Routh-Hurwitz Criterion Research Articles

Hopf Bifurcation, Approximate Periodic Solutions and Multistability of Some Nonautonomous Delayed Differential Equations

Research on nonautonomous delayed differential equations (DDEs) is crucial and very difficult due to nonautonomy and time delay in many fields. The main work of the present paper is to discuss complex dynamics of nonautonomous DDEs, such as Hopf bifurcation, periodic solutions and multistability. We consider three examples of nonautonomous DDEs with time-varying coefficients: a harmonically forced Duffing oscillator with time delayed state feedback and periodic disturbance, generalized van der Pol oscillator with delayed displacement difference feedback and periodic disturbance, and an electro-mechanical system with delayed and periodic disturbance. Firstly, we obtain the amplitude equations of these three examples by the method of multiple scales (MMS), and then analyze the stability of approximate solutions by the Routh–Hurwitz criterion. The obtained amplitude equations are used to construct the bifurcation diagrams, so that we can observe the occurrence of the Hopf bifurcation and judge its type (super- or subcritical) from the bifurcation diagrams. We discover rich dynamic phenomena of the three systems under consideration, such as Hopf bifurcation, quasi-periodic solutions and the coexistence of multiple stable solutions, and then discuss the impact of some parameter changes on the system dynamics. Finally, we validate the correctness of these theoretical conclusions by software WinPP, and the numerical simulations are consistent with our theoretical findings. Therefore, the MMS we use to analyze the dynamics of nonautonomous DDEs is effective, which is of great significance to the research of nonautonomous DDEs in many fields.

Mathematical investigation of a two-strain disease model with double dose vaccination control policies

Antimicrobial resistance (AMR) is the microorganism's resistance to an antimicrobial drug developed and designed to handle an infection. A recent study showed that there were 1.27 million global deaths in 2019 attributable to AMR. In current decades, vaccination is the most effective treatment that can reduce the growing AMR by preventing the incidence of infectious diseases (directly or indirectly). Vaccination has become a popular treatment as this can reduce the number of circulating AMR strains and reduce the need for antimicrobial use. With a double-dose vaccination campaign, we analyzed drug-susceptible and drug-resistant pathogens for this survey. We considered the circumstances of positive invariances and boundedness with appropriate primary conditions. We have determined two basic reproduction numbers: one associated with drug-susceptible strain (R01) and the other with drug-resistant strain (R02). We also revealed that at any stage, one of the strain could spread in a population if max[R01, R02] > 1 and the disease fade-out if both are <1 (i.e., max[R01, R02] < 1). In this study, We evaluated that our model contains four equilibrium points: the disease-free equilibrium, mono-existent endemic equilibrium 1, mono-existent endemic equilibrium 2, and a disease-endemic equilibrium. In particular, we applied the Routh-Hurwitz criteria to generalize the stability of the equilibria. We also found that the drug-resistant strain prevalence increases when the drug-susceptible strain is vaccinated on time due to the poor quality of the vaccine. We carried out the sensitivity analysis through the partial rank correlation coefficient (PRCC) technique to identify most important parameters. We found that the transmission rate of both strains had the biggest stimulus on disease outbreak. We also demonstrate how vaccination rates affected the equilibrium level of prevalence and found that a proper vaccination program is more effective for reducing two-strain disease burden. Numerical simulations carried out using MATLAB routines to support the analytical conclusions.

Disturbance rejection control of shaking table based on adaptive-model-based state estimator

Real-time hybrid simulation (RTHS) is an important method for the seismic testing of building structures. This method requires high control accuracy for the waveform reproduction of a shaking table. An active disturbance rejection control (ADRC) method is introduced to control the shaking table in this research. In view of the difficulty of parameters setting caused by the large deviation of the extended state observer (ESO) to the state estimation of the shaking table, a model-based adaptive state estimator (AMSE) is designed that uses the model information to estimate the state, and introduces an adaptive strategy to dynamically compensate for the amplitude and phase deviation of the model. The total disturbance is observed by the ESO. Additionally, an AMSE-based ADRC is proposed to achieve high-precision seismic waveform reproduction. The separation principle of the designed system is proved by the Routh–Hurwitz criterion, and the pole assignment bandwidth method is proposed to improve the control performance of the system by flexibly assigning the controller poles. Finally, the shaking table experiments are designed for different specimens. These experiments verify the advantages of the proposed method for state estimation and parameter setting and can effectively improve the stability and control accuracy of shaking table control.

