Stability analysis of the classical car-following model

In this paper some local stability results for an autonomous Long Short-Term Memory neural network model with respect to the origin are provided. In particular, it is shown through linearization that the local asymptotic stability conditions with respect to the origin only depend on one of the weight matrices. Simulations indicate that these local stability conditions greatly influence the behavior of the autonomous four-dimensional neural network in the region where each variable's values vary between minus one and one. Finally, some sufficient stability conditions for the nonlinear model are formulated as a convex program involving linear matrix inequalities.

In this paper, the robustness of a new visual servoing scheme with respect to camera calibration errors is analyzed. This scheme, called 2D 1/2 visual servoing, is based on the estimation of the partial camera displacement from the current to the desired camera poses at each iteration of the control law. Visual features and data extracted from the partial displacement allow as to design a decoupled control law controlling the six camera DOF. The necessary and sufficient conditions for local asymptotic stability in presence of camera calibration errors are easily obtained. Then, thanks to the simple structure of the system, sufficient conditions for global asymptotic stability are proposed. Finally, experimental results show the validity of our approach and its robustness not only with respect camera calibration errors but also to robot calibration errors.

A necessary and sufficient condition for local asymptotic stability of a class of nonlinear systems in the critical case

In this paper we formulated and analyzed a predator-prey model with sparssing effect, analysis of the existing conditions of equilibrium point, and the sufficient condition of the local asymptotical stability of the equilibrium was studied with the method of latent root, and furthermore, by constructing a Liapunov function to get the boundary equilibrium and the positive equilibrium sufficient conditions for the globally asymptotical stability.

Asymptotic stabilization of two dimensional real analytic systems

Today unemployment has become a global phenomenon which may be instrumental in forcing unemployed persons to earn their livelihood in an illegal manner resulting in a crime. It is possible that unemployed individuals may become more prone to develop a tendency of committing a crime when they come in contact with persons involved in criminal activities but are still unexposed. Further, when unemployed individuals are exposed to have committed a crime, they are captivated and finally awarded imprisonment if the offence charged on them is proved under the existing criminal laws. In this paper, a nonlinear mathematical model is developed to study the role of unemployment in inducing crime by taking into account four dependent variables representing the unemployment class, the employment class, the criminal class and the jail class. The model analysis, using stability theory of ordinary differential equations, provides some local and nonlinear stability conditions regarding stability of equilibrium of the model system. It is inferred that the endemic equilibrium is locally asymptotically stable as well as nonlinearly stable. Numerical simulations of the model system have also been carried to support the analytical findings and showing the effect of certain key parameters on different variables. It is observed that the increase in unemployment rate induces crime in the community leading to increase the burden on jail class.

For general finite-dimensional time-varying nonlinear systems we derive sufficient conditions for closed-loop asymptotic controllability and asymptotic stabilization by means of a smooth ordinary time-varying feedback integrator. The main hypotheses are based on the existence of an almost smooth time-varying control Lyapunov function. The corresponding results generalize earlier works in the literature concerning feedback stabilization for nonlinear autonomous systems that in general cannot be stabilized by continuous time-varying state feedback.

A general system of difference equations is presented for multispecies communities with density dependent population growth and delayed maturity. Interspecific competition, mutualism, predation, commensalism, and amensalism are accommodated. A sufficient condition for the local asymptotic stability of a coexistence equilibrium in this system is then proven. Using this system, the generalisation of the Beverton-Holt and Leslie-Gower models of competition to multispecies systems with possible maturation delays is presented and shown to yield interesting stability properties. The stability of coexistence depends on the relative abundances of the species at the unique interior equilibrium. A sufficient condition for local stability is derived that only requires intraspecific competition to outweigh interspecific competition. The condition does not depend on maturation delays. The derived stability properties are used to develop a novel estimation approach for the coefficients of interspecific competition. This approach finds an optimal configuration given two conjectures. First, coexisting species strive to outcompete competitors. Second, persisting species are more likely in stable systems with strong dampening of perturbations and high ecological resilience. The optimal solution is compared to estimates of niche overlap using an empirical example of malaria mosquito vectors with delayed maturity in the Anopheles gambiae sensu lato species complex.

We consider a system of delay differential equations which represents the general model of a Hopfield neural networks type. We construct some new sufficient conditions for local asymptotic stability about the trivial equilibrium based on the connection weights and delays of the neural system. We also investigate the occurrence of an Andronov-Hopf bifurcation about the trivial equilibrium. Finally, the simulating results demonstrate the validity and feasibility of our theoretical results.

