Synchronization of the Lohe model on the hyperboloid under a directed graph

A functional version of LaSalle's invariance principle is derived, i.e., rather than the usual pointwise Lyapunov-like functions it uses specially constructed functionals along system trajectories. This modification enables the principle to handle even nonautonomous systems to which the classical LaSalle's principle is not directly applicable. The new theoretical results are then used to study robust synchronization of general Lie/spl acute/nard type of systems. The developed technique is finally applied to chaotic oscillators synchronization. Numerical simulation is included to demonstrate the effectiveness of the proposed methodology.

For the purpose of investigating two complex networks' hybrid synchronization, a controller with fractional-order is provided. Regarding hybrid synchronization which includes the outer synchronization and inner synchronization, some hybrid synchronization's sufficient conditions according to the Lyapunov stability theorem and the LaSalle invariance principle are proposed. Theoretical analysis suggests that, only when the state of driving-response networks is outer synchronization and each network is in inner synchronization, two coupled networks' hybrid synchronization under some suitable conditions could be reached. Finally, theoretical results are illustrated and validated with the given numerical simulations.

We investigate the problem of cluster anticonsensus of multiagent systems. For multiagent continuous systems, a new control protocol is designed based on theQ-theory. Then by LaSalle's invariance principle we prove that if the graph is connected and bipartite, then the cluster anticonsensus is achieved by the proposed control protocol. On the other hand, a similar control protocol is designed for multiagent discrete-time systems. Then, sufficient conditions are given to guarantee the cluster anticonsensus of multiagent discrete-time systems by using theQ-theory and LaSalle's invariance principle. Numerical simulations show the effectiveness of our theoretical results.

Cholera is a common infectious disease caused by Vibrio cholerae, which has different infectivity. In this paper, we propose a cholera model with hyperinfectious and hypoinfectious vibrios, in which both human-to-human and environment-to-human transmissions are considered. By analyzing the characteristic equations, the local stability of disease-free and endemic equilibria is established. By using Lyapunov functionals and LaSalle's invariance principle, it is verified that the global threshold dynamics of the model can be completely determined by the basic reproduction number. Numerical simulations are carried out to illustrate the corresponding theoretical results and describe the cholera outbreak in Haiti. The study of optimal control helps us seek cost-effective solutions of time-dependent control strategies against cholera outbreaks, which shows that control strategies, such as vaccination and sanitation, should be taken at the very beginning of the outbreak and become less necessary after a certain period.

This paper studies an (n + 4)-dimensional nonlinear viral infection model that characterizes the interactions of the viruses, susceptible host cells, n-stages of infected cells, CTL cells and B cells. Both viral and cellular infections have been incorporated into the model. The well-posedness of the model is justified. The model admits five equilibria which are determined by five threshold parameters. The global stability of each equilibrium is proven by utilizing Lyapunov function and LaSalle's invariance principle. The theoretical results are illustrated by numerical simulations.

This paper formulates a virus dynamics model with impairment of B‐cell functions. The model incorporates two modes of viral transmission: cell‐free and cell‐to‐cell. The cell‐free and cell‐cell incidence rates are modeled by general functions. The model incorporates both, latently and actively, infected cells as well as three distributed time delays. Nonnegativity and boundedness properties of the solutions are proven to show the well‐posedness of the model. The model admits two equilibria that are determined by the basic reproduction numberR0. The global stability of each equilibrium is proven by utilizing Lyapunov function and LaSalle's invariance principle. The theoretical results are illustrated by numerical simulations. The effect of impairment of B‐cell functions and time delays on the virus dynamics are studied. We have shown that if the functions of B‐cell is impaired, then the concentration of viruses is increased in the plasma. Moreover, we have observed that increasing the time delay will suppress the viral replication.

ABSTRACTIn this paper, an age-structured cholera model with multiple transmissions, saturation incidence and imperfect vaccination is proposed. In the model, we consider both the infection age of infected individuals and the biological age of Vibrio cholerae in the aquatic environment. Asymptotic smoothness is verified as a necessary argument. By analysing the characteristic equations, the local stability of disease-free and endemic steady states is established. By using Lyapunov functionals and LaSalle's invariance principle, it is proved that the global dynamics of the model can be completely determined by basic reproduction number. The study of optimal control helps us seek cost-effective solutions of time-dependent vaccination strategy against cholera outbreaks. Numerical simulations are carried out to illustrate the corresponding theoretical results.

In this paper, we propose the SEIR epidemic model with delay and nonlinear incidence rate. The resulting model has two possible equilibria: if $R_{0} \leq 1,$ then the SEIR epidemic model has a disease-free equilibrium and if $R_{0} > 1,$ then the SEIR epidemic model admits a unique endemic equilibrium. By using suitable Lyapunov functionals and LaSalle's invariance principle, the global stability of a disease-free equilibrium is established. Our main contribution affirms the existence of non constant periodic solutions which bifurcate from the endemic equilibrium when the delay crosses some critical values. Finally, some numerical simulations are presented to illustrate our theoretical results.