Local and global stability of an HCV viral dynamics model with two routes of infection and adaptive immunity

The aim of this article is to formulate and study a mathematical model describing hepatitis C virus (HCV) infection dynamics. The model includes two essential modes of infection transmission, namely, virus-to-cell and cell-to-cell. The effect of therapy and adaptive immunity are incorporated in the suggested model. The adaptive immunity is represented by its two categories, namely, the humoral and cellular immune responses. Our article begins by establishing some mathematical results through proving the model’s well-posedness in terms of existence, positivity and boundedness of solutions. We present all the steady states of the problem that depend on specific reproduction numbers. It moves then to the theoretical investigation of the local and global stability analysis of the free disease equilibrium and the four disease equilibria. The local and global stability analysis of the HCV mathematical model are established via the Routh-Hurwitz criteria and Lyapunov-LaSalle invariance principle, respectively. Finally, our article presents some numerical simulations to validate the analytical study of the global stability. Numerical simulations have shown the effect of the drug therapies on the system’s dynamical behavior.

Local Stability Analysis of Onchocerciasis Transmission Dynamics With Nonlinear Incidence Functions in Two Interacting Populations

A deterministic compartmental model for the transmission dynamics of onchocerciasis with nonlinear incidence functions in two interacting populations is studied. The model is qualitatively analyzed to investigate its local asymptotic behavior with respect to disease-free and endemic equilibria. It is shown, using Routh-Hurwitz criteria, that the disease-free equilibrium is locally asymptotically stable when the associated basic reproduction number is less than the unity. When the basic reproduction number is greater than the unity, we prove the existence of a locally asymptotically stable endemic equilibrium.

Analyzing the stability and sensitivity of the deterministic SVEQ1IQ2R model for the spread of COVID-19

Covid-19 has become a pandemic in various countries since 2019. The spread of Covid-19 is easy and fast through droplets that come out when coughing, sneezing, or talking. Several ways to overcome the spread are conducting home quarantine, medical quarantine, carrying out health protocols, and vaccinating. This study proposed a deterministic model called SVEQ1IQ2R, which considers home and medical quarantine and vaccination. The model divides the population into seven compartments, namely susceptible (S), exposed (E), vaccinated (V), home quarantine (Q1), infected (I), medical quarantine (Q2), and recovered (R). This study aims to analyze the stability of equilibrium points and the sensitivity of the proposed model. The stability analysis results show that disease-free and endemic equilibrium points are locally asymptotically stable using Routh-Hurwitz criteria and Castillo-Chavez and Song theorems, respectively. Moreover, the sensitivity analysis shows that reducing the contact rate and increasing vaccine effectiveness are the most appropriate steps to take to reduce the spread of Covid-19. Based on numerical simulations, increasing the vaccine effectiveness and the vaccination rate as well as reducing the contact rate can make the transmission of Covid-19 resolved more quickly.

Dinamika dan Stabilitas Populasi pada Model Penyebaran COVID-19 dengan Vaksinasi dan Migrasi Penduduk

This article discusses the mathematical model of the spread of COVID-19 by considering vaccination and population migration. The former model is analyzed by determining the equilibrium point, basic reproduction number, analyzing the stability of the equilibrium point, sensitivity analysis, and accompanied by numerical simulation. Analysis of the stability of disease-free and endemic equilibrium points using the Routh-Hurwitz Criteria and the Castillo-Chaves and Song theorems. The results of the analysis show that there are two equilibrium points, namely a disease-free equilibrium point (T1), which is locally asymptotically stable when R0 1, and an endemic equilibrium point (T2), which is locally asymptotically stable when R0 1. Furthermore, the sensitivity analysis showed that the most sensitive parameters to changes in the basic reproduction number were the emigration rate parameter (m2) and the infection probability parameter after contact between infected and susceptible individuals without vaccination (h). In addition, the numerical simulation results show that the sensitive parameter values, namely m2, h, zse, g, and # have a significant effect on the basic reproduction numbers. Suppressing the chance of infection in susceptible individuals and the rate of contact between susceptible and exposed individuals, as well as increasing the number of individuals who emigrate and who are vaccinated, can reduce the transmission of COVID-19.

