A two dimensional two-delays differential system modeling the dynamics of stem-like cells and white-blood cells in Chronic Myelogenous Leukemia is considered. All three types of stem cell division (asymmetric division, symmetric renewal and symmetric differentiation) are present in the model. Stability of equilibria is investigated and emergence of periodic solutions of limit cycle type, as a result of a Hopf bifurcation, is eventually shown. The stability of these limit cycles is studied using the first Lyapunov coefficient.

Stability of the zero solution is analyzed for a family of switched systems indexed by a parameter, each system having λ = 0 in the spectrum of the Jacobian matrix calculated in zero. It is proved that existence of a common quadratic Lyapunov function for some lower dimensional linear systems is sufficient to ensure local uniform stability of the zero solution of the switched non-linear system and a regular asymptotic behaviour. An application to control synthesis for stabilizing equilibria in a switched non-linear control system modelling an electrohydraulic servomechanism is given.

Heat equations with memory of Gurtin-Pipkin type (i.e. Eq. (1) with $ \alpha=0 $) have controllability properties which strongly resemble those of the wave equation. Instead, recent counterexamples show that when $ \alpha>0 $ the control properties do not parallel those of the (memoryless) heat equation, in the sense that there are square integrable initial conditions which cannot be controlled to zero. The proof of this fact, in previous papers, consists in the construction of two quite special examples of systems with memory which cannot be controlled to zero. Here we prove that lack of controllability holds in general, for every smooth memory kernel $ M(t) $.

Existence and stability of periodic solutions are studied for a system of delay differential equations with two delays, with periodic coefficients. It models the evolution of hematopoietic stem cells and mature neutrophil cells in chronic myelogenous leukemia under a periodic treatment that acts only on mature cells. Existence of a guiding function leads to the proof of the existence of a strictly positive periodic solution by a theorem of Krasnoselskii. The stability of this solution is analysed.