Periodic Solutions for a Class of Neutral Functional Differential Equations with Distributed and Discrete Delays

Based on Krasnoselskii's fixed point theorem,matrix measure and functional analysis methods,some new sufficient conditions for the existence of periodic solutions of neutral functional differential equations with distributed and discrete delays are obtained. Moreover,we construct an example to illustrate the feasibility of our results.

Hopf bifurcation and chaos in a Leslie–Gower prey–predator model with discrete delays

In this work,a Leslie–Gower prey-predator model with two discrete delays has been investigated.The positivity,boundedness and persistence of the delayed system have been discussed.The system exhibits the phenomenon of Hopf bifurcation with respect to both delays.The conditions for occurrence of Hopf bifurcation are obtained for different combinations of delays.It is shown that delay induces the complexity in the system and brings the periodic oscillations,quasi-periodic oscillations and chaos.The properties of periodic solution have been determined using central manifold and normal form theory.Further,the global stability of the system has been established for different cases of discrete delays.The numerical computation has also been performed to verify analytical results.

A descriptor system approach to robust stabilityof uncertain degenerate systems with discreteand distribute delays

The robust stability of uncertain linear degenerate systems with discrete and distributed delays is studied in this paper. The uncertainties under consideration are norm bounded, and possibly time varying. A novel robust stability criterion of the system is derived by constructing Lyapunov functions. The degenerate systems are transformed to a descriptor system and the stability criteria are formulated in the form of a linear matrix inequality （LMI）. Therefore, it is easy to check the robust stability of the degenerate systems by using this method. Numerical examples are also worked out to illustrate the obtained results.

Hopf bifurcation analysis in a turbidostat model with Beddington.DeAngelis functional response and discrete delay

In this paper, regarding the time delay as a bifurcation parameter, the stability and Hopf bifurcation of the model of competition between two species in a turbidostat with Beddington-DeAngelis functional response and discrete delay are studied. The Hopf bifurcations can be shown when the delay crosses the critical value. Furthermore, based on the normal form and the center manifold theorem, the type, stability and other properties of the bifurcating periodic solutions are determined. Finally, some numerical simulations are given to illustrate the results.

Stabilization of a class of nonlinear discrete time systems with time varying delay

The stability and stabilization of a class of nonlinear discrete time delayed systems(NDTDS) with time-varying delay and norm-bounded nonlinearity are investigated. Based on discrete time Lyapunov–Krasovskii functional method, a sufficient delaydependent condition for asymptotic stability of nonlinear systems is offered. Then, this condition is used to design a new efficient delayed state feedback controller(DSFC) for stabilization of such systems. These conditions are in the linear matrix inequality(LMI) framework. Illustrative examples confirm the improvement of the proposed approach over the similar cases. Furthermore, the obtained stability and stabilization conditions will be extended to uncertain discrete time delayed systems(UDTDS) with polytopic parameter uncertainties and also with norm-bounded parameter uncertainties.

The Lotka-Volterra Competition System with Discrete and Distributed Time Delays

In this paper, the Lotka-Volterra competition system with discrete and distributed time delays is considered. By analyzing the characteristic equation of the linearized system, the local asymptotic stability of the positive equilibrium is investigated. Moreover, we discover the delays don＇t effect the stability of the equilibrium in the delay system. Finally, we can conclude that the positive equilibrium is global asymptotically stable in the delay system.

Synchronization of stochastically hybrid coupled neural networks with coupling discrete and distributed time-varying delays

A general model of linearly stochastically coupled identical connected neural networks with hybrid coupling is proposed, which is composed of constant coupling, coupling discrete time-varying delay and coupling distributed timevarying delay. All the coupling terms are subjected to stochastic disturbances described in terms of Brownian motion, which reflects a more realistic dynamical behaviour of coupled systems in practice. Based on a simple adaptive feedback controller and stochastic stability theory, several sufficient criteria are presented to ensure the synchronization of linearly stochastically coupled complex networks with coupling mixed time-varying delays. Finally, numerical simulations illustrated by scale-free complex networks verify the effectiveness of the proposed controllers.

Stability of discrete Hopfield neural networks with delay

Discrete Hopfield neural network with delay is an extension of discrete Hopfield neural network. As it is well known, the stability of neural networks is not only the most basic and important problem but also foundation of the network＇s applications. The stability of discrete HJopfield neural networks with delay is mainly investigated by using Lyapunov function. The sufficient conditions for the networks with delay converging towards a limit cycle of length 4 are obtained. Also, some sufficient criteria are given to ensure the networks having neither a stable state nor a limit cycle with length 2. The obtained results here generalize the previous results on stability of discrete Hopfield neural network with delay and without delay.

Dynamics of a Two Species Competitive System with Pure Delays

A class of non-autonomous two species Lotka-Volterra competitive system with pure discrete time delays is discussed. Some sufficient conditions on the boundedness, permanence, periodic solution and global attractivity of the system are established by means of the comparison method and Liapunov functional.

PERMANENCE AND GLOBAL ATTRACTIVITY OF N-SPECIES COOPERATION SYSTEM WITH DISCRETE TIME DELAYS AND FEEDBACK CONTROLS

In this paper, we consider the N-species cooperation system with discrete time delays and feedback controls. By using the differential inequality theory and constructing a suitable Lyapunov functional, we obtain sufficient conditions which guarantee the permanence and the global attractivity of the system.

