Stability Of Equilibrium Point Research Articles

Dynamic behavior of enzyme kinetics cooperative chemical reactions

This article uses computational mathematics to investigate the dynamics of cooperative occurrences in chemical reactions inside living organisms. We study the dynamics of complex systems using mathematical models based on ordinary differential equations, paying special attention to chemical equilibrium and reaction speed. Explanations of cooperation, non-cooperation, and positive cooperation are presented in our study. We analyze the stabilities of equilibrium points by a systematic analysis that makes use of the Jacobian matrix and the threshold parameter R0. We next extend our investigation to evaluate global stability and the probability of the model. Variations in k3 have a notable effect on substrate concentration probabilities, indicating that it plays an important role in reaction kinetics. Reducing k3 highlights the substrate's critical contribution to the system by extending its presence in the concentration. We find that different results were obtained for cooperative behavior: higher reaction rates at different binding sites are correlated with positive cooperativity, while slower reactions are induced by negative cooperativity. The Adams–Bashforth method is used to show numerical and graphical solutions with the help of MATLAB. Tables and graphs are used to further explain the effects of the parameters. This study underlines how well ordinary differential equations may represent the complicated system dynamics found in chemical reactions. It also provides elusive insights into cooperative occurrences, which develops our understanding of the phenomenon and serves as a foundation for future research.

Carrying Capacity of Low Earth Orbit Computed Using Source-Sink Models

The low-Earth-orbit (LEO) environment will likely experience an exponential growth of the anthropogenic space object population, also due to the many proposed large constellations, which could negatively affect the sustainability of the space environment if no proper regulatory actions are introduced. This paper introduces a methodology to analyze high-capacity sustainable solutions of the LEO region from 200 to 900 km altitude. Sustainability is assessed through the stable equilibrium points of a source-sink model called MIT Orbital Capacity Assessment Tool 3 (MOCAT-3). Capacity is estimated using an optimization procedure that aims to compute the optimal launch rate that maximizes the number of satellites in LEO, subject to a risk-rate constraint. Results show that the sustainable number of satellites increases with the risk rate constraint. Moreover, the resilience of the proposed solutions is tested against a more accurate time-varying atmospheric density model and perturbations of the initial equilibrium population. The compatibility of the proposed solutions with future traffic launches and a strategy to accommodate future traffic needs are presented. Limitations of the current work are discussed, chiefly the importance of validation of model coefficients and behavior before resulting capacity estimates are used to guide actual decision-making.

A 5-D memristive hyperchaotic system with extreme multistability and its application in image encryption

By coupling memristors to nonlinear circuits, more complex dynamical behaviors can be induced. However, to date, there has been insufficient attention given to high-dimensional chaotic systems based on memristors. In this paper, a magnetic-controlled memristor is combined with a three-dimensional chaotic system, resulting in a five-dimensional memristive chaotic system. Through dynamic analysis and numerical simulations, the chaotic nature of the system is elucidated based on fundamental system behaviors, including Lyapunov dimension, dissipativity, stability of equilibrium points, 0–1 test, and Poincaré mapping. During the complex dynamical analysis of this system, unique dynamical behaviors are discovered, including intermittent chaos, transient chaos, extreme multistability, and offset-boosting. Moreover, the consistency between numerical calculations and the physical implementation of the actual system is verified through equivalent circuit design. Finally, this system is applied to image encryption, leading to the design of an efficient and secure hyper-chaotic image encryption algorithm, whose effectiveness is confirmed through several security tests.

Relationship between the College Student and the Campus Club: An Evolutionary Game Theory Analysis.

In this paper, we use an evolutionary game theory approach to build a relationship model of students and clubs for the purpose of improving student enthusiasm for participating in club activities. First, the process of the model building is introduced, which mainly includes the basic assumptions and the equilibrium point stability analysis. Based on this analysis, we find that the motivation adjustment of students and clubs is a dynamic process and that unilateral efforts alone cannot achieve an ideal result. Then, we use real data from Yanshan University to evaluate the model, the results of which indicate that the model can analyze the relationship between students and clubs effectively. Finally, we provide relevant suggestions based on the model established in this study, whereby we contribute a theoretical basis and practical guidance for how students can actively participate in clubs, as well as how clubs can better develop themselves.