A new linear quadratic regulator model to mitigate frequency disturbances in the power system during cyber-attack

&lt;span lang="EN-US"&gt;This paper proposes a new model integrating a linear quadratic regulator (LQR) controller to mitigate frequency disturbances in the power system during cyber-attack, called as linear quadratic regulator to mitigate frequency disturbances (LQRMFD). As we know, most of the existing models have a common problem with achieving significant performances in mitigating dynamic response parameters, such as frequency deviation and settling time. However, the key aspect of LQRMFD is to mitigate the above issues with remarkable performance improvements. An uncommon and stable power system model has been considered in LQRMFD first to reach such a goal. A numerical problem has been solved to derive a certain characteristic equation, where the Routh-Hurwitz array criterion is applied for determining the stability of such a power system. After that, a state-space equation is developed from the power system to activate the LQR controller. Thus, achieving diversity and eliminating the redundancy of the power system considered can be obtained in LQRMFD. To evaluate the performance of LQRMFD, a series of experiments was conducted using the MATLAB-Simulink tool. Rigorous comparisons were also made among the results of LQRMFD, self-implemented and existing models. Furthermore, a detailed analysis was reported among those models to find the performance improvement of LQRMFD in percentage.&lt;/span&gt;

Stability analysis and control parameter optimization of multi-VSG parallel grid-connected system

The virtual synchronous generator (VSG) is emerging as an attractive solution for controlling the grid-connected inverter when the renewable energy has a high penetration level into the grid. This study delves into the stability analysis and parameter optimization of multi-VSG parallel grid-connected systems. Consequently, a small-signal model (SSM) for the multi-VSG parallel grid-connected system is developed to facilitate system analysis. Utilizing this model, the feasible parameter regions are determined by applying the Routh-Hurwitz criterion, enabling the identification of key factors contributing to the system's oscillation mode. Subsequently, a time-domain model is employed to ascertain the dynamic feasible region index, which aids in delineating the scope of parameter optimization. Furthermore, the impact of deviation angle, damping, and eigenvalues on system stability is incorporated into the construction of the objective function. To optimize system parameters within the feasible region, the simulated annealing-particle swarm optimization algorithm (SA-PSO) is utilized. The effectiveness of the theoretical analysis and the proposed parameter optimization method is demonstrated through simulations.

Analysis Dynamics Two Prey of a Predator-Prey Model with Crowley–Martin Response Function

The predator-prey model has been extensively developed in recent research. This research is an applied literature study with a proposed dynamics model using the Crowley–Martin response function, namely the development of the Beddington-DeAngelis response function. The aim of this research is to construct a mathematical model of the predator-prey model, equilibrium analysis and population trajectories analysis. The results showed that the predator-prey model produced seven non-negative equilibrium points, but only one equilibrium point was tested for stability. Stability analysis produces negative eigenvalues indicating fulfillment of the Routh-Hurwitz criteria so that the equilibrium point is locally asymtotically stable. Analysis of the stability of the equilibrium point indicates stable population growth over a long period of time. Numerical simulation is also given to see the trajectories of the population growth movement. The population growth of first prey and second prey is not much different, significant growth occurs at the beginning of the growth period, while after reaching the peak the trajectory growth slopes towards a stable point. Different growth is shown by the predator population, which grows linearly with time. The growth of predators is very significant because predators have the freedom to eat resources. Various types of trajectory patterns in ecological parameters show good results for population growth with the given assumptions.

