Conditions For Local Stability Research Articles

Bifurcation Analysis of a Wind Turbine Generator Drive System with Stochastic Excitation Under Both Displacement and Velocity Delayed Feedback

In this paper, a wind turbine generator drive system with stochastic excitation under both displacement and velocity delayed feedback is considered. Firstly, the center manifold method is used to approximate the delay term of the system, so that the Itô-stochastic differential equation can be obtained by random average method. Through the maximal Lyapunov exponential method, the local stochastic stability and random D-bifurcation conditions of the system are obtained. Secondly, it is verified that the increase of noise intensity and delay value induces the occurrence of random P-bifurcation of the system through Monte Carlo numerical simulations. In addition, the theoretical chaos threshold of the system is derived by the random Melnikov method. The results show that the chaos threshold decreases as the noise intensity increases, and the increase in time delay leads to a delay in the chaotic behavior of the system. Finally, the correctness and effectiveness of the chaos-theoretic analysis are verified based on the one-parameter bifurcation diagrams and the two-parameter bifurcation diagrams.

On the Local Stability of a Certain Class of Polynomial Differential System

We state and prove a condition for the local stability of a certain class of two dimensional system of polynomial differential equations. We give some examples of polynomial differential systems of equations to demonstrate that this local stability condition established for the trivial equilibrium point (0,0) is quite sharp and compare our result with the well known Lyapunov local stability criterion (Lyapunov's second method).

On the Optimal Control of Intervention Strategies for Hepatitis B Model

Hepatitis B is one of the leading causes of morbidity and mortality, affecting hundreds of millions of people worldwide. Thus, this paper focuses on three control measures as the best way to intervene against the hepatitis B viral infection. These measures are condom use, testing and treatment, and vaccination to stop the disease from spreading over a community. The model comprises seven (7) compartments that include susceptible individuals, latent individuals, acute-infected individuals, chronic-infected individuals, infected by carrier individuals, recovered individuals from the disease, and the vaccinated population. We mathematically established a nonlinear differential equation to study the dynamics of the model. The disease-free equilibrium (DFE) and endemic equilibrium (EE) are reached. The basic reproduction numbers, R 0 A , R 0 H , and R 0 C , determine the transmission of the disease and thus are gotten. We perform sensitivity analysis on the reproduction numbers to identify the factors that affect the reproduction numbers. The results of the sensitivity analysis paved a way for introducing a controlled system which was solved using Pontryagin’s maximum principle (PMP) and the optimality system got. The optimality system was then solved numerically using the forward and backward sweep approach, and graphs were generated, establishing the conditions for local and global stability of the disease-free equilibrium using the Routh-Hurwitz criterion and Castillo-Chavez approach, respectively. We also used Pontryagin’s maximum principle to determine the optimality system. The result of the analysis of the stability of the disease-free equilibrium states that hepatitis B virus can be completely wiped out if the rate of infection is kept at a number less than unity. A numerical simulation of the model was carried out and showed that hepatitis B virus transmission can best be controlled when condom use, testing and treatment, and vaccination are implemented.

An ecological‐economic fishery model: Maximizing the societal benefit through an integrated approach of fishing and ecotourism

One of the essential tasks of the modern fishery is to integrate fishery‐based tourism with it as a part of sustainable development. Here, we propose and study a two‐species fishery model where the ecological and economic concepts are integrated. The objectives are to demonstrate the effect of fishing tax and tourist entrance fees to stabilize fishery dynamics and maximize revenue generation. For this, we consider a predator–prey fishery model coupled with dynamic harvesting and time‐dependent fish price variation. The prey fish is commercially harvested, and the predatory fish is used for recreation purposes of the visitors with an entry fee. A fishing tax is levied on the fishers to restrict the overfishing of this renewable resource. We provide the local and global stability conditions of different equilibrium points of the system and unveil the broader dynamics through bifurcation analysis. Pontryagin's maximum principle shows the existence of an optimal fishing tax that maximizes overall revenue generation.

