional-order derivative is used to measure the neuron’s memory decay rate and then the stabilities of the models are evaluated by the eigenvalues of the Jacobian matrix at the equilibrium point of the fractional-order models. By choosing the memory decay rate (or the order of the fractional-order derivative) as the bifurcation parameter, it is proved that Hopf bifurcation occurs in the fractional-order tabu learning single-neuron model where the value of bifurcation point in the fractional-order model is smaller than the integer-order model’s. By numerical simulations, it is shown that the fractional-order network with a lower memory decay rate is capable of producing tangent bifurcation as the learning rate increases from 0 to 0.4. When the learning rate is fixed and the memory decay increases, the fractional-order network enters into frequency synchronization firstly and then enters into amplitude synchronization. During the synchronization process, the oscillation frequency of the fractional-order tabu learning two-neuron network increases with an increase in the memory decay rate. This implies that the higher the memory decay rate of neurons, the higher the learning frequency will be.

An extreme multistable circuit based on a new memcapacitor and a new meminductor

Expanding the definition of memristor gets memcapacitor and meminductor. Because there are very few physical memory elements, it is very meaningful to apply its equivalent model to the study of physical circuits. In this paper, based on the definition of the memory element, a new memcapacitor and a new meminductor are proposed, and a chaotic oscillator circuit is constructed based on them. Whether the two memory elements have memory characteristics is analyzed. The range of equilibrium points of the chaotic oscillator circuit is determined. The dynamical behavior of this chaotic oscillator circuit is analyzed by the Lyapunov exponent spectrum (LEs), bifurcation diagram (BD) and spectral entropy complexity (SE). Finally, Multisim software is used to verify the correctness of the simulation, and the physical realizability of the system is verified by the DSP platform. The results of the research help to better utilize this chaotic oscillator circuit in the field of communication encryption.

Dynamic analysis of an infected predator-prey system with prey herd behavior

ABSTRACT To protect themselves from predation, sometimes prey starts roaming in groups and shows herd behavior. This paper addresses the predator-prey system dynamics with prey herd behavior and infection in predators. We explore the conditions under which all equilibria exist and the stability of each equilibrium point of the system. The outcomes of this work suggest that prey herd behavior is beneficial for the prey population when the predators are diseased. Under suitable conditions, the three populations can coexist and maintain the ecological balance of nature.

Bifurcation analysis with chaotic attractor for a special case of jerk system

This article focuses on investigating local bifurcations in a special type of chaotic jerk system. It examines the occurrence and non-occurrence of saddle-node, transcritical, zero-Hopf, Hopf, and pitchfork bifurcations at the origin. The parameters that result in a zero-Hopf equilibrium point at the origin are characterized for the proposed system. Additionally, a demonstration is provided to show that the utilization of the first-order averaging theory leads to the emergence of a single periodic solution branching out from the zero-Hopf equilibrium located at the origin. Furthermore, the focus quantities method is applied to explore the periodicity of the cubic part of the system. This method helps determine the number of periodic solutions that can emerge from the Hopf point. Due to the computational load for computing singular quantities, only three singular quantities are found. Under specific conditions, it is shown that three periodic solutions can bifurcate from the origin of the system. Finally, the study also examines the chaotic attractors of the system.

Event-triggered controllers using contraction analysis

In this article, we introduce an event-triggered control method that relies on contraction analysis for linear time-varying systems (LTV) and then extend it to a certain class of nonlinear systems. Contraction analysis considers stability with respect to a nominal trajectory rather than an equilibrium point. If two neighbouring trajectories of a system are located in a contraction region, then they will tend to each other and to a nominal trajectory. In the event-triggered control algorithm that we introduce, we suggest to update the control law whenever the system trajectory is about to leave the contraction region. We show that such a scheduling of the control law guarantees system stability, and we show that a minimum inter-event time exists between consecutive updates of the control law. We also show how to place the system trajectory in a contraction region and demonstrate that the classical controllability assumption on LTV systems is enough to ensure the existence of the required transformation to perform that.

Global Dynamics of a Holling-II Amensalism System with a Strong Allee Effect on the Harmful Species

In this study, we discuss the global dynamics of the Holling-II amensalism model for a strong Allee effect of harmful species. We discuss the existence and stabilization of the extinction equilibria, exclusion equilibria, coexistence equilibria, and infinite singularities by analyzing the presence and stabilization of the system characteristics in terms of the possibilities and correspondences in the model when the death rate of the injured species is used as a threshold value. Also, we find that the two equilibrium points in the first quadrant are effective in proving that the model does not have globally stabilizing features and obtain two critical conditions and their corresponding global phase diagrams. Finally, we explore the weak Allee effect of the victim species, and using the analysis from numerical simulations, we recapitulate the analysis and dynamics of the model in equilibrium.

