Equilibrium Point Analysis Research Articles

Analisis Stabilitas dan Kontrol Optimal Model Matematika Kecanduan Game Online

Online games as multimedia can create high interest in users, especially online games that use interaction. This if done in the long term will result in negative impacts. Currently, there are many online game competitions known as esports. The players who are involved with esports are referred to as professionals, although their playing time is the same as online game addicts, they are different from ordinary online game addicts. So, in this article, the author constructs a mathematical model of online game addiction into four compartments (susceptible, infective, professional and quitting compartments) by considering the direct transfer of the vulnerable population to the professional population. The author conducted a stability analysis of the model equilibrium point where there are two equilibrium points and also obtained the basic reproduction number . By using Pontriagin's maximum principle, a solution for optimal control is obtained using control variables and in order to obtain an optimal condition.

Market Equilibrium Point Analysis by a Fuzzy Approach

Market Equilibrium Point Analysis by a Fuzzy Approach

The dynamics of Coronavirus pandemic disease model in the existence of a curfew strategy

The dynamical behavior of the Coronavirus disease-2019 COVID-19 pandemic model is proposed and analyzed. It is assumed that a curfew strategy is applied to control the outbreak of the disease in addition to a social distancing between the individuals. All the basic properties of the solution including existence, uniqueness, and boundedness are discussed. The basic reproduction number , is determined. The local stability analysis is studied. While the global stability analysis of disease-free equilibrium point is investigated using the method of Castillo-Chavez, however for the endemic equilibrium point, the method of Lyapunov function is used. Furthermore, the local bifurcation of the model at the disease-free equilibrium point is discussed. Finally, the numerical simulations are performed in order to show validation of the theoretical results and determine how changes in parameters affect the dynamical behavior of the system.

Bifurcation analysis for Shil'nikov Chaos Electro-dissolution of Copper.

This paper is devoted to study the local bifurcations and stability of three dimensional systems that representing a Shil'nikov chaos during copper electro-dissolution. The local stability analysis of equilibrium points has been studied. It is shown that transcritical bifurcation can appears in the system. Also, the existence of Hopf bifurcation of the system around the equilibrium points is studied when the parameter passes through the critical value. Normal form theory is used to study bifurcating periodic solutions.

A mathematical model of COVID-19 and the multi fears of the community during the epidemiological stage

Within two years, the world has experienced a pandemic phenomenon that changed almost everything in the macro and micro-environment; the economy, the community’s social life, education, and many other fields. Governments started to collaborate with health institutions and the WHO to control the pandemic spread, followed by many regulations such as wearing masks, maintaining social distance, and home office work. While the virus has a high transmission rate and shows many mutated forms, another discussion appeared in the community: the fear of getting infected and the side effects of the produced vaccines. The community started to face uncertain information spread through some networks keeping the discussions of side effects on-trend. However, this pollution spread confused the community more and activated multi fears related to the virus and the vaccines.This paper establishes a mathematical model of COVID-19, including the community’s fear of getting infected and the possible side effects of the vaccines. These fears appeared from uncertain information spread through some social sources. Our primary target is to show the psychological effect on the community during the pandemic stage. The theoretical study contains the existence and uniqueness of the IVP and, after that, the local stability analysis of both equilibrium points, the disease-free and the positive equilibrium point. Finally, we show the global asymptotic stability holds under specific conditions using a suitable Lyapunov function. In the end, we conclude our theoretical findings with some simulations.

Floquet stability analysis of equilibrium points and numerical exploration of pulsating zero-velocity curves and Newton–Raphson basins of attraction

An important and pivot aspect of a dynamical system is the stability property and its range, which play a crucial role towards the stabilisation of a mission design. Present paper deals with Floquet stability analysis of equilibrium points and estimation of pulsating zero velocity curves and Newton–Raphson basins of attraction associated to these equilibrium points under the radiation pressure effect in the photo-gravitational planar elliptic restricted four body problem. Floquet stability test for the equilibrium points is performed with the help of the transition curves, which are generated under the influence of radiation pressure. It is noticed that stability range of the respective equilibrium points have deviated due to radiation pressure. To observe the possible regions of motion for restricted body, the pulsating zero velocity curves are numerically explored after establishing an invariant relation. It is found that not only the radiation parameter but also eccentricity and true anomaly reflect a considerable impact on the shape and size of forbidden regions. To see the tendency of randomly selected points in the phase space, Newton–Raphson basins of attraction associated to each attractor are estimated and effects of perturbing parameters are analysed. It is observed that effects of radiation and mass parameters are considerable however, that of eccentricity and true anomaly are negligible. Moreover, from probability distribution bar diagram, most probable number of iterations is 10 for the case when two primaries are of equal masses and third one is different, whereas it is 20 for the case when all primaries are of identical mass. These results will be helpful to study the generalised problem along with other kind of perturbations.