Applying Cognitive Methodology in Designing On-Line Auto-Tuning Robust PID Controller for the Real Heating System

A novel design and implementation of a cognitive methodology for the on-line auto-tuning robust PID controller in a real heating system is presented in this paper. The aim of the proposed work is to construct a cognitive control methodology that gives optimal control signal to the heating system, which achieve the following objectives: fast and precise search efficiency in finding the on- line optimal PID controller parameters in order to find the optimal output temperature response for the heating system. The cognitive methodology (CM) consists of three engines: breeding engine based Routh-Hurwitz criterion stability, search engine based particle swarm optimization (PSO) and aggregation knowledge engine based cultural algorithm (CA). Matlab simulation package is used to carry out the proposed methodology that finds and tunes the optimal values of the robust PID parameters on-line. In real-time, the LabVIEW package is guided to design the on-line robust PID controller for the heating system. Numerical simulations and experimental results are compared with each other and showed the effectiveness of the proposed control methodology in terms of fast and smooth dynamic response for the heating system, especially when the control methodology considers the external disturbance attenuation problem.

Dynamical analysis of a class of SEIR models through delayed strategies

In recent decades, the mathematical modeling of infectious diseases, real-world problems, non-linear dynamical complex systems, etc., has increased significantly. According to World Health Organization, tobacco use is the cause of about 22% of cancer deaths. Another 10% are due to obesity, poor diet, lack of physical activity, and excessive drinking of alcohol. Approximately 5%–10% of cancers are due to inherited genetic defects. The objective is to investigate the impact of time delays in implementing control measures on the epidemic dynamics. The classification of cell population has four compartments: susceptible cells (x), cancer-infected cells (y), virus-free cells (v), and immune cells (z). Our focus is to find the equilibria of the problem and their stability. The stability of the solutions is of two types: locally asymptotic and globally asymptotic. The Routh–Hurwitz criterion, Volterra-type Lyapunov function, and LaSalle’s invariance principle are used to verify the stability of solutions. The graphical behavior depicts the stable solutions to a real-world problem and supports the stability analysis of the problem. The findings contribute to the understanding of epidemic dynamics and provide valuable information for designing and implementing effective intervention strategies in public health systems.

The Routh Hurwitz Criteria for Studying The Stability and Bifurcation in Multispiral Chua Chaotic Attractor

This article discusses the multispiral Chua Chaotic attractor’s hidden bifurcations that are generated by the sine function. The number of spirals (also known as a multiscroll attractor) that are controlled by the integer parameter c can be used to describe the basic shape of chaotic attractors. Since this parameter is an integer, increasing it by one does not allow the observation of bifurcations from n to n + 2 spirals. The method of hidden bifurcations, however, enables the observation of such bifurcations by adding a real parameter ε. Chaotic attractors with either an even or an odd number of spirals are visible along the marked paths of bifurcation. Moreover, this additional hidden parameter allows finding the bifurcation of the multispiral Chua attractor from a stable state to a chaotic state. Furthermore, the Routh-Hurwitz criteria are used to study the stability of the original equilibrium point of the Chua attractor.

The impact of standard and nonstandard finite difference schemes on HIV nonlinear dynamical model

Mathematical models are enormously valuable in recognition the characteristics of infectious afflictions. The present study describes and analyses a nonlinear Susceptible-Infected (S·I) type mathematical model for HIV/AIDS. To better comprehend the dynamics of disease diffusion, it is assumed that by giving AIDS patients timely Anti Retroviral Therapy (ART), their transition into HIV infected class is attainable. The ART treatment can reduce or manage the spread of disease among individuals that can extend their life for some more years. For the model, the basic reproduction number is formed which provides a base to study the stability of disease free and endemic equilibria. To understand the entire dynamical behavior of the model, standard finite difference (SFD) schemes such as Runge-Kutta of order four (RK-4) and forward Euler schemes and nonstandard finite difference (NSFD) scheme are implemented. The goal of constructing the NSFD scheme for differential equations is to ensure that it is dynamically reliable, while maintaining important dynamical properties like the positivity of the solutions and its convergence to equilibria of continuous model for all finite step sizes. However, the essential characteristics of the continuous model cannot be properly maintained by the Euler and RK-4 schemes, leading to the possibility of numerical solutions that are not entirely similar to those of the original model. For the NSFD scheme, the Routh-Hurwitz criterion is used to assess the local stability of disease-free and endemic equilibria. To explain the global stability of both the equilibria, Lyapunov functions are offered. To verify the theoretical findings and validate the dynamical aspects of the abovementioned schemes, numerical simulations are also provided. The outcomes offered in this study may be engaged as an effective tool for forecasting the progression of HIV/AIDS epidemic diseases.

