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This thesis contains some results on the behavior of a delay differential equation (DDE) with two delays, at a Hopf bifurcation, for the nonzero equilibrium, using the growth rate, r, as bifurcation parameter. This DDE is a model for population growth, incorporating a maturation delay, and a second delay in the harvesting term. Considering a Taylor expansion of the non-dimensionalized model, we find a region of stability for the nonzero equilibrium, after which we find a pair of ODEs which help define the flow on the center manifold. We then find an expression for the first Lypapunov coefficient, which changes sign, so we also find the second Lyapunov coefficient, allowing us to predict multi-stability in the model. Numerical simulations provide examples of the behavior expected. For a similar model with one delay (PMC model), we prove the Hopf bifurcation at the nonzero equilibrium is always supercritical.

In this paper we explore bifurcations, in particular the Hopf bifurcation. We study this especially in connection with the Brusselator, which is a model of certain chemical reaction-diffusion systems. After a thorough exploration of what a bifurcation is and what classifications there are, we give graphic representations of an occurring Hopf bifurcation in the Brusselator. When an additional forcing term is added, behavior changes dramatically. This includes the introduction of a horseshoe in the time map as well as a strange attractor in the system.

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This thesis presents a study of the centre bifurcation and chaotic behaviour of three dimensional Lotka-Volterra systems. In two dimensional systems, Christopher (2005) considered a simple computational approach to estimate the cyclicity bifurcating from the centre. We generalized the technique to estimate the cyclicity of the centre in three dimensional systems. A lower bounds is given for the cyclicity of a hopf point in the three dimensional Lotka-Volterra systems via centre bifurcations. Sufficient conditions for the existence of a centre are obtained via the Darboux method using inverse Jacobi multiplier functions. For a given centre, the cyclicity is bounded from below by considering the linear parts of the corresponding Liapunov quantities of the perturbed system. Although the number obtained is not new, the technique is fast and can easily be adapted to other systems. The same technique is applied to estimate the cyclicity of a three dimensional system with a plane of singularities. As a result, eight limit cycles are shown to bifurcate from the centre by considering the quadratic parts of the corresponding Liapunov quantities of the perturbed system. This thesis also examines the chaotic behaviour of three dimensional Lotka-Volterra systems. For studying the chaotic behaviour, a geometric method is used. We construct an example of a three dimensional Lotka-Volterra system with a saddle-focus critical point of Shilnikov type as well as a loop. A construction of the heteroclinic cycle that joins the critical point with two other critical points of type planar saddle and axial saddle is undertaken. Furthermore, the local behaviour of trajectories in a small neighbourhood of the critical points is investigated. The dynamics of the Poincare map around the heteroclinic cycle can exhibit chaos by demonstrating the existence of a horseshoe map. The proof uses a Shilnikov-type structure adapted to the geometry of these systems. For a good understanding of the global dynamics of the system, the behaviour at infinity is also examined. This helps us to draw the global phase portrait of the system. The last part of this thesis is devoted to a study of the zero-Hopf bifurcation of the three dimensional Lotka-Volterra systems. Explicit conditions for the existence of two first integrals for the system and a line of singularity with zero eigenvalue are given. We characteristic the parameters for which a zero-Hopf equilibrium point takes place at any points on the line. We prove that there are three 3-parameter families exhibiting such equilibria. First order of averaging theory is also applied but we show that it gives no information about the possible periodic orbits bifurcating from the zero-Hopf equilibria.