A two-dimensional single-species stage-structured model is proposed and analyzed to study the effect of intraguild predation with harvesting of the adult species. Time lags in reproduction and maturation of the organism are introduced in the system and conditions for local asymptotic stability of steady states are derived. The length of the delay preserving the stability is also estimated. It is found that there are stability switches for time delays, and Hopf bifurcations when time delays cross through some critical values. The stability and direction of the Hopf bifurcations are determined by applying the normal form method and the center manifold theory. Computer simulations have been carried out to illustrate various analytical results.

This paper deals with the dynamic behaviors of a discrete-time fractional-order predator–prey model in the presence of both the Allee effect and immigration on the prey population and in the presence of harvesting on the predator population. The existence and uniqueness and parametric conditions for local asymptotic stability of fixed points of the discrete-time fractional-order model are studied. Moreover, using the center manifold theorem and bifurcation theory, it is shown that the considered model undergoes flip and Neimark–Sacker bifurcations in a small neighborhood of the interior fixed point. Then, the direction of bifurcation is calculated. A feedback controller is implemented in the proposed model to control chaos thanks to the emergence of the Neimark–Sacker bifurcation. Furthermore, numerical analysis confirms the theoretical analysis with the help of Matlab software.

Asymptotic stability of stationary temporary equilibria and changes in expectations

This paper proposed a delayed fractional-order SEIHR-M model incorporating media influence to investigate the transmission dynamics of COVID-19 in Malaysia. By integrating fractional-order dynamics and time-delay media influence into a unified epidemic framework, this novel structure more accurately captures both memory effects and behavioral response lags in the context of COVID-19. Theoretical analysis verified the existence, non-negativity, and boundedness of the solutions, ensuring the biological feasibility of the model. The basic reproduction number was derived using the next-generation matrix method, serving as a key metric for evaluating disease transmission and model stability. Furthermore, when , the disease-free equilibrium is locally asymptotically stable regardless of the value of the delay parameter . When 1$$\\end{document}]]>, the stability of the endemic equilibrium exhibits two scenarios: if , sufficient conditions for local asymptotic stability are provided; if 0$$\\end{document}]]>, there exists a critical delay . The endemic equilibrium remains locally asymptotically stable for but becomes unstable for \ au _0$$\\end{document}]]>, undergoing a Hopf bifurcation at , leading to periodic oscillations. The numerical simulation results not only validate the theoretical analysis but also show that as the fractional-order parameter increases, the system exhibits more pronounced oscillations; furthermore, longer delay times facilitate the emergence of these oscillatory behaviors, making the epidemic more prone to recurrent and periodic fluctuations. By fitting the model with early COVID-19 data from Malaysia, the feasibility and applicability of the model are further validated, and the superior fitting performance of the fractional-order delay model compared to the corresponding integer-order model is highlighted. Finally, sensitivity analysis results show that media interventions have a significant impact on epidemic spread, further demonstrating that timely and effective information dissemination plays a crucial role in reducing the peak of infections and controlling the epidemic.

In this paper, we investigate the existence and stability properties of both classical and fractional-order generalized Brusselator reaction-diffusion systems. By analyzing the system in both ODEs and PDEs forms, we establish the global existence of solutions in the classical case by using an appropriate Lyapunov functional, we also show the local existence of solutions in the fractional-order case. Additionally, we derive sufficient conditions for local asymptotic stability. To illustrate our theoretical findings, we consider the Brusselator reaction-diffusion model as a particular case of our generalized system and employ numerical methods to explore its solutions under the proposed conditions.

In mathematical biology, discrete time models governed by difference equations are more suitable to study the evolutionary growths of species or taxonomic group of organisms over the course of time. These discrete time population models are appropriate for non-overlapping generations and are suitable to analyse the nonlinear dynamics and their chaotic behaviour. In this research article, the complexity of intra species competition between two species with carrying capacity is investigated in discrete time, under the assumption that average growth rates in prey and predator populations change at regular intervals of time. The method of piecewise constant arguments for differential equations is implemented to obtain the discrete time counterpart of continuous case. Results related to uniform boundedness of solutions, and existence and uniqueness of positive fixed points are established. The parametric conditions for local asymptotic stability of fixed points are obtained. Also, the existence criteria and direction of flip bifurcation and hopf bifurcation are derived. Moreover, feedback control strategy is employed to control the chaos under the influence of bifurcations. To validate the analytical results, numerical illustrations are presented. The existence of chaos is confirmed from calculating Maximal Lyapunov Exponent.