Sufficient Conditions For Global Stability Research Articles

Switching-based stabilization of aperiodic sampled-data Boolean control networks with all subsystems unstable

We aim to further study the global stability of Boolean control networks (BCNs) under aperiodic sampled-data control (ASDC). According to our previous work, it is known that a BCN under ASDC can be transformed into a switched Boolean network (SBN), and further global stability of the BCN under ASDC can be obtained by studying the global stability of the transformed SBN. Unfortunately, since the major idea of our previous work is to use stable subsystems to offset the state divergence caused by unstable subsystems, the SBN considered has at least one stable subsystem. The central thought in this paper is that switching behavior also has good stabilization; i.e., the SBN can also be stable with appropriate switching laws designed, even if all subsystems are unstable. This is completely different from that in our previous work. Specifically, for this case, the dwell time (DT) should be limited within a pair of upper and lower bounds. By means of the discretized Lyapunov function and DT, a sufficient condition for global stability is obtained. Finally, the above results are demonstrated by a biological example.

A Mathematical Model for the Transmission Dynamics of Lymphatic Filariasis with Intervention Strategies.

This manuscript considers the transmission dynamics of lymphatic filariasis with some intervention strategies in place. Unlike previously developed models, our model takes into account both the exposed and infected classes in both the human and mosquito populations, respectively. We also consider vaccinated, treated and recovered humans in the presented model. The global dynamics of the proposed model are completely determined by the basic ([Formula: see text]) and effective reproduction numbers ([Formula: see text]). We then use Lyapunov function theory to find the sufficient conditions for global stability of both the disease-free equilibrium and endemic equilibrium. The Lyapunov functions show that when the basic reproduction number is less than or equal to unity, the disease-free equilibrium is globally asymptotically stable, and when it is greater than unity then the endemic equilibrium is also globally asymptotically stable. Finally, numerical simulations are carried out to investigate the effects of the intervention strategies and key parameters to the spread of lymphatic filariasis. The numerical simulations support the analytical results and illustrate possible model behavioral scenarios.

On Matrix Norms, Stability and Stabilization of a Class of Discrete Takagi–Sugeno Fuzzy Systems

Based on a result of stability for linear time varying systems, a sufficient condition for global stability that does not use Lyapunov theory has been established for a class of discrete Takagi–Sugeno fuzzy systems. The condition involves testing the norms of products of the matrices of the consequences. The approach seeks to determine if at an instant of time the system becomes a contraction mapping. Because the computation burden associated with these tests may become prohibitive, a necessary and sufficient condition for local stability is derived. In a way, this decreases the computational burden and allows to customize the test. A two-step iterative stabilization algorithm is then introduced. The problem involves norm minimization and a posteriori stability test. Numerical examples are presented to demonstrate the applicability of the approach.

Mathematical Modeling and Characterization of the Spread of Chikungunya in Colombia

The Chikungunya virus is the cause of an emerging disease in Asia and Africa, and also in America, where the virus was first detected in 2006. In this paper, we present a mathematical model of the Chikungunya epidemic at the population level that incorporates the transmission vector. The epidemic threshold parameter R 0 for the extinction of disease is computed using the method of the next generation matrix, which allows for insights about what are the most relevant model parameters. Using Lyapunov function theory, some sufficient conditions for global stability of the the disease-free equilibrium are obtained. The proposed mathematical model of the Chikungunya epidemic is used to investigate and understand the importance of some specific model parameters and to give some explanation and understanding about the real infected cases with Chikungunya virus in Colombia for data belonging to the year 2015. In this study, we were able to estimate the value of the basic reproduction number R 0 . We use bootstrapping and Markov chain Monte Carlo techniques in order to study parameters’ identifiability. Finally, important policies and insights are provided that could help government health institutions in reducing the number of cases of Chikungunya in Colombia.

Stability and Hopf Bifurcation of a Delayed Epidemic Model of Computer Virus with Impact of Antivirus Software

In this paper, we investigate an SLBRS computer virus model with time delay and impact of antivirus software. The proposed model considers the entering rates of all computers since every computer can enter or leave the Internet easily. It has been observed that there is a stability switch and the system becomes unstable due to the effect of the time delay. Conditions under which the system remains locally stable and Hopf bifurcation occurs are found. Sufficient conditions for global stability of endemic equilibrium are derived by constructing a Lyapunov function. Formulae for the direction, stability, and period of the bifurcating periodic solutions are conducted with the aid of the normal form theory and center manifold theorem. Numerical simulations are carried out to analyze the effect of some of the parameters in the system on the dynamic behavior of the system.

Permanency in predator–prey models of Leslie type with ratio-dependent simplified Holling type-IV functional response

We consider a predator–prey model of Leslie type with ratio-dependent simplified Holling type-IV functional response. The novelty of the model is that the functional response simulates group defense of the prey kind, which in turn, affects permanency of the system and existence of limit cycles. We show that permanency of the system holds automatically for some values of parameters and provides sufficient conditions for global stability of interior equilibrium by constructing a Lyapunov function. We prove that for some values of parameters the system exhibits a Hopf cycle and provides conditions by which the corresponding stable Hopf cycle is the only cycle that model may have. Numerical simulations show that if the conditions are broken, the model may have more than one limit cycle, which is phenomenal among the predator–prey models with one interior equilibrium.