Analysis of Equilibrium Points in Quantized Hill System

In this work, the quantized Hill problem is considered in order for us to study the existence and stability of equilibrium points. In particular, we have studied three different cases which give all whole possible locations in which two points are emerging from the first case and four points from the second case, while the third case does not provide a realistic locations. Hence, we have obtained four new equilibrium points related to the quantum perturbations. Furthermore, the allowed and forbidden regions of motion of the first case are investigated numerically. We demonstrate that the obtained result in the first case is a generalization to the classical one and it can be reduced to the classical result in the absence of quantum perturbation, while the four new points will disappear. The regions of allowed motions decrease as the value of the Jacobian constant increases, and these regions will form three separate areas. Thus, the infinitesimal body can never move from one allowed region to another, and it will be trapped inside one of the possible regions of motion with the relative large values of the Jacobian constant.

Complex dynamics and control analysis of an epidemic model with non-monotone incidence and saturated treatment.

In this manuscript, we consider an epidemic model having constant recruitment of susceptible individuals with non-monotone disease transmission rate and saturated-type treatment rate. Two types of disease control strategies are taken here, namely vaccination for susceptible individuals and treatment for infected individuals to minimize the impact of the disease. We study local as well as global stability analysis of the disease-free equilibrium point and also endemic equilibrium point based on the values of basic reproduction number R_0. Therefore, disease eradicates from the population if basic reproduction number less than unity and disease persists in the population if basic reproduction number greater than unity. We use center manifold theorem to study the dynamical behavior of the disease-free equilibrium point for R_0 = 1. We investigate different bifurcations such as transcritical bifurcation, backward bifurcation, saddle-node bifurcation, Hopf bifurcation and Bogdanov–Takens bifurcation of co-dimension 2. The biological significance of all types of bifurcations are described. Some numerical simulations are performed to check the reliability of our theoretical approach. Sensitivity analysis is performed to identify the influential model parameters which have most impact on the basic reproduction number of the proposed model. To control or eradicate the influence of the emerging disease, we need to control the most sensitive model parameters using necessary preventive measures. We study optimal control problem using Pontryagin’s maximum principle. Finally using efficiency analysis, we determine most effective control strategy among applied controls.

The Dynamics of the Aquatic Food Chain System in the Contaminated Environment

In this paper, the aquatic food chain model, consisting of Phytoplankton, Zooplankton, and Fish, in the contaminated environment is proposed and studied. Modified Leslie–Gower model with Holling type IV functional response are used to describe the growth of Fish and the food transition throughout the food chain, respectively. The toxic substance affects directly the Phytoplankton and indirectly the other species. The local stability analysis of all possible equilibrium points is done. The persistence conditions of the model are established. The basin of attraction for each point is specified using the Lyapunov function. Bifurcation analysis near the coexistence equilibrium point is investigated. Detecting the existence of chaos is carried out using bifurcation diagrams. Numerical simulation shows that the food chain has rich dynamics including chaos. Moreover, the existence of toxic substances works as a stabilizing factor in the model.

Dynamic investigations in a Stackelberg model with differentiated products and bounded rationality

This work focuses on a Stackelberg duopoly game characterized by firms with differentiated products and bounded rationality. Both firms adopt myopic mechanism to update their production decisions in each discrete period. For the purpose of stressing hierarchy between both firms, the leader firm is supposed to estimate marginal profit by only thinking over his own interfering factors. Since global stability analysis of Nash equilibrium point becomes desirable in complex behaviors research, we cover it as well besides local stability. Numerical simulations are carried out to verify some essential appearances of previous theoretical analyses. Such aspects include maximum Lyapunov exponent, strange attractor, basin of attraction and sensitivity on initial conditions. In addition, we calculate fractal dimension of the non-integral dimension shapes in chaotic state. As to the chaos emerged under certain conditions, effective control method has been utilized to force them back towards stability.

DYNAMIC ANALYSIS OF THE MATHEMATICAL MODEL OF THE SPREAD OF CHOLERA WITH VACCINATION STRATEGIES

This research discusses the math model of spreading cholera disease with a mathematical strategy of math model constructed by considering a vaccination strategy. In addition, there is a population of hyperinfectious and lessinfectious bacteria so that the model of SVIR-BhiBli type, by. The model formed in the form of determination of fixed point, determination of basic reproductions numbers, analyzing the equilibrium point and sensitivity analysis. The equilibrium analysis produces two equilibrium points of a immediate-free equilibrium point of aceletotic local if and endemic equilibrium points will be stable local asymptotics if . Furthermore, numerical simulation that the increase in vaccination rate influences on the decline in value while increased rate of vaccine depreciation can increase the value of . In addition, sensitivity analysis shows that if the parameter is enhanced while other contrast parameters will contribute to the increase in value, as a result can increase the rate of transmission of cholera disease. Whereas if the parameter is enhanced while other contrast parameters will contribute to the decrease in value, as a result of the dissemination of the disease can be pressed very significantly.