Nonlinear parametric resonances of a rotating composite thin-walled beam under aerodynamic force and hygrothermal environment

Considering combined effects of aerodynamic force and hygrothermal environment, the nonlinear parametric resonance of a composite thin-walled beam subjected to a time-dependent rotating speed is investigated in this work. The rotating speed is presumed to comprise a small disturbance harmonically changing one with a steady one. The dynamical model of the system is obtained with the help of established dynamic model in existing research. Then the nonlinear partial differential equation of the system is discretized by the Galerkin method. Afterwards, the low-frequency and high-frequency resonances of the system are analyzed via the multiple scales method to discuss the combined effect of rotating speed, pitch angle, setting angle, fiber orientation angle, temperature-humidity and inflow ratio on the amplitude-frequency responses and the unstable region. In addition, the Routh-Hurwitz criterion is employed to determine the stability conditions of the trivial and non-trivial solutions. The results show that both sub- and super-critical pitchfork bifurcations will occur to the system in parametric resonances regime, accompanied with stable and unstable non-trivial solutions simultaneously. Particularly, the mechanism of the influence of hygrothermal environment is deeply studied, which reveals a competitive relation between the change of mechanical properties and additional stress brought by the change of temperature and humidity.

Dynamics of Covid-19 model with public awareness, quarantine, and isolation

This paper presents a new COVID-19 model that contains public awareness, quarantine, and isolation. The model includes eight compartments: susceptible aware (SA), susceptible unaware (SU), exposed (E), asymptomatic infected (A), symptomatic infected (I), recovered (R), quarantined (Q), and isolated (J). The introduction will be shown in the first section, followed by the model simulation. The equilibrium points, basic reproduction number, and stability of the equilibrium points are then determined. The model has two equilibrium points: disease-free equilibrium point and endemic equilibrium point. The next-generation matrix is used to calculate the basic reproduction number R0. The disease-free equilibrium point always exists and is locally stable if R0 1, whereas the endemic equilibrium point exists when R0 1 and is locally stable if satisfying the Routh-Hurwitz criterion. Stability properties of the equilibrium confirmed by numerical simulation also show that quarantine rate and isolation rate have an impact in the transmission of COVID-19

STABILITY ANALYSIS OF MATHEMATICAL MODEL AND OPTIMAL CONTROL STRATEGIES FOR REDUCING THE COVID-19 SPREAD

The respiratory infectious disease Coronavirus Disease 2019 caused by the SARS-CoV-2 virus has become a world pandemic. This paper aims to develop a mathematical model of the spread of Covid-19 SEIQR (Susceptible-Exposed-Infected-Quarantine-Recovered), then analyse the stability of the model and its optimal control. Based on the local stability analysis using the Routh-Hurwitz criteria obtained two equilibrium points, namely the local asymptotically stable disease-free equilibrium points if reproduction number less than one and the local asymptotically stable endemic equilibrium point if reproduction number more than one. Next, to analyze global stability at the equilibrium points is used the Lyapunov method. Further, four controls, namely vaccination, physical distancing, treatment of infected individuals, and self-prevention (wearing masks and hand sanitisers) are applied to reduce the spread of covid-19. We applied Pontryagin’s Maximum Principle to obtain the optimal solution for the control. Finally, based on numerical result is found that the value of reproduction number is 9,44183. By using data from province of East Java, Indonesia, it is obtained that the physical distancing control has a better level effectiveness than the others three controls.

