Classification of coupled dynamical systems with multiple delays: Finding the minimal number of delays

In this article we study networks of coupled dynamical systems with time-delayed connections. If two such networks hold different delays on the connections it is in general possible that they exhibit different dynamical behavior as well. We prove that for particular sets of delays this is not the case. To this aim we introduce a componentwise timeshift transformation (CTT) which allows to classify systems which possess equivalent dynamics, though possibly different sets of connection delays. In particular, we show for a large class of semiflows (including the case of delay differential equations) that the stability of attractors is invariant under this transformation. Moreover we show that each equivalence class which is mediated by the CTT possesses a representative system in which the number of different delays is not larger than the cycle space dimension of the underlying graph. We conclude that the 'true' dimension of the corresponding parameter space of delays is in general smaller than it appears at first glance

Behaviors and global dynamics of population models living in periodically fluctuating environments

Dans ce projet, nous étudions certaines classes d équations non-linéaires aux différences pour toutes les valeurs non-négatives admissibles de paramètres et de conditions initiales. Dans la première partie, nous présentons quelques définitions initiales et nécessaires de la stabilité dans la littérature des équations aux différences. Nous exprimons également plusieurs résultats connus et quelques théorèmes qui seront utiles dans notre recherche à la suite. Dans le deuxièllle chapitre, on étudie l'intervalle invariant, le caractère des sellli-cycles, la stabilité globale, et le "boundedness" de l'équation aux différences [Formule mathématique]. Dans la deuxième partie de ce travail, nous étudions la famille de modèles de la population, d'ordre supérieur, de l'équation logistique non-autonorne [Formule mathématique]. En particulier, le comporternent périodique, l'attractivité des solutions et la stabilité de solutions sont examinés en détail. Ce modèle représente les différentes croissances de population telles que la plupart des saumons et "spruce budworms" ayant assez de nourriture (feuillage). L'impact du caractère saisonnier , qui est indubitablenlent présent dans la croissance de la population, sur le comporternent des solutions (croissance de la population) est également envisagé. Enfin, nous présentons un modèle déterministe de l'infection par le VIH (virus d'immunodéficience hurnaine) en présence de la trithérapie. Puis la stabilité globale asymptotique de l'équilibre "disease-free" et l'équilibre endémique sont étudiés pour le modèle continu. En outre, le modèle est implicitement estimé par la méthode des différences finies, un système d'équations aux différences, afin de résoudre le système d 'IVP (problème aux valeurs initiales) . La dynamique du modèle estimé est complètement déterminée par une quantité de seuil R appelée le "basic reproduction number". Lorsque le nombre associé à la reproduction est inférieur à l'unité, le résultat indique que l'infection par le VIH peut être éliminée de la personne infectée. Un équilibre stable endémique existe lorsque le nombre associé à la reproduction est supérieur à l'unité (conduisant à la persistance et l'existence du VIH au sein de la personne infectée, puis de la communauté)

Sensitivity analysis of oscillator models in the space of phase-response curves: Oscillators as open systems

Oscillator models are central to the study of system properties such as entrainment or synchronization. Due to their nonlinear nature, few system-theoretic tools exist to analyze those models. The paper develops a sensitivity analysis for phase-response curves, a fundamental one-dimensional phase reduction of oscillator models. The proposed theoretical and numerical analysis tools are illustrated on several system-theoretic questions and models arising in the biology of cellular rhythms

Analysis of two types of cyclic biological system models with time delays

Ankara : The Department of Electrical and Electronics Engineering and the Institute of Engineering and Sciences of Bilkent University, 2011.Thesis (Master's) -- Bilkent University, 2011.Includes bibliographical references leaves 95-97.In this thesis, we perform the stability analysis of two types of cyclic biological processes involving time delays. We analyze the genetic regulatory network having nonlinearities with negative Schwarzian derivatives. Using preliminary results on Schwarzian derivatives, we present necessary conditions implying the global stability and existence of periodic solutions regarding the genetic regulatory network. We also analyze homogenous genetic regulatory network and prove some stability conditions which only depend on the parameters of the nonlinearity function. In the thesis, we also perform a local stability analysis of a dynamical model of erythropoiesis which is another type of cyclic system involving time delay. We prove that the system has a unique fixed point which is locally stable if the time delay is less than a certain critical value, which is analytically computed from the parameters of the model. By the help of simulations, existence of periodic solutions are shown for delays greater than this critical value.Ahsen, Mehmet ErenM.S

Asymptotic Stability and Asymptotic Synchronization of Memristive Regulatory-Type Networks

Memristive regulatory-type networks are recently emerging as a potential successor to traditional complementary resistive switch models. Qualitative analysis is useful in designing and synthesizing memristive regulatory-type networks. In this paper, we propose several succinct criteria to ensure global asymptotic stability and global asymptotic synchronization for a general class of memristive regulatory-type networks. The experimental simulations also show the performance of theoretical results

Monotone and near-monotone biochemical networks

Monotone subsystems have appealing properties as components of larger networks, since they exhibit robust dynamical stability and predictability of responses to perturbations. This suggests that natural biological systems may have evolved to be, if not monotone, at least close to monotone in the sense of being decomposable into a “small” number of monotone components, In addition, recent research has shown that much insight can be attained from decomposing networks into monotone subsystems and the analysis of the resulting interconnections using tools from control theory. This paper provides an expository introduction to monotone systems and their interconnections, describing the basic concepts and some of the main mathematical results in a largely informal fashion