A Method to Improve Both Frequency Stability and Transient Stability of Virtual Synchronous Generators during Grid Faults

Transient stability and frequency stability of virtual synchronous generator (VSG)-controlled converters during grid faults are critical for the stable operation of high-share renewable energy power systems. However, the existing transient stability methods often overlook the consideration of frequency stability. This paper presents a method to improve both frequency stability and transient stability of VSGs during grid faults. The influence mechanism of inertia and damping on transient stability and frequency stability is comprehensively analyzed under two types of faults, with or without stable equilibrium points. To address the conflict between frequency stability and transient stability of VSGs with large inertia under the first type of fault, an improved VSG control with angular frequency deviation feedback is proposed. The transient response of this control method is analyzed using small-signal and large-signal models. Furthermore, to achieve optimal dynamic performance, this improved VSG control introduces the optimal damping ratio and presents a parameter design method. Finally, the correctness of the proposed method and its theoretical analysis is verified through experimental results.

How stakeholders influence MaaS implementation? An analysis based on evolutionary game theory

Mobility as a service (MaaS) is an emerging mobility service that has received high expectations from the transportation sector in recent years. MaaS integrates multiple transportation services into a single platform, enhancing the accessibility of various transportation resources. However, the complex integration involved in MaaS leads to uncertainty in stakeholder interaction and system implementation. For the successful deployment and development of MaaS, this study aims to identify stakeholder enablers and barriers as well as to solve the uncertainty in MaaS implementation. We construct a tripartite evolutionary game model among government, transportation service providers (TSPs) and travelers. By synthesizing the evolutionary stability of each participant through simulation, we explored the impact of factors on stakeholder strategy choices and further analyzed the stability of equilibrium points in the system. The results show that: 1) Government subsidies and penalties, TSPs' costs and revenues, travelers' utility and expenses all impact the strategy choices of stakeholders. 2) The costs of government policy-making and TSPs will influence the evolution of MaaS system into two different development modes: free-market model and policy-supported model. 3) Travelers’ utilities play a crucial role in promoting the stability of travelers' adoption of MaaS. This study provides new insights into identifying stakeholder demands. The findings contribute to formulating specific policy recommendations and incentive measurements to help in the strengthen and forward movement of MaaS development in the future.

Exploring the complex dynamics of a diffusive epidemic model: Stability and bifurcation analysis.

The recent pandemic has highlighted the need to understand how we resist infections and their causes, which may differ from the ways we often think about treating epidemic diseases. The current study presents an improved version of the susceptible-infected-recovered (SIR) epidemic model, to better comprehend the community's overall dynamics of diseases, involving numerous infectious agents. The model deals with a non-monotone incidence rate that exhibits psychological or inhibitory influence and a saturation treatment rate. It has been identified that depending on the measure of medical resources and the effectiveness of their supply, the model exposes both forward and backward bifurcations where two endemic equilibria coexist with infection-free equilibrium. The model also experiences local and global bifurcations of codimension two, including saddle-node, Hopf, and Bogdanov-Takens bifurcations. Additionally, the stability of equilibrium points is investigated. For a spatially extended SIR model system, we have shown that cross-diffusion allows S and I populations to coexist in a habitat. Also, the Turing instability requirements and Turing bifurcation regime are derived. The relationship between distinct role-playing model parameters and various pattern formations like spot and stripe patterns is validated by carrying out in-depth numerical simulations. The findings in the vicinity of the endemic equilibrium solution demonstrate the significance of positive and negative valued cross-diffusion coefficients in regulating the genesis of spatial patterns in susceptible as well as diseased individuals. The discussion of the findings of epidemiological ramifications concludes the manuscript.

Finite-time stability of fractional-order nonlinear systems.

The paper studies the finite-time stability (FTS) of equilibrium points (EPs) in fractional-order nonlinear systems (FONSs). Classifying the EPs into initial EP and finite-time EP innovatively, equations of the EPs in FONSs are investigated comprehensively, and a unified definition depicting the EPs in the FONSs is proposed. Furthermore, sufficient conditions of the FTS of EPs in FONSs are given. The proposed results are verified with an illustrative example. Thus, different from existing works that declare non-existence of finite-time stable equilibria in FONSs, the existence of finite-time stable EPs in FONSs is confirmed in this paper.