Fractional HCV infection model with adaptive immunity and treatment is suggested and studied in this paper. The adaptive immunity includes the CTL response and antibodies. This model contains five ordinary differential equations. We will start our study by proving the existence, uniqueness, and boundedness of the positive solutions. The model has free-equilibrium points and other endemic equilibria. By using Lyapunov functional and LaSalle's invariance principle, we have shown the global stability of these equilibrium points. Finally, some numerical simulations will be given to validate our theoretical results and show the effect of the fractional derivative order parameter and the other treatment parameters.

The plug-and-play function is one of the features of future smart grid. Distributed economic dispatch algorithm will pave the way for the achievement of this function. Most existing distributed economic dispatch algorithms only achieve asymptotic or exponential convergence and work under time-invariant communication topology. In this work, a consensus based distributed economic dispatch algorithm, which achieves finite-time convergence under jointly connected topology condition, is proposed to calculate the optimal active power for each generator. By virtue of Lyapunov theory, LaSalle's invariance principle and homogeneous property, the convergence and optimality of the proposed algorithm are proved. Several case studies are performed to illustrate the effectiveness of the proposed algorithm.

In this paper, we investigate the dynamical behaviors of three human immunodeficiency virus infection models with two types of cocirculating target cells and distributed intracellular delay. The models take into account both short‐lived infected cells and long‐lived chronically infected cells. In the two types of target cells, the drug efficacy is assumed to be different. The incidence rate of infection is given by bilinear and saturation functional responses in the first and second models, respectively, while it is given by a general function in the third model. Lyapunov functionals are constructed and LaSalle invariance principle is applied to prove the global asymptotic stability of all equilibria of the models. We have derived the basic reproduction numberR0for the three models. For the first two models, we have proven that the disease‐free equilibrium is globally asymptotically stable (GAS) whenR0≤1, and the endemic equilibrium is GAS whenR0>1. For the third model, we have established a set of sufficient conditions for global stability of both equilibria of the model. We have checked our theoretical results with numerical simulations. Copyright © 2015 John Wiley & Sons, Ltd.

This article proposes and analyzes a fractional-order African Swine Fever model with saturation incidence. Firstly, the existence and uniqueness of a positive solution is proven. Secondly, the basic reproduction number and the sufficient conditions for the existence of two equilibriums are obtained. Thirdly, the local and global stability of disease-free equilibrium is studied using the LaSalle invariance principle. Next, some numerical simulations are conducted based on the Adams-type predictor-corrector method to verify the theoretical results, and sensitivity analysis is performed on some parameters. Finally, discussions and conclusions are presented. The theoretical results show that the value of the fractional derivative α will affect both the coordinates of the equilibriums and the speed at which the equilibriums move towards stabilization. When the value of α becomes larger or smaller, the stability of the equilibriums will be changed, which shows the difference between the fractional-order systems and the classical integer-order system.

Based on the classic SEIR infectious disease model, a new type of new coronary pneumonia transmission model containing a population with a history of basic diseases was established, the basic reproduction number of its transmission was obtained, and the existence of the equilibrium of the model was determined. Through construction of the Lyapunov function and with the LaSalle invariance principle, the global stability of the equilibrium was proved, and the theoretical research results were also verified by numerical simulation. At the same time, the influence of the transformation rate coefficient from no basic medical disease to basic disease on the disease transmission was discussed. It is found that, the mathematical model considering no basic disease will underestimate the basic reproduction number of disease transmission and the scale of infection. Numerical simulations also show the definite impact of the transformation rate coefficient from no basic disease to basic disease on the peak number of the infected population.

During the past eras, many mathematicians have paid their attentions to model the dynamics of dengue virus (DENV) infection but without taking into account the mobility of the cells and DENV particles. In this study, we develop and investigate a partial differential equations (PDEs) model that describes the dynamics of secondary DENV infection taking into account the spatial mobility of DENV particles and cells. The model includes five nonlinear PDEs describing the interaction among the target cells, DENV-infected cells, DENV particles, heterologous antibodies, and homologous antibodies. In the beginning, the well-posedness of solutions, including the existence of global solutions and the boundedness, is justified. We derive three threshold parameters which govern the existence and stability of the four equilibria of the model. We study the global stability of all equilibria based on the construction of suitable Lyapunov functions and usage of Lyapunov–LaSalle’s invariance principle (LLIP). Last, numerical simulations are carried out in order to verify the validity of our theoretical results.

In the paper, an extension of LaSalle's Invariance Principle to a class of switched linear systems is studied. One of the motivations is the consensus problem in multi-agent systems. Unlike most existing results in which each switching mode in the system needs to be asymptotically stable, this paper allows that the switching modes are only Lyapunov stable. Under certain ergodicity assumptions, an extension of LaSalle's Invariance Principle for global asymptotic stability is obtained. Then it is used to solve the consensus reaching problem of certain multi-agent systems in which each agent is modeled by a double integrator, and the associated interaction graph is switching and is assumed to be only jointly connected.