The New Criteria Robust Stability of Uncertain Neutral Systems with Discrete and Distributed Delays

The robust stability of uncertain linear neutral systems with discrete and distributed delays is investigated. The uncertainties under consideration are norm bounded, and possibly time varying. By means of the equivalent equation of zero in the derivative of the Lyapunov-Krasovskli function, the proposed stability criteria are formulated in the form of a linear matrix inequality and it is easy to check the robust stability of the considered systems. Numerical examples demonstrate that the proposed criteria are effective.

Impact of time delay and cooperation strategy on the stability of a predator-prey model with Holling type III functional response

In this paper,we propose and analyze a delayed predator-prey model with Holling type III functional response taking into account cooperation behavior in predators.The time delay is introduced in the attack rate to represent the time necessary to trigger the attack.Each analytical result is followed by an ecological interpretation.We investigate the effect of hunting cooperation on both the number and the level of positive steady states.We observe that the level of the positive equilibrium decreases when increasing the hunting cooperation parameter.Then,we study the impact of the delay as well as the cooperation in hunting on the dynamics of the system.We prove that the presence of delay in the attack rate induces stability switches around the coexisting equilibrium when predators cooperate.In addition,we consider the discrete delay as a bifurcation parameter and prove that the model undergoes a Hopf-bifurcation at the coexisting equilibrium when the delay crosses some critical values.Numerical simulations are presented to confirm our analytical findings.

STABILITY ANALYSIS OF A DISCRETE NONLINEAR DELAY SURVIVAL RED BLOOD CELLS MODEL

In this article we consider the kth-order discrete delay survival red blood cells model. The general form of the discrete dynamical system is rewritten as Xn＋l = f（Pn,δn,xn,... ,xn＋1） where Pn,δn converge to the parametric values P and 6. We show that when the parameters are replaced by sequences, the stability results of the original system still hold.

Stability and multi-parametric Hopf bifurcation analyses of viral infection model with time delay

Ever since HIV was first diagnosed in human, a great number of scientific works have been undertaken to explore the biological mechanisms involved in the infection and progression of the disease. This paper deals with stability and bifurcation analyses of mathematical model that represents the dynamics of HIV infection of thymus. The existence and stability of the equilibria are investigated. The model is described by a system of delay differential equations with logistic growth term, cure rate and discrete type of time delay. Choosing the time delay as a bifurcation parameter, the analysis is mainly focused on the Hopf bifurcation problem to predict the existence of a limit cycle bifurcating from the infected steady state.Further, using center manifold theory and normal form method we derive explicit formulae to determine the stability and direction of the limit cycles. Moreover the mitosis rate r also plays a vital role in the model, so we fix it as second bifurcation parameter in the incidence of viral infection. Our analysis shows that, while both the bifurcation parameters can destabilize the equilibrium E＊ and cause limit cycles. Numerical simulations are performed to investigate the qualitative behaviors of the inherent model.

Adaptive Fuzzy Dynamic Surface Control for a Class of Perturbed Nonlinear Time-varying Delay Systems with Unknown Dead-zone

In this paper,adaptive dynamic surface control(DSC) is developed for a class of nonlinear systems with unknown discrete and distributed time-varying delays and unknown dead-zone.Fuzzy logic systems are used to approximate the unknown nonlinear functions.Then,by combining the backstepping technique and the appropriate Lyapunov-Krasovskii functionals with the dynamic surface control approach,the adaptive fuzzy tracking controller is designed.Our development is able to eliminate the problem of 'explosion of complexity' inherent in the existing backstepping-based methods.The main advantages of our approach include:1) for the n-th-order nonlinear systems,only one parameter needs to be adjusted online in the controller design procedure,which reduces the computation burden greatly.Moreover,the input of the dead-zone with only one adjusted parameter is much simpler than the ones in the existing results;2) the proposed control scheme does not need to know the time delays and their upper bounds.It is proven that the proposed design method is able to guarantee that all the signals in the closed-loop system are bounded and the tracking error is smaller than a prescribed error bound,Finally,simulation results demonstrate the effectiveness of the proposed approach.

EXISTENCE OF POSITIVE PERIODIC SOLUTIONS TO A DISCRETE TIME NON-AUTONOMOUS COMPETING SYSTEM WITH FEEDBACK CONTROLS

In this paper,a class of discrete time non-autonomous competing system with feedback controls is considered. With the help of differential equations with piecewise constant arguments,we first propose a discrete model of a continuous non-autonomous competing system with feedback controls. Then,using the coincidence degree and the related continuation theorem as well as some priori estimations,a suficient condition for the existence of positive solutions to difference equations is obtained.

Global stability for multi-species Lotka-Volterra cooperative systems： One hyper-connected mutualistic-species

We propose two models of one hyper-connected mutualistic-species described by delay differential equations of Lotka-Volterra type. An hyper-connected model comprises a central species interacting with a number of peripheral species around it, that is to say, one animal species （pollinators or dispersers） that interacts with several plant species （flowering plants or fruit trees）, or several animal species that interact with one plant species. We derive a necessary and sufficient condition for the global asymptotic stability of the unique coexisting steady state of hyper-connected systems by means of novel Lyapunov functionals.