Estimating the globally attractive set and positively invariant set of a unified chaotic system

By constructing two suitable generalized Lyapunov functions,we derived a generalized ellipsoidal estimate of the globally attractive set and positively invariant set of the unified chaotic system with the parameters α=1/29 and 1/29<α<2/29,respectively,which extends some related results of Li,et al. [Li DM,Lu JA,Wu XQ,Chen GR,Estimating the global basin of attraction and positively invariant set for the Lorenz system and a unified chaotic system,Journal of Mathematical Analysis and Applications,2006,323(2): 844-853]. The theoretical results obtained in this paper will find wide application in chaos control and synchronization.

GLOBALLY ATTRACTING SOLUTIONS TO IMPULSIVE FRACTIONAL DIFFERENTIAL INCLUSIONS OF SOBOLEV TYPE

We study a generalized Cauchy problem associated with a class of impulsive fractional differential inclusions of Sobolev type in Banach spaces. Our aim is to prove the existence of a compact set of globally attracting solutions to the problem in question. An application to fractional partial differential equations subject to impulsive effects is given to illustrate our results.

Global Attractivity of a Kind of Delay Difference Equations

Several new sufficient conditions are given for the global attractivity of solutions of a kind of delay difference equations. They either include or improve some known results and put the study of Ladas' conjecture forward.

Existence and global attractivity of unique positive almost periodic solution for a model of hematopoiesis

Sufficient conditions are obtained which guarantee the uniform persistence and global attractivity of solutions for the model of hematopoiesis. Then some criteria are established for the existence, uniqueness and global attractivity of almost periodic solutions of almost periodic system.

Differential Susceptibility SIR Epidemic Model with Time Delay and Pulse Vaccination

Differential susceptibility SIR epidemic models with time delay and pulse vaccination is studied in this paper. We show that there exists an infection-free periodic solution by using the comparison method, which is globally attractive provided that R1 〈 1, and that R2 〉 1 implies the disease is permanent, which means that after some period of time the disease will become endemic.

Globally exponentially attractive sets of the family of Lorenz systems

In this paper, the concept of globally exponentially attractive set is proposed and used to consider the ultimate bounds of the family of Lorenz systems with varying parameters. Explicit estimations of the ultimate bounds are derived. The results presented in this paper contain all the existing results as special cases. In particular, the critical cases, b→ 1^＋ and a→0^＋, for which the previous methods failed, have been solved using a unified formula.

GLOBAL ATTRACTIVITY AND GLOBAL EXPONENTIAL STABILITY FOR DELAYED HOPFIELD NEURAL NETWORK MODELS

Some global properties such as global attractivity and global exponential stability for delayed Hopfield neural networks model, under the weaker assumptions on nonlinear activation functions, are concerned. By constructing suitable Liapunov function, some simpler criteria for global attractivity and global exponential stability for Hopfield continuous neural network,; with time delays are presented.

GLOBAL ATTRACTIVITY IN A CLASS OF HIGHER-ORDER NONLINEAR DIFFERENCE EQUATION

In this paper the global attractivity of the nonlinear difference equationis investigated, where a,b, A ∈ (0,∞), k is an positive integer and the initial conditions x- k, …, x-1 and x0 are arbitrary positive numbers. It is shown that the unique positive equilibrium of the equation is global attractive. As a corollary, the result gives a positive confirmation on the conjecture presented by Kocic and Ladas [1,p154].

GLOBAL ATTRACTIVITY OF NEUTRAL DIFFERENCE EQUATIONS WITH POSITIVE AND NEGATIVE COEFFICIENTS

A neutral difference equation with positive and negative coefficientsΔ(x n-c nx n-k )+p nx n-l -q nx n-r =0, n=0,1,2,...,is considered and a sufficient condition for the global attractivity of the zero solution of this equation is obtained, which improves and extends the all known results in the literature.

Delayed stage-structured predator-prey model with impulsive perturbations on predator and chemical control on prey

We consider a delayed stage-structured pest management predator-prey system with impulsive transmitting on predator and chemical control on prey. Sufficient conditions of the global attractiveness of the pest-extinction boundary periodic solution and permanence of the system are obtained. We also prove that all solutions of the system are uniformly ultimately bounded. Our results provide reliable tactical basis for practical pest management.

Global dynamics behaviors for new delay SEIR epidemic disease model with vertical transmission and pulse vaccination

A robust SEIR epidemic disease model with a profitless delay and vertical transmission is formulated, and the dynamics behaviors of the model under pulse vaccination are analyzed. By use of the discrete dynamical system determined by the stroboscopic map, an ‘infection-free＇ periodic solution is obtained, further, it is shown that the ‘infection-free＇ periodic solution is globally attractive when some parameters of the model are under appropriate conditions. Using the theory on delay functional and impulsive differential equation, the sufficient condition with time delay for the permanence of the system is obtained, and it is proved that time delays, pulse vaccination and vertical transmission can bring obvious effects on the dynamics behaviors of the model. The results indicate that the delay is ‘profitless’.