Mathematical model and optimal control analysis of coffee berry borer with temperature and rainfall variability

This study focuses on a nonlinear deterministic mathematical model for coffee berry borer (Hypothenemus hampei) with temperature and rainfall variability. In the model analysis, CBB free and coexistence equilibria are computed. The basic reproduction numbers at a minimum and maximum temperature and rainfall are derived. The qualitative analysis of the model revealed the scenario for equilibria together with basic reproduction numbers. The local stability of equilibria is established through the Jacobian matrix and the Routh–Hurwitz criteria, while the global stability of equilibria is demonstrated using an appropriate Lyapunov function. The normalized sensitivity analysis has also been performed to observe the impact of different parameters on basic reproduction numbers. The proposed model is extended into an optimal control problem by incorporating two control variables, namely, the preventive measure variable based on the separation of susceptible coffee berries from contacting the pests based on biological control and an increase in the death rate of colonizing females of CBB based on chemical control. Optimal disease control analysis is examined using Pontryagin’s minimum principle. Finally, the numerical simulations are performed based on analytical results and are discussed quantitatively. Furthermore, the cost-effectiveness of control strategies to determine the best approach to minimize the CBB burden was studied. The study is significant in providing reliable information on how one can use mathematical modeling to improve the roles of control strategies and prevention in CBB transmission in a coffee farm. The outcome of the study may guide public agriculture policymakers on optimal control strategies to control the pests. In particular, using chemical pesticides is very effective to combat pests with minimum costs.

Contingent partitioning and adaptation in hydrological systems

AbstractThe question of whether the concept of adaptation can be applied to Earth surface systems (independently of biological adaptation) is addressed by examining hydrological flow systems. Hydrological systems are represented in terms of a partitioning of water inputs among various flux and storage components and outflows or outputs of the system. Partitioning is contingent on the flow system in question and the synoptic situation (i.e., drier, low‐input vs. wetter, high‐input conditions). The general allocation among inputs, flows through or within the system, storage and outputs is examined via analysis of 20 scenarios for soil hydrology, a fluvial channel‐wetland complex and a fluviokarst landscape representing different combinations of positive, negative and zero (neutral) relationships among these elements, and positive self‐reinforcing and negative self‐limiting effects. Conditions for stability were determined using the Routh–Hurwitz criteria and linked to the two fundamental roles or ‘jobs’ of hydrological flow systems. The ecological job is to support biota and biogeochemical fluxes and transformations necessary for ecosystem functions. The geophysical job is to remove excess water. Results show that low‐input scenarios for the soil, fluvial wetland and fluviokarst scenarios are marked by dynamical instability. During drier periods the geophysical job is irrelevant and the ecological functions are suboptimal. Instability allows for rapid state changes when moisture inputs increase, to system states that support ecosystem functions. High‐input, excess moisture and flood scenarios, by contrast, are generally dynamically stable. In wetter conditions, the ecological functions are not moisture‐stressed, and the geophysical job becomes paramount. The high‐input stability is associated with activation of ‘spillway’ mechanisms that allow the systems to maintain themselves by efficient export and augmented storage of excess water. Contingent partitioning indeed appears to be an adaptation mechanism in hydrological systems and suggests the possibility of adaptation in other Earth surface systems with important abiotic components.

Parametric resonance of fractional viscoelastic webs under time-dependent tension

The focus of this paper is on the parametric resonance of fractional viscoelastic webs under time-dependent tension. The novelty of this work lies in the introduction of a fractional order model for viscoelastic webs and the time-dependent tension in roll-to-roll manufacturing. The time-dependent tension is induced by the web treatment processes in roll-to-roll mass production. The rheological properties of viscoelastic webs are represented by the fractional Kelvin-Voigt relation. Considering the geometric nonlinearity, the governing equation is derived based on the Hamilton principle, and then the resulting equation is reduced to a fractional ordinary differential equation by using the Bubnov-Galerkin method. The multiple-scale method is applied to analyze the occurrence condition and dynamic responses of parametric resonance, and the solution stability is demarcated by the Routh-Hurwitz criterion. The effects of the tension variation coefficient, viscoelastic parameter and initial tension on the resonance responses are discussed. The results reveal that the increase in tension variation coefficient and initial tension not only widen the unstable zone of the response curve, but also enhance the resonance amplitude. Moreover, the resonance amplitude decreases with a larger fractional order. This investigation aims to explore the cause of the instability phenomena and thus avoid the unexpected failure of flexible fabrication.