Dynamics of opinion polarization in a population

This article addresses the polarization of the population around an idea and proposes a simple model that describes its dynamics in a society characterized by asymmetries in freedom of expression. The model considers a population divided into followers of an idea, consisting of moderate sympathizers and staunch defenders, and a group of opponents who try to spread their position but are also susceptible to opinion change. An analysis of stability of the proposed system of differential equations is conducted to examine policies that prevent homogenization around the idea. The results reveal conditionally stable equilibrium points that represent the coexistence of opinions and the extinction of polarization. Two threshold values are proposed to determine the persistence of polarization over time. Furthermore, the results of the model are validated through the analysis of two real cases of polarization in Colombian society, demonstrating its ability to reproduce the general behavior of opinion divergence and its utility in polarization control.

Attitude tracking of rigid bodies using exponential coordinates and a disturbance observer

In this paper, we study the problem of attitude trajectory tracking of rigid bodies subject to external disturbances. The attitude control problem has been studied in the last decades and novel solutions have been proposed. Nevertheless, most of the proposed controllers only achieve local or almost global asymptotic stability without considering external disturbances. Contrary to other works, we focus on the problem of achieving almost global exponential stability of the attitude tracking errors in the presence of external disturbances using continuous control laws. We propose an attitude trajectory tracking controller based on the exponential coordinates of rotation in combination with a disturbance observer that estimates the exogenous signals. To solve this problem, we adopt a hierarchical approach, where the proposed control law is divided into a kinematic controller (outer control loop) and a velocity tracking controller (inner control loop). To design the disturbance observer, it is assumed that exogenous signals can be generated by a linear exosystem, i.e. the magnitude and phase of the disturbance are unknown. The almost global exponential stability of the closed-loop dynamics' equilibrium point was proved by a strict Lyapunov function. The performance of the proposed approach is assessed by numerical simulations and experimental tests on a low-cost quadrotor.

μ-stability and instability of multiple equilibrium points in delayed neural networks with general discontinuous activation functions

This paper proposes a general type of discontinuous activation functions (AFs) and studies the μ-stability of multiple equilibrium points (EPs) in delayed neural networks (DNNs). By judging 4n algebraic inequalities and the nonsingularity of an M-matrix, DNNs with the general discontinuous AFs can be shown to have 5n EPs, therein 3n EPs are locally μ-stable. Moreover, these 3n EPs are located at the points of continuity of the AF. Compared with the existing general continuous AF, DNNs with the general discontinuous AF can have more total/stable EPs. Hence, DNNs with the general discontinuous AF could store a much larger number of memory patterns if they are applied to associative memory. In addition, the attraction basin (AB) of each stable EP in DNNs is estimated. Two numerical examples are shown to testify the validity of the obtained results.

Global stability analysis of two strains epidemic model with imperfect vaccination and immunity waning in a complex network

For some epidemic diseases, there exist several strains of the pathogen, which can pose a challenge in controlling the disease and result in complicated dynamics, such as dengue fever and HIV. However, we lack the deep understanding of the dynamic features of interacting multiple strains. In order to reduce the final size of infections and achieve the herd immunity, vaccination is the most effective way of controlling an outbreak. To this end, we put forward the two-strain epidemic model in which vaccination is not perfect and immunity wanes in a complex network. It can be revealed that the disease free equilibrium point of two strains is global asymptotically stable when max{R1,R1}<1, but the coexistence equilibrium point remains and is stable when min{R1,R1}>1. Then, to verify the global stability of each strain dominance equilibrium point, the critical values are further derived. Finally, numerical simulations have been performed to visualize the results of the theoretical analysis. The current results are beneficial to comprehending the dynamics behaviors of multi-strain epidemics.