COEXISTENCE FOR MULTIPLE LARGEST REPRODUCTION RATIOS OF A MULTI-STRAIN SIS EPIDEMIC MODEL

In this paper, to complete the global dynamics of a multi-strains SIS epidemic model, we establish a precise result on coexistence for the cases of the partial and complete duplicated multiple largest reproduction ratios for this model.

Permanence and global attractivity of stage-structured predator-prey model with continuous harvesting on predator and impulsive stocking on prey

We investigate a stage-structured delayed predator-prey model with impulsive stocking on prey and continuous harvesting on predator. According to the fact of biological resource management, we improve the assumption of a predator-prey model with stage structure for predator population that each individual predator has the same ability to capture prey. It is assumed that the immature and mature individuals of the predator population are divided by a fixed age, and immature predator population does not have the ability to attach prey. Sufficient conditions are obtained, which guarantee the global attractivity of predator-extinction periodic solution and the permanence of the system. Our results show that the behavior of impulsive stocking on prey plays an important role for the permanence of the system, and provide tactical basis for the biological resource management. Numerical analysis is presented to illuminate the dynamics of the system.

A Stage-structured Predator-prey Model with Time Delay and Impulsive Effects

A delayed predator-prey model concerning impulsive spraying pesticides and releasing natural enemies is proposed and investigated,in which both the prey and the predator have a history that takes them through two stages:immature and mature.The global attractiveness of the pest-eradication periodic solution is discussed,and sufficient condition is obtained for the permanence of the system.Further,numerical simulations show that there is a characteristic sequence of bifurcations leading to a chaotic dynamics,which implies that the system with constant periodic impulsive perturbations admits rich and complex dynamics.

GLOBAL ATTRACTIVITY IN NICHOLSON＇S BLOWFLIES

In this paper we further study the delay differential equation N .(t)=-δN(t)+pN(t-τ)e -aN(t-τ) , t0(*) used in describing the dynamics of Nicholson’s blowflies. When p】δ ,we establish new sufficient conditions for the positive equilibrium N * of (*) which is a global attractor.

Dynamics of a Nonautonomous SIR Model with Time-Varying Impulsive Release and General Nonlinear Incidence Rate in a Polluted Environment

In a polluted environment, considering the biological population infected with a kind of disease and hunted by human beings, we formulate a nonautonomous SIR population-epidemic model with time-varying impulsive release and general nonlinear incidence rate and investigate dynamical behaviors of the model. Under the reasonable assumptions, the sufficient conditions which guarantee the globally attractive of the disease-free periodic solution and the permanence of the infected fish are established, that is, the infected fish dies out if , whereas the disease persists if . To substantiate our theoretical results, extensive numerical simulations are performed for a hypothetical set of parameter values.

GLOBAL ATTRACTIVITY FOR A CLASS OF NONLINEARDELAY DIFFERENCE EQUATIONS

The global attractivity of the delay difference equation Deltax(n) + a(n)x(n) + f(n, Sigma(s = -k)(0)(q) over bar (s), (n) x(s + n)) = 0, which includes the discrete type of many mathematical ecological equations, was discussed. The sufficient conditions that guarantee every solution to converge to zero are obtained. Many known results are improved and generated.

PERMANENCE AND ASYMPTOTIC PROPERTIES OF NONLINEAR DELAY DIFFERENCE EQUATIONS

The asymptotic behavior of a class of nonlinear delay difference equation wax studied . Some sufficient conditions are obtained for permanence and global attractivity . The results can be applied to a claxs of nonlinear delay difference equations and to the delay discrete Logistic model and some known results are included.

Partial extinction, permanence and global attractivity innonautonomous n-species Lotka-Volterra competitivesystems with impulses

In this paper, the qualitative properties of general nonautonomous Lotka-Volterran-species competitive systems with impulsive e？ects are studied. Some new criteria on thepermanence, extinction and global attractivity of partial species are established by used themethods of inequalities estimate and Liapunov functions. As applications, nonautonomous twospecies Lotka-Volterra systems with impulses are discussed.

A reaction-difusion model with nonlinearity driven difusion

In this paper, we deal with the model with a very general growth law and an M- driven diffusion For the general case of time dependent functions M and #, the existence and uniqueness for positive solution is obtained. If M and # are T0-periodic functions in t, then there is an attractive positive periodic solution. Furthermore, if M and # are time-independent, then the non-constant stationary solution M（x） is globally stable. Thus, we can easily formulate the conditions deriving the above behaviors for specific population models with the logistic growth law, Gilpin-Ayala growth law and Gompertz growth law, respectively. We answer an open problem proposed by L. Korobenko and E. Braverman in [Can. Appl. Math. Quart. 17（2009） 85-104].