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Abstract: This paper continues the work contained in two previous papers of the authors, devoted to the qualitative study of the dynamical system generated by a delay differential equation that models leukemia. The problem depends on five parameters and has two equilibria. As already known, at the non-zero equilibrium solution, for certain values of the parameters, Hopf bifurcation occurs. Our aim here is to investigate the Bautin bifurcation for the considered model. By using a second order approximation of the center manifold, we compute the first Lyapunov coefficient for Hopf bifurcation points. We explore the space of parameters in a zone of biological interest and find, by direct computation, points where the first Lyapunov coefficient equals zero. These are candidates for Bautin type bifurcation. For these points we compute the second Lyapunov coefficient, that determines the type of Bautin bifurcation. The computation of the second Lyapunov coefficient requires a fourth order approximation of the center manifold, that we determine. The points with null first Lyapunov coefficient obtained are given in tables and are also plotted on surfaces of Hopf bifurcation obtained by fixing two parameters. Next we vary two parameters around a point in the space of parameters with and numerically explore the behavior of the solution. The results confirm the Bautin type-bifurcation. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the stability of -dimensional stochastic differential delay systems with nonlinear impulsive effects. First, the equivalent relation between the solution of the -dimensional stochastic differential delay system with nonlinear impulsive effects and that of a corresponding -dimensional stochastic differential delay system without impulsive effects is established. Then, some stability criteria for the -dimensional stochastic differential delay systems with nonlinear impulsive effects are obtained. Finally, the stability criteria are applied to uncertain impulsive stochastic neural networks with time-varying delay. The results show that, this convenient and efficient method will provide a new approach to study the stability of impulsive stochastic neural networks. Some examples are also discussed to illustrate the effectiveness of our theoretical results. [Copyright &y& Elsevier]

Abstract: This paper investigates control for a class of nonlinear systems with time-varying delays and convex polytopic uncertainties. A new type of parameter-dependent Lyapunov–Krasovskii functional is introduced to derive delay-dependent sufficient conditions for the optimal control with exponential stability. All the conditions developed in this paper are formulated in terms of linear matrix inequalities. Finally, a numerical example shows the effectiveness of the proposed methodology. [Copyright &y& Elsevier]

Abstract: The problem of Lyapunov stability for functional differential equations in Hilbert spaces is studied. The system to be considered is non-autonomous and the delay is time-varying. Known results on this problem are based on the Gronwall inequality yielding relative conservative bounds on nonlinear perturbations. In this paper, using more general Lyapunov–Krasovskii functional, neither model variable transformation nor bounding restriction on nonlinear perturbations is required to obtain improved conditions for the global exponential stability of the system. The conditions given in terms of the solution of standard Riccati differential equations allow to compute simultaneously the two bounds that characterize the stability rate of the solution. The proposed method can be easily applied to some control problems of nonlinear non-autonomous control time-delay systems. [Copyright &y& Elsevier]

Abstract: A sex-structured mathematical model for heterosexual transmission of HIV/AIDS with explicit incubation period is presented as a system of discrete delay differential equations. The epidemic threshold and equilibria for the model are determined and stabilities are examined. The disease-free equilibrium is shown to be locally and globally stable when the basic reproductive number is less than unity. We use the Lyapunov functional approach to show that the endemic equilibrium is locally asymptotically stable. Further comprehensive qualitative analysis of the model including persistence and permanence are investigated. [Copyright &y& Elsevier]

Abstract: In this paper we consider a nonlinear scalar delay differential equation with variable delays and give some new conditions for the boundedness and stability by means of Krasnoselskii’s fixed point theory. A stability theorem with a necessary and sufficient condition is proved. The results in [T.A. Burton, Stability by fixed point theory or Liapunov’s theory: A comparison, Fixed Point Theory 4 (2003) 15–32; T.A. Burton, T. Furumochi, Asymptotic behavior of solutions of functional differential equations by fixed point theorems, Dynamic Systems and Applications 11 (2002) 499–519; B. Zhang, Fixed points and stability in differential equations with variable delays, Nonlinear Analysis 63 (2005) e233–e242] are improved and generalized. Some examples are given to illustrate our theory. [Copyright &y& Elsevier]

Global asymptotic stability of delayed differential equation of neutral type with off-diagonally increasing right-hand side is considered. First, it is shown that by using a special form of variable transformation, the considered system can be reduced to a differential difference inequality of retarded type. The stability of the trivial solution of the given system is studied with the help of thus obtained inequality. This is a development of the approach previously presented by Ma et al.However, the condition so far obtained on the nonlinear cases by Ma et al. (Nonlinear Anal. Theory Methods Appl. 33 (4) (1998) 367) is rather restrictive and these restrictions are somewhat similar to those for linear systems. In short, the nonlinear cone of invariant region, with the origin as its head, constructed by nonlinear functions, must contain a linear cone inside it. The object of this report is to remove such restriction. [Copyright &y& Elsevier]

Abstract: This paper is devoted to linear parameter systems under linear fractional representations (LFR) of parameter-dependent nonlinear systems with real–rational nonlinearities and point-delayed dynamics. The robust global asymptotic stability of the system either independent of or dependent on the delay sizes is investigated. The associate matrix inequalities are related to the time-derivatives of appropriate Lyapunov functions at all the vertices of the polytope which contains the parameterized uncertainties. [Copyright &y& Elsevier]