“Zero-sum game” or “win-win cooperation”: an analysis of the evolution effect of competition neutrality based on the participation of four parties

To effectively grasp the interactive evolution mechanism of relevant entities in the process of competitive neutrality and reveal the impact of the implementation effect of the principle of competitive neutrality on the evolution path of the game system is the key to guiding the development of the government, industry associations and enterprises with reasonable policies. The equilibrium point of the game is analyzed by the four-party evolutionary game model, and the influence of the change of relevant parameters on the evolution result of the game system is further simulated by numerical simulation. The results show that the central government supervision plays an important role in guiding the strategy selection of different game players. By raising the penalty standards for non-neutral competitive behavior of market players and increasing the non-neutral penalty for local governments, the system can evolve faster to the ideal state of “loose supervision by the central government, neutral implementation by local governments, strong supervision by industry associations, and active cooperation by enterprises.” Therefore, to improve the execution effect of competitive neutrality, the central government shall formulate a reasonable regulatory system, and to maximize inspire the enthusiasm of local government, industry associations and enterprises, to promote stability in the sustainable development of the market.

Nonlinear SIRS Fractional-Order Model: Analysing the Impact of Public Attitudes towards Vaccination, Government Actions, and Social Behavior on Disease Spread

This present work develops a nonlinear SIRS fractional-order model with a system of four equations in the Caputo sense. This study examines the impact of positive and negative attitudes towards vaccination, as well as the role of government actions, social behavior and public reaction on the spread of infectious diseases. The local stability of the equilibrium points is analyzed. Sensitivity analysis is conducted to calculate and discuss the sensitivity index of various parameters. It has been established that the illness would spread across this system when the basic reproduction number is larger than 1, the system becomes infection-free when the reproduction number lies below its threshold value of 1. Numerical figures depict the effects of positive and negative attitudes towards vaccination to make the system disease-free sooner. A comprehensive study regarding various values of the order of fractional derivatives together with integer-order derivatives has been discussed in the numerical section to obtain some useful insights into the intricate dynamics of the proposed system. The Pontryagin principle is used in the formulation and subsequent discussion of an optimum control issue. The study also reveals the significant role of government actions in controlling the epidemic. A numerical analysis has been conducted to compare the system’s behavior under optimal control and without optimal control, aiming to discern their differences. The policies implemented by the government are regarded as the most adequate control strategy, and it is determined that the execution of control mechanisms considerably diminishes the ailment burden.

Enhancing Water Pollution Management Through a Comprehensive Fractional Modeling Framework and Optimal Control Techniques

Water pollution poses a significant threat to public health, particularly in developing nations. This study aims to enhance our understanding of water pollution transmission dynamics by introducing a novel mathematical model within a general fractional framework. The model offers a comprehensive analysis, including assessing solution feasibility and the stability of equilibrium points. To effectively implement this model, we develop an efficient numerical scheme based on the trapezoidal method. Furthermore, we conduct a thorough error discussion and convergence analysis. Notably, we devise and evaluate a new strategy to effectively tackle the associated fractional optimal control problem concerning water pollutant transmission. Applying this approach to a real case demonstrates its substantial impact, notably in reducing non-soluble contaminants within the water transmission system.

Dynamical analysis of a discrete Aihara neuron under a locally active memristor as electromagnetic radiation and its DSP implementation

The study of neuron model under electromagnetic radiation is significant for learning brain principles and treating mental diseases. In this paper, a novel discrete locally active memristor(DLAM) model is designed and its characteristics are investigated thoroughly. Then, the DLAM is used to imitate electromagnetic radiation to stimulate Aihara neuron, called EMR-Aihara neuron model. The equilibrium point of this discrete model is analyzed. Dynamical characteristics are studied by means of phase diagram, iteration sequence, bifurcation diagram, Lyapunov Exponent spectrum(LEs), Kolmogorow entropy(KE) and Spectral Entropy(SE) complexity. With these analysis methods, rich dynamical behaviors and neuron firing patterns are discovered from the EMR-Aihara neuron map, including hyperchaos, chaos and period. In addition, complex multistability and state transition phenomena concerning various attractors and neuron firing modes are observed. This EMR-Aihara neuron map is implemented in digital circuit by DSP platform as well, confirming the physical availability of the model. The EMR-Aihara neuron model can simulate biological neuron under electromagnetic radiation and apply to image encryption.

Optimal design of assembling robot considering different limb topologies and layouts

Abstract A class of 5 degree-of-freedom (DoF) hybrid robots consisting of one translational and two rotational (1T2R) parallel module and 2T serial module is presented for assembling in the aircraft cabin. The 1T2R parallel modules are with three limb topologies (PRS, RRS and internal closed-loop) and two layouts (symmetrical and “T” shape). Herein, P, R and S denote prismatic, rotational and spherical joints. This paper presents the multi-objective optimization of the hybrid robot regarding different limb topology, limb layout and corresponding dimensions as design variables. Considering demands from the in-cabin assembling, kinematic, linear stiffness along z-axis and total mass of the robot are the objectives. The Pareto fronts and cooperative equilibrium point (CEP) indicate that the robot with 3-PRS and 3-RRS parallel module have better performances than the one with internal closed-loop. The overall performances of robot with symmetrical layout are superior than the one with “T” shape layout. In addition, two optimization methods are compared. One is to separately optimize six robots with specific topology and layout. The other is to optimize all design variables in a model. It is found that six robots have their own performance zones. Therefore, final optimums of two methods are close to each other. But optimization in one model is able to eliminate unfeasible topologies in the early stage of searching and thus is more efficient. Optimal module is 3-PRS with symmetrical layout. Experiments on the physical prototype validates performances of the optimal robot.