Stability analysis of a disease resistance SEIRS model with nonlinear incidence rate

In this paper, we study a new SEIRS epidemic model describing nonlinear incidence with a more general form and the transmission of influenza virus with disease resistance. The basic reproductive number Re_{0} is obtained by using the method of next generating matrix. If Re_{0}<1, the disease-free equilibrium is globally asymptotically stable, and if Re_{0}>1, by using the geometric method, we obtain some sufficient conditions for global stability of the unique endemic equilibrium. Finally, numerical simulations are provided to support our theoretical results.

Stability and $l_1$ Gain Analysis of Boolean Networks With Markovian Jump Parameters

This paper presents some results on stability and $l_1$ gain analysis of Boolean networks with Markovian jump parameters. A necessary and sufficient condition for global stability of the concerned Boolean networks is given in terms of linear programming by utilizing the semi-tensor product of matrices and some properties of linear positive systems. Then, the definition of Lyapunov function for stochastic Boolean networks is presented and Lyapunov theorem is derived. Moreover, an $l_1$ gain problem for stochastic Boolean networks with external disturbances is formulated and solved by a sufficient condition. Examples are shown to illustrate the effectiveness of the obtained results.

Global Stability Analysis of Healthy Situation for a Coupled Model of Healthy and Cancerous Cells Dynamics in Acute Myeloid Leukemia

In this paper we aim to study the global stability of a coupled model of healthy and cancerous cells dynamics in healthy situation of Acute Myeloid Leukemia. We also clarify the effect of interconnection between healthy and cancerous cells dynamics on the global stability The interconnected model is obtained by transforming the PDE-based model into a nonlinear distributed delay system. Using Lyapunov approach, we derive necessary and sufficient conditions for global stability for a selected equilibrium point of particular interest (healthy situation). Simulations are conducted to illustrate the obtained results.

Multigroup deterministic and stochastic SEIRI epidemic models with nonlinear incidence rates and distributed delays: A stability analysis

In this paper, we investigate the dynamics of a multigroup disease propagation model with distributed delays and nonlinear incidence rates, which accounts for the relapse of recovered individuals. The main concern is the stability of the equilibria, sufficient conditions for global stability being obtained by applying Lyapunov‐LaSalle invariance principle and using Lyapunov functionals, which are constructed using their single‐group counterparts. The situation in which the deterministic model is subject to perturbations of white noise type is also investigated from a stability viewpoint.

Formation control with disturbance rejection for a class of Lipschitz nonlinear systems

In this paper, we consider the leader-follower formation control problem for general multi-agent systems with Lipschitz nonlinearity and unknown disturbances. To deal with the disturbances, a disturbance observer-based control strategy is developed for each follower. Then, a time-varying formation protocol is proposed based on the relative state of the neighbouring agents and sufficient conditions for global stability of the formation control are identified using Lyapunov method in the time domain. The proposed strategy and analysis guarantee that all signals in the closed-loop dynamics are uniformly ultimately bounded and the formation tracking errors converge to an arbitrarily small residual set. Finally, the validity of the proposed controller is demonstrated through a numerical example.

Global stability of discrete-time competitive population models

We develop practical tests for the global asymptotic stability of interior fixed points for discrete-time competitive population models. Our method constitutes the extension to maps of the Split Lyapunov method developed for differential equations. We give ecologically-motivated sufficient conditions for global stability of an interior fixed point of a map of Kolmogorov form. We introduce the concept of a principal reproductive mode, which is linked to a normal at the interior fixed point of a hypersurface of vanishing weighted-average growth. Our method is applied to establish new global stability results for 3-species competitive systems of May-Leonard type, where previously only parameter values for local stability was known.

PID controller design for second order nonlinear uncertain systems

Although the classical PID (proportional-integral-derivative) controller is most widely and successfully used in engineering systems which are typically nonlinear with various uncertainties, almost all the existing investigations on PID controller focus on linear systems. The aim of this paper is to present a theory on PID controller for nonlinear uncertain systems, by giving a simple and analytic design method for the PID parameters together with a mathematic proof for the global stability and asymptotic regulation of the closed-loop control systems. To be specific, we will construct a 3-dimensional manifold within which the three PID parameters can be chosen arbitrarily to globally stabilize a wide class of second order nonlinear uncertain dynamical systems, as long as some knowledge on the upper bound of the derivatives of the nonlinear uncertain function is available. We will also try to make the feedback gains as small as possible by investigating the necessity of the manifold from which the PID parameters are chosen, and to establish some necessary and sufficient conditions for global stabilization of several special classes of nonlinear uncertain systems.