Tripartite evolutionary game analysis of governance mechanism in Chinese WEEE recycling industry

The rapidly increase in waste electrical and electronic equipment (WEEE) poses challenges to the government to manage the WEEE recycling industry. In order to better understand and improve the governance mechanism in the WEEE recycling industry, taking China as an example, we construct a tripartite evolutionary game model consisting of the manufacturer, the collector, and the government by underscoring product eco-design and qualified disassembly. First, we obtain the evolutionary stability strategy (ESS) by calculating the replicator dynamic equation and the stability analysis of equilibrium points. Then, we examine the feasibility and rationality of the evolutionary game model by analyzing ESSs corresponding to different stages of the WEEE recycling industry. Finally, we explore the ramifications of the governance mechanism on the behaviors of the manufacturer’s product eco-design and the collector’s qualified disassembly through a numerical example. We find that the evolutionary game model exists eleven equilibrium points and six possible ESSs, and the choice of each ESS depends mainly on the trade-off between costs and revenues for each stakeholder. Through the numerical study, we observe that the current Chinese WEEE governance mechanism is not effective. We also observe that the manufacturer’s willingness to implement product eco-design is decreasing in the tax rate but increasing in the environmental tex, and the collector’s willingness to recycle WEEE through a compliance process is increasing in both the subsidy and the penalty. We suggest that the government focuses on improving the reward-penalty mechanism, establishing appropriate WEEE disposal fund levy and subsidy standards, and removing informal collectors to promote the healthy development of WEEE recycling industry.

Qualitative and quantitative analysis of the COVID-19 pandemic by a two-side fractional-order compartmental model

Global efforts are focused on discussing effective measures for minimizing the impact of COVID-19 on global community. It is clear that the ongoing pandemic of this virus caused an immense threat to public health and economic development. Mathematical models with real data simulations are powerful tools that can identify key factors of pandemic and improve control or mitigation strategies. Compared with integer-order and left-hand side fractional models, two-side fractional models can better capture the state of pandemic spreading. In this paper, two-side fractional models are first proposed to qualitative and quantitative analysis of the COVID-19 pandemic. A basic framework are given for the prediction and analysis of infectious diseases by these types of models. By means of asymptotic stability analysis of disease-free and endemic equilibrium points, basic reproduction number R0 can be obtained, which is helpful for estimating the severity of an outbreak qualitatively. Sensitivity analysis of R0 is performed to identify and rank key epidemiological parameters. Based on the real data of the United States, numerical tests reveal that the model with both left-hand side fractional derivative and right-hand side fractional integral terms has a better forecast ability for the epidemic trend in the next ten days. Our extensive computational results also quantitatively reveal that non-pharmaceutical interventions, such as isolation, stay at home, strict control of social distancing, and rapid testing can play an important role in preventing the pandemic of the disease. Thus, the two-side fractional models are proposed in this paper can successfully capture the change rule of COVID-19, which provide a strong tool for understanding and analyzing the trend of the outbreak.

The identifiability of gene regulatory networks: the role of observation data.

Identifying gene regulatory networks (GRN) from observation data is significant to understand biological systems. Conventional studies focus on improving the performance of identification algorithms. However, besides algorithm performance, the GRN identification is strongly depended on the observation data. In this work, for three GRN S-system models, three observation data collection schemes are used to perform the identifiability test procedure. A modified genetic algorithm-particle swarm optimization algorithm is proposed to implement this task, including the multi-level mutation operation and velocity limitation strategy. The results show that, in scheme 1 (starting from a special initial condition), the GRN systems are of identifiability using the sufficient transient observation data. In scheme 2, the observation data are short of sufficient system dynamic. The GRN systems are not of identifiability even though the state trajectories can be reproduced. As a special case of scheme 2, i.e., the steady-state observation data, the equilibrium point analysis is given to explain why it is infeasible for GRN identification. In schemes 1 and 2, the observation data are obtained from zero-input GRN systems, which will evolve to the steady state at last. The sufficient transient observation data in scheme 1 can be obtained by changing the experimental conditions. Additionally, the valid observation data can be also obtained by means of adding impulse excitation signal into GRN systems (scheme 3). Consequently, the GRN systems are identifiable using scheme 3. Owing to its universality and simplicity, these results provide a guide for biologists to collect valid observation data for identifying GRNs and to further understand GRN dynamics.

