Controlling Mackey--Glass chaos

The Mackey--Glass equation, which was proposed to illustrate nonlinear phenomena in physiological control systems, is a classical example of a simple looking time delay system with very complicated behavior. Here we use a novel approach for chaos control: we prove that with well chosen control parameters, all solutions of the system can be forced into a domain where the feedback is monotone, and by the powerful theory of delay differential equations with monotone feedback we can guarantee that the system is not chaotic any more. We show that this domain decomposition method is applicable with the most common control terms. Furthermore, we propose an other chaos control scheme based on state dependent delays.Comment: accepted in Chaos: An Interdisciplinary Journal of Nonlinear Scienc

A Global Attractivity in a Nonmonotone Age-Structured Model with Age Dependent Diffusion and Death Rates

In this paper, we investigated the global attractivity of the positive constant steady state solution of the mature population $w(t,x)$ governed by the age-structured model: \begin{equation*} \left\{\begin{array}{ll} \frac{\partial u}{\partial t}+\frac{\partial u}{\partial a}=D(a)\frac{\partial ^2 u}{\partial x^2} - d(a)u, & t\geq t_0\geq A_l,\;a\geq 0,\; 0< x< \pi,\\ w(t,x)=\int_r^{A_l}u(t,a,x)da,& t\geq t_0\geq A_l,\; 0<x<\pi,\\ u(t,0,x)=f(w(t,x)), & t\geq t_0\geq A_l,\; 0<x<\pi,\\ u_x(t,a,0)=u_x(t,a,\pi)=0,\;& t\geq t_0\geq A_l,\; a \geq 0, \end{array} \right. \end{equation*} when the diffusion rate $D(a)$ and the death rate $d(a)$ are age dependent, and when the birth function $f(w)$ is nonmonotone. We also presented some illustrative examples.Comment: 11 page

Oscillations in I/O monotone systems under negative feedback

Oscillatory behavior is a key property of many biological systems. The Small-Gain Theorem (SGT) for input/output monotone systems provides a sufficient condition for global asymptotic stability of an equilibrium and hence its violation is a necessary condition for the existence of periodic solutions. One advantage of the use of the monotone SGT technique is its robustness with respect to all perturbations that preserve monotonicity and stability properties of a very low-dimensional (in many interesting examples, just one-dimensional) model reduction. This robustness makes the technique useful in the analysis of molecular biological models in which there is large uncertainty regarding the values of kinetic and other parameters. However, verifying the conditions needed in order to apply the SGT is not always easy. This paper provides an approach to the verification of the needed properties, and illustrates the approach through an application to a classical model of circadian oscillations, as a nontrivial ``case study,'' and also provides a theorem in the converse direction of predicting oscillations when the SGT conditions fail.Comment: Related work can be retrieved from second author's websit

The exponential ordering for nonautonomous delay systems with applications to compartmental Nicholson systems

Producción CientíficaThe exponential ordering is exploited in the context of nonautonomous delay systems, inducing monotone skew-product semiflows under less restrictive conditions than usual. Some dynamical concepts linked to the order, such as semiequilibria, are considered for the exponential ordering, with implications for the determination of the presence of uniform persistence or the existence of global attractors. Also, some important conclusions on the long-term dynamics and attraction are obtained for monotone and sublinear delay systems for this ordering. The results are then applied to almost periodic Nicholson systems and new conditions are given for the existence of a unique almost periodic positive solution which asymptotically attracts every other positive solution.The first three authors were partly supported by MICIIN/FEDER project RTI2018- 096523-B-I00 and by Universidad de Valladolid under project PIP-TCESC-2020. The fourth author was partly supported by MICINN/FEDER under projects RTI2018-096523-B-I00 and PGC2018-097565-B-I0

Attractors in almost periodic Nicholson systems and some numerical simulations

Producción CientíficaThe existence of a global attractor is proved for the skew-product semiflow induced by almost periodic Nicholson systems and new conditions are given for the existence of a unique almost periodic positive solution which exponentially attracts every other positive solution. Besides, some numerical simulations are included to illustrate our results in some concrete Nicholson systems.Ana M. Sanz: MICIIN/FEDER under project PID2021-125446NB-I00 and by Universidad de Valladolid under project PIP-TCESC-2020Víctor M. Villarragut: MICIIN/FEDER under project PID2021- 125446NB-I0

Delay-independent asymptotic stability in monotone systems

Monotone systems comprise an important class of dynamical systems that are of interest both for their wide applicability and because of their interesting mathematical properties. It is known that under the property of quasimono-tonicity time-delayed systems become monotone, and some remarkable properties have been reported for such systems. These include, for example, the fact that for linear systems global asymptotic stability of the undelayed system implies global asymptotic stability for the delayed system under arbitrary bounded delays. Nevertheless, extensions to nonlinear systems have thus far relied on various restrictive conditions, such as homogeneity and subhomogeneity, and it has been conjectured that these can be relaxed. Our aim in this paper is to show that this is feasible for a general class of nonlinear monotone systems, by deriving asymptotic stability results in which simple properties of the undelayed system lead to delay-independent stability. In particular, one of our results is to show that if the undelayed system has a convergent trajectory that is unbounded in all components as t → -∞ then the system is globally asymptotically stable for arbitrary time-varying delays. This follows from a more general result derived in the paper where delay-independent regions of attraction are quantified from the asymptotic behavior of individual trajectories of the undelayed system. This result recovers various known delay-independent stability results, and several examples are included in the paper to illustrate the significance of the proposed stability conditions.This is the author accepted manuscript. The final version is available from IEEE via http://dx.doi.org/10.1109/ACC.2015.717206