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The nonlinear delay differential equation x ˙ ( t ) = r ( t ) [ g ( t , x t ) − h ( x ( t ) ) ] , t ≥ 0 is considered. Sufficient conditions are established for the uniform permanence of the positive solutions of the equation. In several particular cases, explicit formulas are given for the upper and lower limit of the solutions. In some special cases, we give conditions which imply that all solutions have the same asymptotic behavior, in particular, when they converge to a periodic or constant steady-state. [ABSTRACT FROM AUTHOR]

• Delay differential equations with impulses. • Self-support condition. • Numerical approximation using equations with piecewise constant arguments. • Existence of periodic solutions. In this paper we consider a scalar linear delay equation with constant delay associated with an impulsive self-support condition. We define a numerical approximation scheme using a sequence of approximate delay equations with piecewise constant arguments, and we show its theoretical convergence. We present numerical examples to illustrate the applicability of the method, and we also observe existence of periodic solutions of the impulsive delay equation using numerical studies. [ABSTRACT FROM AUTHOR]

Abstract: Unwanted relative vibrations between the tool and the workpiece represent significant challenges in high-speed machining. In order to avoid this problem, one needs to specify ranges for system parameters (spindle speed, depth of cut) for which the process is stable, i.e., to obtain a so-called stability chart. Such stability charts usually can only be given by numerical means which is illustrated in the paper for a single degree of freedom model of milling. In this paper, we establish the convergence of the semi-discretization approximation method for a class of delay equations modeling the milling process. Moreover, we show that semi-discretization preserves asymptotic stability of the original equation, thus it can be used to obtain good approximations for the stability charts. [Copyright &y& Elsevier]

Abstract: This paper investigates the bounded input bounded output (BIBO) stability in a class of control system of nonlinear differential equations with time-delay. The proofs are based on our studies on the boundedness of the solutions of a general class of nonlinear Volterra integral equations. [Copyright &y& Elsevier]