Mathematical analysis of a COVID-19 model with different types of quarantine and isolation.

A COVID-19 deterministic compartmental mathematical model with different types of quarantine and isolation is proposed to investigate their role in the disease transmission dynamics. The quarantine compartment is subdivided into short and long quarantine classes, and the isolation compartment is subdivided into tested and non-tested home-isolated individuals and institutionally isolated individuals. The proposed model has been fully analyzed. The analysis includes the positivity and boundedness of solutions, calculation of the control reproduction number and its relation to all transmission routes, existence and stability analysis of disease-free and endemic equilibrium points and bifurcation analysis. The model parameters have been estimated using a dataset for Oman. Using the fitted parameters, the estimated values of the control reproduction number and the contribution of all transmission routes to the reproduction number have been calculated. Sensitivity analysis of the control reproduction number to model parameters has also been performed. Finally, numerical simulations to demonstrate the effect of some model parameters related to the different types of quarantine and isolation on the disease transmission dynamics have been carried out, and the results have been demonstrated graphically.

A system of two competitive prey species in presence of predator under the influence of toxic substances

In this article, a two prey - one predator model has been studied where two prey species are competitive in nature and also uses toxic substances for own existence. Biologically well posedness of the model system has been shown through positivity and boundedness of solutions. Existence criterion and stability analysis of the non-negative equilibrium points have been discussed. The sufficient conditions for existence of Hopf bifurcation and stability switches induced by delay are investigated. The direction and the stability criteria of the bifurcating periodic solutions are determined with the help of the normal form theory and the center manifold theorem. Numerical simulations are performed to illustrate the theoretical analysis results.

Potential-Based Large-Signal Stability Analysis in DC Power Grids With Multiple Constant Power Loads

The increasing adoption of power electronic devices may lead to large disturbance in DC power grids. Traditionally, the Brayton-Moser mixed potential theory is utilized to address large-signal stability (LSS) analysis. However, this theory proposes sufficient criteria only for large-signal global stability, which provides general but fuzzy information about the stability of DC microgrids. To solve this issue, we propose for the first time a comprehensive hyperlocal large-signal stability analysis (HL-LSS) that is adaptable to large-scale complex power grids. The novel proposed analysis studies the complex situation in nonlinear DC grids with multiple equilibrium points, including stability analysis of each equilibrium point and convergence analysis of state trajectories. The region of attraction (ROA) of microgrids is also estimated.

A Simple Conservative Chaotic Oscillator with Line of Equilibria: Bifurcation Plot, Basin Analysis, and Multistability

Here, a novel conservative chaotic oscillator is presented. Various dynamics of the oscillator are examined. Studying the dynamical properties of the oscillator reveals its unique behaviors. The oscillator is multistable with symmetric dynamics. Equilibrium points of the oscillator are investigated. Bifurcations, Lyapunov exponents (LEs), and the Poincare section of the oscillator’s dynamics are analyzed. Also, the oscillator is investigated from the viewpoint of initial conditions. The study results show that the oscillator is conservative and has no dissipation. It also has various dynamics, such as equilibrium point and chaos. The stability analysis of equilibrium points shows there are both stable and unstable fixed points.

Analysis of equilibrium point based on nonlinear traffic flow model

In this paper, the traveling wave solution of Kerner konhauser model is obtained by phase plane analysis method, and a new model suitable for stability analysis is obtained by variable transformation. The new model can describe and predict the nonlinear traffic phenomenon on Expressway from the perspective of system global stability. In this paper, the nonlinear system and linear system are obtained respectively by traveling wave substitution and Taylor expansion at the equilibrium point. According to the qualitative theory of differential equations, the equilibrium point type and stability of the linear system are determined. Finally, the simulation results show the consistency between the numerical results and the theoretical analysis.

Analysis of equilibrium point based on viscous traffic flow model

In this paper, the phase plane analysis method is used, which can describe and predict the nonlinear traffic phenomenon on Expressway from the perspective of system global stability. The nonlinear system of the model is obtained by the traveling wave substitution and Taylor expansion of the model, and the equilibrium point of the model is solved by specifying the model parameters. According to the definite theory of differential equation, the model is further analyzed to judge the type and stability of equilibrium point. Finally, numerical verification is carried out according to the simulation diagram. Through numerical verification, the simulation results of the model are consistent with the conclusions of theoretical analysis. It can more clearly describe the change of density or speed with time or road section.