Right fractional calculus to inverse-time chaotic maps and asymptotic stability analysis

Inverse or terminal value problems of fractional differential equations become popular recently. But memory effects or non-locality of fractional operators cause many difficulties for theoretical analysis. This study suggests a right fractional calculus method for inverse problem modelling and proposes a concept of inverse-time fractional chaotic maps. First, a simple right fractional linear differential equation's terminal value problem and the solutions are investigated. Then, some basics of the right discrete fractional calculus are introduced and the idea is extended to the discrete case. Right fractional sum equations are derived and numerical schemes are provided for dynamical analysis. Discrete chaos does exist in an inverse-time fractional logistic map and Hénon map, respectively. Their local stability conditions are given. It can be concluded that this right fractional calculus method is simpler but more efficient than the left one.

Adaptive slip vectoring for speed and yaw-rate control in electric vehicles with four in-wheel motors

A non model-based control architecture, called adaptive slip vectoring, is presented to regulate speed and yaw-rate in a fully electric vehicle equipped with four in-wheel induction motors. The actuators redundancy is dealt with by a Slip Vectoring strategy which, given a driving force and a driving yaw-moment as control constraints, allocates optimal slip references to each tire, so that four speed references are provided to the four in-wheel motors on the basis of vehicle accelerations, speed and yaw-rate regulation errors. In order to drive independently each in-wheel motor speed to its reference, a decentralized feedback control loop is designed which leads to exponentially convergent load torque estimates so that the torque–slip operating conditions can be monitored online and the load torque requirements are matched for each tire. As a result four decentralized current-pair reference inputs are provided to each motor. Structural sufficient conditions for local exponential stability of the speed and yaw-rate regulation are obtained for uniform rectilinear references. A new overall stability index is introduced and monitored online, and an automatic longitudinal speed reduction can be performed with the goal of keeping such a stability index within a predetermined range so that the structural sufficient conditions are met. Realistic CarSim simulations are presented: (i) a snowy uphill path illustrates the non model-based speed reduction driven by the stability index monitoring; (ii) standard moose-test maneuvers on dry and wet asphalt illustrate uniform yaw-rate tracking performance due to the continuous differential actions of the four in-wheel torques.

Interlayer antisynchronization in degree-biased duplex networks.

With synchronization being one of nature's most ubiquitous collective behaviors, the field of network synchronization has experienced tremendous growth, leading to significant theoretical developments. However, most previous studies consider uniform connection weights and undirected networks with positive coupling. In the present article, we incorporate the asymmetry in a two-layer multiplex network by assigning the ratio of the adjacent nodes' degrees as the weights to the intralayer edges. Despite the presence of degree-biased weighting mechanism and attractive-repulsive coupling strengths, we are able to find the necessary conditions for intralayer synchronization and interlayer antisynchronization and test whether these two macroscopic states can withstand demultiplexing in a network. During the occurrence of these two states, we analytically calculate the oscillator's amplitude. In addition to deriving the local stability conditions for interlayer antisynchronization via the master stability function approach, we also construct a suitable Lyapunov function to determine a sufficient condition for global stability. We provide numerical evidence to show the necessity of negative interlayer coupling strength for the occurrence of antisynchronization, and such repulsive interlayer coupling coefficients cannot destroy intralayer synchronization.

Spatiotemporal Tipping Induced by Turing Instability and Hopf Bifurcation in a Population Ecosystem Model with the Fear Factor

Population ecosystems can display the tipping points at which extinctions of species happens. To predict the appearance of tipping points and to understand their evolution mechanism are of uttermost importance for ecological balance. Using techniques from bifurcation theory, we can predict the emergence of tipping points based on a spatiotemporal predator-prey system having a fear effect before an instability is encountered. In the case of no time delay, the tipping induced by Turing instability is studied. The conditions for Turing instability and local asymptotic stability of coexisting equilibrium points are given. It is ascertained that the introduction of diffusion causes the ecosystem to change from stable to unstable, and then the tipping occurs. Then, we investigate another tipping due to Hopf bifurcation. The delay-dependent stability criterion and Hopf bifurcation condition are derived, and the onset of Hopf bifurcations (tipping point) is also determined. In order to further probe into the mechanism of tipping evolution, explicit formulae are derived to ascertain the stability of bifurcated oscillations and the direction of bifurcation via the center manifold reduction. It is revealed that many tipping points may exist in ecological competition systems, and the tipping occurs many times as the fear delay increases. Finally, several simulation examples are provided to substantiate the analytical results.