Persistence and global stability of a Leslie-Gower predator-prey refuge system with a competitor for the prey

This paper analyzes the effect of refuge on the dynamics of a Leslie-Gower predator-prey model in which one predator feeds on one of two competing species. Existence conditions for equilibrium points are discussed. By using differential inequality argument, we developed persistence criterion. Sufficient condition for global stability of the unique positive equilibrium point is derived. Different type of local bifurcation near the equilibrium points has been investigated. The role of refuges have been shown on equilibrium densities of prey, competitor for prey and predator respectively. The results establish the fact that the effects of refuges used by prey increase the equilibrium density of prey population under certain restrictions, whereas opposite hold for competitor of prey population. However equilibrium density of predator may decrease or increase by increasing the amount of prey refuge. Some numerical simulations are performed to validate the results obtained.https://doi.org/10.28919/cmbn/3382

Dynamics of a predator–prey model with double Allee effects and impulse

In this paper, a predator–prey model with double Allee effects and impulse is studied. The existence and stability of the prey-free periodic solution are investigated. The sufficient conditions for global stability of the prey-free periodic solution are obtained. We also find a critical threshold that the predator and prey populations will coexist. The existence of the transcritical bifurcations is considered by means of the bifurcation theory when the prey population is not subject to Allee effect. Combining mathematical analysis and numerical simulations, we show that the double Allee effects and impulse greatly alter the outcome of the survival of both species.

Stability and traveling fronts for a food chain reaction-diffusion systems with nonlocal delays

This paper is purported to investigate a food chain reaction-diffusion predator-prey system with nonlocal delays in a bounded domain with no flux boundary condition. We investigate the global stability and find the sufficient conditions of global stability of the unique positive equilibrium for this system. The derived results show that delays often restrain stability. Using the method of linearizing this system, we see that the zero equilibrium is unstable. Moreover, by constructing upper-lower solutions, we find that there exist traveling wavefronts which connect the zero equilibrium and positive equilibrium when the wave speed is large enough and the prey intrinsic growth rate and the death rate of the predator are relatively big.

Effective competition determines the global stability of model ecosystems

We investigate the stability of Lotka-Volterra (LV) models constituted by two groups of species such as plants and animals in terms of the intragroup effective competition matrix, which allows separating the equilibrium equations of the two groups. In matrix analysis, the effective competition matrix represents the Schur complement of the species interaction matrix. It has been previously shown that the main eigenvalue of this effective competition matrix strongly influences the structural stability of the model ecosystem. Here, we show that the spectral properties of the effective competition matrix also strongly influence the dynamical stability of the model ecosystem. In particular, a necessary condition for diagonal stability of the full system, which guarantees global stability, is that the effective competition matrix is diagonally stable, which means that intergroup interactions must be weaker than intra-group competition in appropriate units. For mutualistic or competitive interactions, diagonal stability of the effective competition is a sufficient condition for global stability if the inter-group interactions are suitably correlated, in the sense that the biomass that each species provides to (removes from) the other group must be proportional to the biomass that it receives from (is removed by) it. For a non-LV mutualistic system with saturating interactions, we show that the diagonal stability of the corresponding LV system close to the fixed point is a sufficient condition for global stability.

Successive lag synchronization on dynamical networks with communication delay**Project supported by the National Natural Science Foundation of China (Grant No. 61004101), the Natural Science Foundation Program of Guangxi Province, China (Grant No. 2015GXNSFBB139002), the Graduate Innovation Project of Guilin University of Electronic Technology, China (Grant No. GDYCSZ201472), and the Guangxi Colleges and Universities Key Laboratory of Data Analysis and Computation,

In this paper, successive lag synchronization (SLS) on a dynamical network with communication delay is investigated. In order to achieve SLS on the dynamical network with communication delay, we design linear feedback control and adaptive control, respectively. By using the Lyapunov function method, we obtain some sufficient conditions for global stability of SLS. To verify these results, some numerical examples are further presented. This work may find potential applications in consensus of multi-agent systems.

Sufficient and necessary conditions for global stability of genetic regulator networks with time delays

This paper is concerned with the global stability of the nonlinear model for genetic regulator networks (GRNs) with time delays. Four new sufficient and necessary conditions for global asymptotic stability and global exponential stability of the equilibrium point of GRNs are derived. Specifically, using comparing theorem and Dini derivation method, three weak sufficient conditions for global stability of GRNs with constant time delays are proposed. Finally, a general GRN model is used to illustrate the effectiveness of the proposed theoretical results. Compared with the previous results, some sufficient and necessary conditions for Lyapunov stability of GRNs are proposed, which are not seen before.

Consensus Control of a Class of Lipschitz Nonlinear Systems With Input Delay

This paper deals with the consensus control design for Lipschitz nonlinear multi-agent systems with input delay. The Artstein-Kwon-Pearson reduction method is employed to deal with the input delay and the integral term that remains in the transformed system is analyzed by using Krasovskii functional. Upon exploring certain features of the Laplacian matrix, sufficient conditions for global stability of the consensus control are identified using Lyapunov method in the time domain. The proposed control only uses relative state information of the agents. The effectiveness of the proposed control design is demonstrated through a simulation study.