Nonautonomous chemostats with variable delays

The appearance of delay terms in a chemostat model can be fully justified since the future behavior of a dynamical system does not in general depend only on the present but also on its history. Sometimes only a short piece of history provides the relevant influence (bounded or finite delay), while in other cases it is the whole history that has to be taken into account (unbounded or infinite delay). In this paper a chemostat model with time variable delays and wall growth, hence a nonautonomous problem, is investigated. The analysis provides sufficient conditions for the asymptotic stability of nontrivial equilibria of the chemostat with variable delays, as well as for the existence of nonautonomous pullback attractors

On Nonoscillation of Mixed Advanced-Delay Differential Equations with Positive and Negative Coefficients

For a mixed (advanced--delay) differential equation with variable delays and coefficients $$\dot{x}(t) \pm a(t)x(g(t)) \mp b(t)x(h(t)) = 0, t\geq t_0$$ where $$a(t)\geq 0, b(t)\geq 0, g(t)\leq t, h(t)\geq t$$ explicit nonoscillation conditions are obtained.Comment: 17 pages; 2 figures; to appear in Computers & Mathematics with Application

Attractors for Differential Equations with Multiple Variable Delays

We establish some results on the existence of pullback attractors for non-autonomous delay di fferential equations with multiple delays. In particular, we generalise some recent works on the existence of pullback attractors for delay di fferential equations

Implementation of an Internet-controlled system under variable delays

International audienceThis work deals with the control and the observation of a remote system using Internet as a communication line. The process consists in a Slave part S, with poor computing capacity, connected to a Master M through the Internet. The internal times of both M and S are synchronized thanks to a common GPS clock. The global system must ensure some speed performance whatever the delay variations. The main technical restriction is that the delays are less than some known bound, even if the framework we consider takes into account the perturbations generated by the Internet communication delays, by the packet losses and by the sampling phenomenon as well. In what concerns the design of the observer and controller, most of the theorems to be used here refer to proofs given in [14], and this will allow for focussing on the implementation aspects

Characterization and modification of cavitation pattern in shock wave lithotripsy

The temporal and spatial dynamics of cavitation bubble cloud growth and collapse in extracorporeal shock wave lithotripsy (ESWL) is studied experimentally. The first objective is obtaining reproducible cloud patterns experimentally and comparing them with FDTD-calculations. Second, we describe a method to modify the cavitation pattern by timing two consecutive pressure waves at variable delays. It is found that the spatial and temporal dynamics of the cavitation bubble can be varied in large ranges. The ability to control cavitation dynamics allows discussing strategies for improvement of medical and biological applications of shock waves such as cell membrane poration and stone fragmentation

(R2066) New Results of Ulam Stabilities of Functional Differential Equations of First Order Including Multiple Retardations

In this study, we pay attention to a functional differential equation (FDE) of first order including N-variable delays. We construct new sufficient conditions in relation to the Hyers-Ulam stability (HUS) and the generalized Hyers-Ulam-Rassias stability (GHURS ) of the FDE of first order including N-variable delays. By using Banach contraction principle (BCP), Picard operator and Gronwall lemma, we confirm two new theorems in relation to the HUS and the GHURS. The results of this study are new and extend, improve some earlier results of the HUS and the GHURS