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We study the effect of population mobility on the transmission dynamics of infectious diseases by considering a susceptible-exposed-infectious-recovered (SEIR) epidemic model with graph Laplacian diffusion, that is, on a weighted network. First, we establish the existence and uniqueness of solutions to the SEIR model defined on a weighed graph. Then by constructing Liapunov functions, we show that the disease-free equilibrium is globally asymptotically stable if the basic reproduction number is less than unity and the endemic equilibrium is globally asymptotically stable if the basic reproduction number is greater than unity. Finally, we apply our generalized weighed graph to Watts–Strogatz network and carry out numerical simulations, which demonstrate that degrees of nodes determine peak numbers of the infectious population as well as the time to reach these peaks. It also indicates that the network has an impact on the transient dynamical behaviour of the epidemic transmission.

This paper is mainly to analyze stability of equilibria in a new predator-prey model, Logistic-Volterra model. It exists three different equilibria for the model. The stability of these equilibria for the model are discussed by Liapunov function or other principle. Through theoretical analysis, (0,0) is unstable and existent in any situations. (N1, 0) is existent and global asymptotic stability if δ2 < k2 / N1, and (x1*, x2*) is existent and uniformly and globally asymptotic stability if δ2 > k2 / N1. Then, some simulation are given and verify the proof of the result. At last, some conclusions about biology were raised.

In a density-dependent single-species population growth model, a simple method is proposed to explicitly and directly derive the analytic expressions of reliable regions for local and global asymptotic stability. Specifically, first, a reliable region ΛLAS is explicitly represented by solving the fixed point and utilizing the asymptotic stability criterion, over which the fixed point is locally asymptotically stable. Then, two types of auxiliary Liapunov functions are constructed, where the variation of the Liapunov function is decomposed into the product of two functions and is always negative at the non-equilibrium state. Finally, based on the Liapunov stability theorem, a closed-form expression of reliable region ΛGAS is obtained, where the fixed point is globally asymptotically stable in the sense that all the solutions tend to fixed point. Numerical results show that our analytic expressions of reliable regions are accurate for both local and global asymptotic stability.

We determine conditions allowing for simplification of the description of the impact of a short and arbitrarily intense laser pulse onto a cold plasma at rest. If both the initial plasma density and pulse profile have plane symmetry, then suitable matched upper bounds on the maximum and the relative variations of the initial density, as well as on the intensity and duration of the pulse, ensure a strictly hydrodynamic evolution of the electron fluid without wave-breaking or vacuum-heating during its whole interaction with the pulse, while ions can be regarded as immobile. We use a recently developed fully relativistic plane model whereby the system of the Lorentz–Maxwell and continuity PDEs is reduced into a family of highly nonlinear but decoupled systems of non-autonomous Hamilton equations with one degree of freedom, the light-like coordinate ξ=ct−z instead of time t as an independent variable, and new a priori estimates (eased by use of a Liapunov function) of the solutions in terms of the input data (i.e., the initial density and pulse profile). If the laser spot radius R is finite and is not too small, the same conclusions hold for the part of the plasma close to the axis z→ of cylindrical symmetry. These results may help in drastically simplifying the study of extreme acceleration mechanisms of electrons.

A difference system of the Lotka–Volterra type is considered. It is assumed that this system can operate both in some program and perturbed modes. The restrictions on the time of the system’s stay in these modes, providing the desired dynamical behavior, are investigated. In particular, the conditions of the ultimate boundedness of solutions and the permanence of the system are obtained. The direct Lyapunov method is used, and different Lyapunov functions are constructed in different parts of the state space. The sizes of the domain of permissible initial values of solutions and the domain of the ultimate bound of solutions corresponding to the required dynamics of the system are estimated. Constraints are set on the size of the digitization step of the system.

Standard power sources have wide applications in fields such as healthcare, measurement and control, education, scientific research, and metrology. This paper proposes a switched control method based on the Lyapunov function for power sources based on digital power amplifiers run as inverters and a new selection method for switching states. Combining the two methods in power sources ensures and improves the system’s stability. In addition, the proposed switching state selection method reduces the computation of the power source system, and the switching loss of IGBTs decreases and is distributed. A MATLAB/Simulink model utilizing the switching state selection method and Lyapunov function is developed, confirming its feasibility.

This paper presents an adaptive control method for a class of uncertain strict-feedback switched nonlinear systems. First, we consider the constraint characteristics in the switched nonlinear systems to ensure that all states in switched systems do not violate the constraint ranges. Second, we design the controller based on the backstepping technique, while integral Barrier Lyapunov functions (iBLFs) are adopted to solve the full state constraint problems in each step in order to realize the direct constraints on state variables. Furthermore, we introduce the Lyapunov stability theory to demonstrate that the adaptive controller achieves the desired control goals. Finally, we perform a numerical simulation, which further verifies the significance and feasibility of the presented control scheme.

Taking into account the effects of patch structure and nonlinear density-dependent mortality terms, we explore a class of almost periodic Nicholson’s blowflies model in this paper. Employing the Lyapunov function method and differential inequality technique, some novel assertions are developed to guarantee the existence and exponential stability of positive almost periodic solutions for the addressed model, which generalize and refine the corresponding results in some recently published literatures. Particularly, an example and its numerical simulations are arranged to support the proposed approach.

In this paper, an impulsive differential system is investigated for the ?rst time for practical stability and boundedness criteria with respect to initial time di?er- ence. The investigations are carried out by perturbing Lyapunov functions and by using comparison results. A generalized Lyapunov function has been used for the investiga- tion. The present results indicate that the stability criteria signi?cantly depend on the moment of impulses.

A novel SMC-based prescribed performance guidance law combined with a base generator (TBG) is proposed. A TBG-based modified Lyapunov function is first proposed, which ensures that the LOS tracking error converges into zero within a predefined time. A novel sliding manifold with adjusting convergent time property is proposed, which brings about a smaller control magnitude and predefined time convergent rate. To ensure a satisfied transit and steady performance, some converted variables are defined based on which a novel prescribed performance guidance law is constructed. The input saturation problem is properly solved by designing a novel TBG-based one-order auxiliary system. Extensive simulation results have verified the effectiveness of the proposed method.