An adaptive time stepping scheme for rate-independent systems with nonconvex energy

Abstract We investigate a local incremental stationary scheme for the numerical solution of rate-independent systems. Such systems are characterized by a (possibly) nonconvex energy and a dissipation potential, which is positively homogeneous of degree one. Due to the nonconvexity of the energy, the system does in general not admit a time-continuous solution. In order to resolve these potential discontinuities, the algorithm produces a sequence of state variables and physical time points as functions of a curve parameter. The main novelty of our approach in comparison to existing methods is an adaptive choice of the step size for the update of the curve parameter, depending on a prescribed tolerance for the residua in the energy-dissipation balance, and in a complementarity relation concerning the so-called local stability condition. It is proven that, for tolerance tending to zero, the piecewise affine approximations generated by the algorithm converge (weakly) to a so-called parametrized balanced viscosity solution. Numerical experiments illustrate the theoretical findings, and show that an adaptive choice of the step size indeed pays off as they lead to a significant increase of the step size during sticking and in viscous jumps.

Stability Analysis and Control Strategy of a Generalized Food Chain Model

The stability analysis and control strategy for a generalized food chain model is considered in this paper. Firstly, we investigate the existence of the positive equilibrium and give the conditions of local asymptotic stability. Meanwhile, we design a feedback controller and prove the stability of the proposed model with the controller by using the Lyapunov function. Due to the addition of control, the unstable food chain system can become stable. At last, the analysis results are verified by numerical simulations. Our results may help to maintain the stability of the biological systems and avoid ecosystem imbalance.

Analytical bifurcation behaviors of a host–parasitoid model with Holling type III functional response

This topic presents a study on a host–parasitoid model with a Holling type III functional response. In population dynamics, when host density rises, the parasitoid response initially accelerates due to the parasitoid’s improved searching efficiency. However, above a certain density threshold, the parasitoid response will reach a saturation level due to the influence of reducing the handling time. Thus, we incorporated a Holling type III functional response into the model to characterize such a phenomenon. The dynamics of the current model are discussed in this paper. We first obtained the existence and local stability conditions of the positive fixed point of the model. Furthermore, we investigated the bifurcation behaviors at the positive fixed point. More specifically, we used bifurcation theory and the center manifold theorem to prove that the model possess both period doubling and Neimark–Sacker bifurcations. Then, the chaotic behavior of the model, in the sense of Marotto, is proven. Furthermore, we apply a state-delayed feedback control strategy to control the complex dynamics of the present model. Finally, numerical examples are provided to support our analytic results.

Mathematical Model for Crimes in Developing Countries with Some Control Strategies

Crime is one among the most challenging problems in most developing countries in which unemployment is among the causes. Not all kind of crimes can be eradicated indeed; this paper is intended to contribute on eradication of unemployment-related crimes in the developing countries by proposing a deterministic mathematical model of unemployment-crime dynamics including vocational training and employment as control measures for crime. The study adopts the epidemiological model concepts on model formulation and model analysis while considering unemployment as main driver of crime. The basic properties of the model are analyzed, and well-posed of the model is established by using the Lipschitz condition. The next-generation matrix is used to obtain the criminal reproduction number which help to derive the conditions for local and global stability of the model. Moreover, the existence of backward and forward bifurcation when the crime reproduction number is equal to one was analyzed by center manifold theory. Simulations of the model are carried out to validate the theoretical part of the model and demonstrate vocational training, and employment strategies are more effective in combating crime when applied simultaneously. The findings suggest that unemployment problem should be addressed in order to reduce the number of unemployed individuals in joining the criminal activities.

Mathematical analysis of a competition model with mutualism

We perform the mathematical analysis of a model describing the interaction of two species in a chemostat, involving competition and mutualism, simultaneously. The model is a five‐dimensional system of differential equations with nonlinear growth functions. We give a comprehensive description of the dynamics of the system by determining analytically the existence and the local stability conditions of all steady‐states, considering a large class of growth rates. We prove that there exists a unique stable coexistence steady‐state and give the conditions under which bistability can occur. We give bifurcation diagrams and operating diagrams showing the rich behavior of the system.