Optimising Express Packaging Recycling: A Tripartite Evolutionary Game Modelling Government Strategies under Extended Producer Responsibility

This study presents a tripartite evolutionary game model examining the interplay among government, producers, and recyclers within the express packaging recycling network. Emphasizing the government's role as an endogenous variable under the extended producer responsibility framework, our study elucidates the behavioural evolution and stability of network subjects under distinct government mechanisms. Leveraging MATLAB simulations, we analyse subject behaviours to validate the model, revealing insightful contrasts between static and dynamic subsidy mechanisms. Our findings highlight that in the context of a static subsidy mechanism, equilibrium among the three subjects remains elusive. Recyclers' disposal fees show an inverse correlation with subsidy limits, yet an enhanced recycling rate directly corresponds to subsidies. However, excessive subsidies risk triggering adverse government regulations. Thus, we advocate for refined punitive policies alongside judicious, moderate subsidy systems to optimize recovery rates and social welfare. Conversely, a dynamic subsidy mechanism drives all subjects towards a stable equilibrium point, demonstrating optimal incentive effects. This study underscores the importance of government policy refinement and moderation in subsidy systems for achieving sustainable express packaging recycling, emphasizing the delicate balance between incentivization and regulatory intervention within the recycling network.

A double time-delay Holling Ⅱ predation model with weak Allee effect and age-structure

&lt;abstract&gt;&lt;p&gt;A double-time-delay Holling Ⅱ predator model with weak Allee effect and age structure was studied in this paper. First, the model was converted into an abstract Cauchy problem. We also discussed the well-posedness of the model and the existence of the equilibrium solution. We analyzed the global stability of boundary equilibrium points, the local stability of positive equilibrium points, and the conditions of the Hopf bifurcation for the system. The conclusion was verified by numerical simulation.&lt;/p&gt;&lt;/abstract&gt;

Quantitative analysis of a fractional order of the $ SEI_{c}\, I_{\eta} VR $ epidemic model with vaccination strategy

&lt;abstract&gt;&lt;p&gt;This work focused on studying the effect of vaccination rate $ \kappa $ on reducing the outbreak of infectious diseases, especially if the infected individuals do not have any symptoms. We employed the fractional order derivative in this study since it has a high degree of accuracy. Recently, a lot of scientists have been interested in fractional-order models. It is considered a modern direction in the mathematical modeling of epidemiology systems. Therefore, a fractional order of the SEIR epidemic model with two types of infected groups and vaccination strategy was formulated and investigated in this paper. The proposed model includes the following classes: susceptible $ \mathrm{S}(t) $, exposed $ \mathrm{E}(t) $, asymptomatic infected $ \mathrm{I_{c}}(t) $, symptomatic infected $ \mathrm{I_{\eta}}(t) $, vaccinated $ \mathrm{V}(t) $, and recovered $ \mathrm{R}(t) $. We began our study by creating the existence, non-negativity, and boundedness of the solutions of the proposed model. Moreover, we established the basic reproduction number $ \mathcal{R}_{0} $, that was used to examine the existence and stability of the equilibrium points for the presented model. By creating appropriate Lyapunov functions, we proved the global stability of the free-disease equilibrium point and endemic equilibrium point. We concluded that the free-disease equilibrium point is globally asymptotically stable (GAS) when $ \mathcal{R}_{0}\, \leq \, 1 $, while the endemic equilibrium point is GAS if $ \mathcal{R}_{0} &amp;gt; 1 $. Therefore, we indicated the increasing vaccination rate $ \kappa $ leads to reducing $ \mathcal{R}_0 $. These findings confirm the important role of vaccination rate $ \kappa $ in fighting the spread of infectious diseases. Moreover, the numerical simulations were introduced to validate theoretical results that are given in this work by applying the predictor-corrector PECE method of Adams-Bashforth-Moulton. Further more, the impact of the vaccination rate $ \kappa $ was explored numerically and we found that, as $ \kappa $ increases, the $ \mathcal{R}_{0} $ is decreased. This means the vaccine can be useful in reducing the spread of infectious diseases.&lt;/p&gt;&lt;/abstract&gt;

A Mathematical Model for the Dynamics of Onchocerciasis With Vector Control and Mass Drug Administration

In this paper, we investigate the transmission dynamics of onchocerciasis with asymptomatic infected humans using a mathematical model. The model incorporates interventions for treatment and vector control to evaluate the impact of these strategies. We analyse the model to determine the existence and stability of equilibrium points. Our results reveal that for the disease to persist in the community, the infection rate must exceed the sum of the treatment rate and the per capita death rate due to the disease. Sensitivity analysis highlights the critical role of the blackfly vector’s average daily biting rate in disease transmission. Numerical simulations indicate that administering highly effective drugs to infected individuals significantly reduces the number of cases. Therefore, in addition to vector control, the use of highly efficient drugs is crucial for controlling the transmission of river blindness.

