On the dominance of roots of characteristic equations for neutral functional differential equations

We present a sufficient condition for a zero of a function that arises typically as the characteristic equation of a linear functional differential equations of neutral type, to be simple and dominant. This knowledge is useful in order to derive the asymptotic behaviour of solutions of such equations. A simple characteristic equation, arisen from the study of delay equations with small delay, is analyzed in greater detail. (C) 2009 Elsevier Inc. All rights reserved.

Cascades of Hopf bifurcations from boundary delay

We consider a 1-dimensional reaction-diffusion equation with nonlinear boundary conditions of logistic type with delay. We deal with non-negative solutions and analyze the stability behavior of its unique positive equilibrium solution, which is given by the constant function u equivalent to 1. We show that if the delay is small, this equilibrium solution is asymptotically stable, similar as in the case without delay. We also show that, as the delay goes to infinity, this equilibrium becomes unstable and undergoes a cascade of Hopf bifurcations. The structure of this cascade will depend on the parameters appearing in the equation. This equation shows some dynamical behavior that differs from the case where the nonlinearity with delay is in the interior of the domain. (C) 2009 Elsevier Inc. All rights reserved.

ON THE SUPERCRITICALITY OF THE FIRST HOPF BIFURCATION IN A DELAY BOUNDARY PROBLEM

The goal of this paper is to analyze the character of the first Hopf bifurcation (subcritical versus supercritical) that appears in a one-dimensional reaction-diffusion equation with nonlinear boundary conditions of logistic type with delay. We showed in the previous work [Arrieta et al., 2010] that if the delay is small, the unique non-negative equilibrium solution is asymptotically stable. We also showed that, as the delay increases and crosses certain critical value, this equilibrium becomes unstable and undergoes a Hopf bifurcation. This bifurcation is the first one of a cascade occurring as the delay goes to infinity. The structure of this cascade will depend on the parameters appearing in the equation. In this paper, we show that the first bifurcation that occurs is supercritical, that is, when the parameter is bigger than the delay bifurcation value, stable periodic orbits branch off from the constant equilibrium.

Implants of polyanionic collagen matrix in bone defects of ovariectomized rats

In recent years, there has been a great interest in the development of biomaterials that could be used in the repair of bone defects. Collagen matrix (CM) has the advantage that it can be modified chemically to improve its mechanical properties. The aim of the present study was to evaluate the effect of three-dimensional membranes of native or anionic (submitted to alkaline treatment for 48 or 96 h) collagen matrix on the consolidation of osteoporosis bone fractures resulting from the gonadal hormone alterations caused by ovariectomy in rats subjected to hormone replacement therapy. The animals received the implants 4 months after ovariectomy and were sacrificed 8 weeks after implantation of the membranes into 4-mm wide bone defects created in the distal third of the femur with a surgical bur. Macroscopic analysis revealed the absence of pathological alterations in the implanted areas, suggesting that the material was biocompatible. Microscopic analysis showed a lower amount of bone ingrowth in the areas receiving the native membrane compared to the bone defects filled with the anionic membranes. In ovariectomized animals receiving anionic membranes, a delay in bone regeneration was observed mainly in animals not subjected to hormone replacement therapy. We conclude that anionic membranes treated with alkaline solution for 48 and 96 h presented better results in terms of bone ingrowth.