Mathematical Modelling of the Spatial Distribution of a COVID-19 Outbreak with Vaccination Using Diffusion Equation.

The formulation of mathematical models using differential equations has become crucial in predicting the evolution of viral diseases in a population in order to take preventive and curative measures. In December 2019, a novel variety of Coronavirus (SARS-CoV-2) was identified in Wuhan, Hubei Province, China, which causes a severe and potentially fatal respiratory syndrome. Since then, it has been declared a pandemic by the World Health Organization and has spread around the globe. A reaction−diffusion system is a mathematical model that describes the evolution of a phenomenon subjected to two processes: a reaction process, in which different substances are transformed, and a diffusion process, which causes their distribution in space. This article provides a mathematical study of the Susceptible, Exposed, Infected, Recovered, and Vaccinated population model of the COVID-19 pandemic using the bias of reaction−diffusion equations. Both local and global asymptotic stability conditions for the equilibria were determined using a Lyapunov function, and the nature of the stability was determined using the Routh−Hurwitz criterion. Furthermore, we consider the conditions for the existence and uniqueness of the model solution and show the spatial distribution of the model compartments when the basic reproduction rate R0<1 and R0>1. Thereafter, we conducted a sensitivity analysis to determine the most sensitive parameters in the proposed model. We demonstrate the model’s effectiveness by performing numerical simulations and investigating the impact of vaccination, together with the significance of spatial distribution parameters in the spread of COVID-19. The findings indicate that reducing contact with an infected person and increasing the proportion of susceptible people who receive high-efficacy vaccination will lessen the burden of COVID-19 in the population. Therefore, we offer to the public health policymakers a better understanding of COVID-19 management.

Dynamical behaviours of discrete amensalism system with fear effects on first species.

Amensalism, a rare yet impactful symbiotic relationship in ecological systems, is the focus of this study. We examine a discrete-time amensalism system by incorporating the fear effect on the first species. We identify the plausible equilibrium points and analyze their local stability conditions. The global attractivity of the positive equilibrium, $ E^* $, and the boundary equilibrium, $ E_1 $, are analyzed by exploring threshold conditions linked to the level of fear. Additionally, we analyze transcritical bifurcations and flip bifurcations exhibited by the boundary equilibrium points analytically. Considering some biologically feasible parameter values, we conduct extensive numerical simulations. From numerical simulations, it is observed that the level of fear has a stabilizing effect on the system dynamics when it increases. It eventually accelerates the extinction process for the first species as the level of fear continues to increase. These findings highlight the complex interplay between external factors and intrinsic system dynamics, enriching potential mechanisms for driving species changes and extinction events.

Stability Conditions for Cluster Synchronization in Directed Networks of Diffusively Coupled Nonlinear Systems

This paper investigates the stability issue of cluster synchronization manifold in networks of diffusively-coupled nonlinear system with directed topology. It is assumed that each cluster subdigraph is strongly connected and contains only cooperative interactions, but there may exist competitions among nodes of different clusters. In the case that the clusters meet the cluster-input-equivalence condition and each nonlinear system has a forward invariant set over which the system possesses a bounded Jacobian, some local stability conditions of cluster synchronization are derived. These conditions are expressed as an inequality of the matrix measure of the system’s Jacobian matrix, the matrix measure of a reduced-order matrix about the digraph among the clusters, the coupling strength, and some eigenvalue conditions of the subdigraphs. Finally, the theoretical findings are validated by two numerical examples about coupled Lorenz-like system and coupled Hopfield neural network, respectively.

The effect of migration on transmission of Wolbachia in Nilaparvata lugens.