Multistability and Robustness of Competitive Neural Networks With Time-Varying Delays.

This article is devoted to analyzing the multistability and robustness of competitive neural networks (NNs) with time-varying delays. Based on the geometrical structure of activation functions, some sufficient conditions are proposed to ascertain the coexistence of ∏i=1n(2Ri+1) equilibrium points, ∏i=1n(Ri+1) of them are locally exponentially stable, where n represents a dimension of system and Ri is the parameter related to activation functions. The derived stability results not only involve exponential stability but also include power stability and logarithmical stability. In addition, the robustness of ∏i=1n(Ri+1) stable equilibrium points is discussed in the presence of perturbations. Compared with previous papers, the conclusions proposed in this article are easy to verify and enrich the existing stability theories of competitive NNs. Finally, numerical examples are provided to support theoretical results.

Tritrophic fractional model with Holling III functional response

&lt;abstract&gt;&lt;p&gt;In this paper, we analyzed the local stability of three species in two fractional tritrophic systems, with Caputo's fractional derivative and Holling type Ⅱ and Ⅲ functional responses, when the prey density has a linear growth. To begin, we obtained the equilibria in the first octant under certain conditions for the parameters. Subsequently, through linearization and applying the Routh-Hurwitz Criterion, we concluded that only the system with Holling type Ⅲ exhibits an asymptotically stable equilibrium point, where the fractional derivative order belongs to the interval $ (0, 1] $. Finally, we obtained the solution of the system with the Holling type Ⅲ functional response, using the multistage homotopic perturbation method, and presented an example that shows the dynamics of the solutions around the stable equilibrium point.&lt;/p&gt;&lt;/abstract&gt;

Stability analysis of solutions of certain May's host-parasitoid model by using KAM theory

&lt;abstract&gt;&lt;p&gt;We use the Kolmogorov-Arnold-Moser (KAM) theory to investigate the stability of solutions of a system of difference equations, a certain class of a generalized May's host-parasitoid model. We show the existence of the extinction, interior, and boundary equilibrium points and examine their stability. When the rate of increase of hosts is less than one, the zero equilibrium is globally asymptotically stable, which means that both populations are extinct. We thoroughly describe the dynamics of 1:1 non-isolated resonance fixed points and have used the KAM theory to determine the stability of interior equilibrium point. Also, we have conducted several numerical simulations to support our findings by using the software package Mathematica.&lt;/p&gt;&lt;/abstract&gt;

Dynamic Analysis of a Mathematical Model of the Anti-Tumor Immune Response

This study discusses the dynamic analysis, the Hopf bifurcation, and numerical simulations. The mathematical model of the anti-tumor immune response consists of three compartments namely Immature T Lymphocytes (L1), Mature T Lymphocytes (L2) and Tumor Cells (T). This research was conducted to represent the behavior between immune cells and tumor cells in the body with five γ conditions. Where γ is the intrinsic growth rate of mature T lymphocytes. This study produces R0 &gt; 1 in conditions 1 to 4 while in condition 5 produces R0 &lt; 1. The disease-free equilibrium point is stable only in condition 5, while the endemic equilibrium point is stable only in conditions 2 and 4. Hopf bifurcation occurs at the endemic equilibrium point. Numerical simulation graph in condition 1 shows that tumor cells will increase uncontrollably. In condition 2 the graph show that the endemic equilibrium point for large tumors is stable. In condition 3 the graph show that there will be a bifurcation from the endemic equilibrium point by the disturbance of the parameter value γ. In condition 4 the graph show the small tumor endemic equilibrium point is stable. Finally, in condition 5, the graph show a stable disease-free equilibrium point.