Use of Nash equilibrium in finding game theoretic robust security bound on quantum bit error rate

Abstract Nash equilibrium is employed to find a game theoretic robust security bound on quantum bit error rate (QBER) for DL04 protocol which is a scheme for quantum secure direct communication that has been experimentally realized recently. The receiver, sender and eavesdropper (Eve) are considered to be quantum players (players having the capability to perform quantum operations). Specifically, Eve is considered to have the capability of performing quantum attacks (e.g., Wójcik's original attack, Wójcik's symmetrized attack and Pavi£i¢ attack) and classical intercept and resend attack. Game theoretic analysis of the security of DL04 protocol in the above scenario is performed by considering several game scenarios. The analysis revealed the absence of a Pareto optimal Nash equilibrium point within these game scenarios. Consequently, mixed strategy Nash equilibrium points are identified and employed to establish both upper and lower bounds for QBER. Further, the vulnerability of the DL04 protocol to Pavicic attack in the message mode is established. In addition, it is observed that the quantum attacks performed by Eve are more powerful than the classical attack, as the QBER value and the probability of detecting Eve's presence are found to be lower in quantum attacks compared to classical ones.

Stability analysis of homogeneous cooperative positive differential systems with time-varying delays and its generalization

This article is devoted to studying the asymptotic behavior of differential systems with bounded delays. We first focus on the analysis of homogeneous cooperative positive systems. Under the assumption that the vector field is homogeneous of a degree greater than 1, we show that the non-trivial solutions of the system converge to the origin at a polynomial rate. In the case when the degree of homogeneity equals 1, we prove that the solutions will decay at an exponential rate. As a generalization of these results, we consider nonlinear non-autonomous differential systems with time-varying delays that are bounded above by stable homogeneous positive systems. By some additional imposed conditions, in light of the comparison principle, we obtain the locally exponential stability and polynomial stability of the equilibrium point to these systems. Finally, specific examples and discussions are provided to illustrate the validity of the proposed theoretical results.

Generating grid double-scroll attractors from magnetized SC-CNN and its application in image encryption

Cellular neural network has attracted widespread attention due to its easy hardware implementation and general application in information processing. In this paper, a new memristive State-Controlled-CNN (MSC-CNN) model is established by introducing two novel flux-controlled memristors into a three-cell network, where one of which simulates the electromagnetic radiation and the other characterizes the coupling weight. The present MSC-CNN can generate multi-double-scroll attractors (MDSAs) in one direction or grid multi-double-scroll attractors (GMDSAs). Stability analysis shows that each asymptotically stable equilibrium point of the memristors serves as index-1 saddle-focus in the MSC-CNN, which induces the corresponding connecting bond orbits between two scrolls. Thus, the number of GMDSAs can be flexibly adjusted by varying the number of the asymptotically stable equilibrium points of the memristors. Moreover, the memristive coupling strength also controls the number of MDSAs. Through numerical simulations, complex dynamic behaviors such as extreme multistability and amplitude control are also observed. Then, an image encryption scheme based on the proposed MSC-CNN is designed from the application perspective. Finally, a microcontroller-based hardware system is implemented to verify the effectiveness of numerical simulations.

Framework design of distributed P2P2G energy transaction considering photovoltaic power and dispatch of manufacturing facilities

In this paper, a distributed peer-to-peer-to-grid (P2P2G) energy transaction framework considering photovoltaic generation and dispatch of manufacturing facilities is designed to increase the penetration of renewable energy resource. The factory with typical manufacturing assembly production task, photovoltaic panels and battery is regarded as a micro-grid (MG) to participate in the energy transaction market. The designed framework contains two levels. In the upper level, each MG determines the scheduling of manufacturing facilities and battery and the quantities of P2P2G energy transactions based on the given P2P energy transaction price and distribution locational marginal price (DLMP), in which the complex typical manufacturing assembly process is formulated as a kind of deferable load and the DLMP is updated according to the optimal power flow (OPF) of the network. In the lower level, peer-to-peer (P2P) energy transaction prices are updated by non-cooperative games and the homologous Rosen-Nash equilibrium points are obtained by using Nikaido-Isoda relaxation and alternating direction method of multipliers (ADMM) algorithm. The efficacy of the designed framework is demonstrated by a case study, and the results indicate the designed framework can improve the utilisation rate of renewable energy.

Dynamics of a restricted (6 + 1)-vortex problem

This paper investigates the dynamics of a restricted (6 + 1)-vortex problem. The existence of nested equilateral triangle configurations depends on the position of two triangles, which can be divided into two situations: staggered arrangement and symmetrical arrangement. Firstly, we provide the existence conditions for the configuration and further discuss the stability of the configuration numerically. Secondly, we have established the global dynamics of the restricted point-vortex problem with configurations under different arrangements, including the dynamic behavior of equilibrium points, periodic orbits, homoclinic orbits, and heteroclinic orbits.