Brown planthopper Nilaparvata lugens, which can transmit rice ragged stunt virus, is a serious and damaging pest to rice plants. Rice plants can protect themselves from the associated diseases of N.lugens by either suppressing or replacing N.lugens by releasing N.lugens infected by a special strain of Wolbachia wStri. The long-distance migration habit of N.lugens is one of the important precursors leading up to the large-scale occurrence of N.lugens. To study the effect of migration on the transmission of Wolbachia in N.lugens, a Wolbachia spreading dynamics model with migration of N.lugens between two patches is put forward. The existence and local stability conditions of equilibrium points of the system and its subsystems are obtained. Moreover, the effects of migration on the dynamic properties and the control of N.lugens are analyzed; the results show that the system can exhibit a bistable phenomenon, and the migration can change the stability of equilibrium infected with wStri from stable to unstable. The quantitative control methods for the migration of the insect N.lugens are proposed, which provide a theoretical guidance for future field experiments. Lastly, we use the Markov chain Monte Carlo (MCMC) method to estimate the parameters of the wild N.lugens migration model based on limited observational data; the numerical simulation results show that migration can increase the quantity of N.lugens, which is consistent with the relevant experimental results.

Wolbachia infected and uninfected mosquito growth model using the Leslie multispecies matrix model

Currently, there is a new strategy for stopping the spread of dengue, namely infecting wild mosquitoes with Wolbachia bacteria. Wolbachia is a bacteria that can lower the rate of infection in a mosquito's body, which lowers the risk of the virus being transmitted when it bites susceptible people. Our aim in this paper is to find the equilibrium points and perform local stability analysis for each equilibrium point of a growth model of uninfected and Wolbachia-infected Aedes aegypti mosquitoes. The growth model of the two types of mosquitoes was modeled using the Leslie multispecies matrix model. We assume that the first species is Wolbachia-uninfected mosquitoes and the second species is Wolbachia-infected mosquitoes. In this study, we obtained four equilibrium points. Then, we obtain asymptotically local stability conditions at the four equilibrium points. Based on these results, this study provides conditions that guarantee that efforts to use Wolbachia can suppress dengue disease.

Ultimate Dynamics Analysis of the 5D Structural Leukemia Model and Partitioning of the Parameter Space

In this paper, we study the global dynamics of the 5D structural leukemia model with 14 parameters as developed by Clapp et al. [2015]. This model describes the interaction between leukemic cell populations and the immune system. Our analysis is based on the localization method of compact invariant sets. We develop this method by introducing the notion of a partitioning of the parameter space and the notion of a localization set corresponding to this partitioning as its parameters change. Further, we obtain ultimate upper and lower bounds for all variables of a state vector without imposing additional restrictions. Local asymptotic stability conditions with respect to the leukemia-free equilibrium point (EP) are given. We deduce formulas describing inner EPs expressed in terms of positive roots of one 7D equation. Based on this equation, it is shown that the number of inner EPs cannot exceed 3 and one case of a global bifurcation of EPs is detected. Next, we prove the existence of the attracting set. Further, in two theorems, the global eradication/extinction leukemia theorems are described. The impact of using parametrically variable localization sets for a qualitative analysis of the ultimate behavior of leukemic cell populations is discussed.

Gravitational form factors of the baryon octet with flavor SU(3) symmetry breaking

We investigate the gravitational form factors of the baryon octet within the framework of the SU(3) chiral quark-soliton model, considering the effects of flavor SU(3) symmetry breaking, and with the corresponding energy-momentum tensor distributions. We examine the effects of flavor SU(3) symmetry breaking to the mass, angular momentum, pressure, and shear force distributions of the baryon octet. We first find that a heavier baryon is energetically more compact than a lighter one. For the spin distributions of the baryon octet, they are properly normalized to their spins and are decomposed into the flavor-singlet axial charge and the orbital angular momentum even when the flavor SU(3) symmetry is broken. While the effects of the flavor SU(3) symmetry breaking differently contribute to the angular momentum distributions for the octet baryons, they are found to be rather small. The spin and orbital angular momentum almost equally contribute to the angular momentum distributions for the octet baryons. We also estimate the effects of the flavor SU(3) symmetry breaking to the pressure and shear force distributions. Interestingly, even if we include the effects of the SU(3) flavor symmetry breaking, then the shear force distributions are kept to be positive over $r$. It indicates that the Polyakov and Schweitzer local stability condition is kept to be intact with the flavor SU(3) symmetry broken. Lastly, we discuss how much the gravitational form factors vary with the effects of flavor SU(3) symmetry breaking considered.