Notre premier objectif dans ce travail est de donner une démonstration du changement<br />de la stabilité de la branche supercritique de solutions périodiques bifurquées<br />dans le cadre des équations diérentielles à retard, en se basant sur les deux étapes<br />suivantes:<br />(i) Réduction de l'équation à un système en dimension deux par la formule de variation<br />de la constante et le théorème de la variété centre.<br />(ii) Estimation de la distance entre la solution de l'équation initiale et la solution pé-<br />riodique bifurquée.<br />Nous obtenons ainsi un domaine de stabilité de la branche supercritique.<br />Le second objectif est d'étudier une équation différentielle à un seul retard issue<br />d'un modèle en dynamique de population cellulaire sanguine (Haematopoiese).<br />Ce modèle, initialement introduit par Mackey (1978) présente une position d'équilibre<br />triviale qui est instable et une famille de positions d'équilibre non triviales dont la<br />stabilité dépend du retard.<br />Nous montrons l'existence d'une valeur critique ¿0 du retard \tau autour de laquelle nous<br />obtenons un changement de stabilité de cette famille de positions d'équilibre en fonction<br />du retard.<br />Nous avons ainsi introduit un modèle approché en fonction de cette valeur critique du<br />retard qui coincide avec celui de Mackey pour la valeur du retard \tau = \tau_{0}. Le modèle<br />approché possède un point d'équilibre trivial et un non trivial ne dépendant pas du<br />retard.<br />Par une étude du modèle approché analogue à celle du modèle de Mackey, nous obtenons<br />en particulier l'existence d'une branche de solutions périodiques bifurquées à<br />partir du point d'équilibre non trivial. Enn nous donnons un algorithme explicite de<br />calcul des éléments de la bifurcation.

Lienard Equations serve as the elegant models for oscillating circuits. Motivated by this fact, this thesis addresses the stability property of a class of delayed Lienard equations. It shows the existence of the Hopf bifurcation around the steady state. It has both practical and theoretical importance in determining the criticality of the Hopf bifurcation. For such purpose, center manifold analysis on the bifurcation line is required. This thesis uses operator differential equation formulation to reduce the infinite dimensional delayed Lienard equation onto a two-dimensional manifold on the critical bifurcation line. Based on the reduced two-dimensional system, the so called Poincare-Lyapunov constant is analytically determined, which determines the criticality of the Hopf bifurcation. Numerics based on a Matlab bifurcation toolbox (DDE-Biftool) and Matlab solver (DDE-23) are given to compare with the theoretical calculation. Two examples are given to illustrate the method.

Cheyne-Stokes respiration is a distinct breathing pattern consisting of periods of hyperpnea followed by apneas, with unknown etiology. One in two patients with congestive heart failure suffer from this condition. Researchers hypothesize that key factors in CSR are the fluid shift from the standing to supine position and the differences between genders. A mathematical model of the cardio-respiratory system was constructed using parameter values from real data. Hopf bifurcation analysis was used to determine regions of stable versus oscillatory breathing patterns. In the model, Cheyne-Stokes respiration is more likely to occur while in the supine position and males are more likely to develop Cheyne-Stokes than females. These findings, which are in agreement with clinical experience, suggest that both gender and fluid shift contribute to the pathogenesis of Cheyne-Stokes respiration, and that physical quantities such as blood volumes and neural feedback may be sufficient to explain the observations of CSR. / Department of Mathematics and Statistics

Circadian rhythms are periodic physiological cycles that recur about every 24 hours, by means of which organisms integrate their physiology and behavior to the daily cycle of light and temperature imposed by the rotation of the earth. Circadian derives from the Latin word circa "about" and dies "day". Circadian rhythms have three noteworthy properties. They are endogenous, that is, they persist in the absence of external cues (in an environment of constant light intensity, temperature, etc.). Secondly, they are temperature compensated, that is, the nearly 24 hour period of the endogenous oscillator is remarkably independent of ambient temperature. Finally, they are phase shifted by light. The circadian rhythm can be either advanced or delayed by applying a pulse of light in constant darkness. Consequently, the circadian rhythm will synchronize to a periodic light-dark cycle, provided the period of the driving stimulus is not too far from the period of the endogenous rhythm. A window on the molecular mechanism of 24-hour rhythms was opened by the identification of circadian rhythm mutants and their cognate genes in Drosophila, Neurospora, and now in other organisms. Since Konopka and Benzer first discovered the period mutant in Drosophila in 1971 (Konopka and Benzer, 1971), there have been remarkable developments. Currently, the consensus opinion of molecular geneticists is that the 24-hour period arises from a negative feedback loop controlling the transcription of clock genes. However, a better understanding of this mechanism requires an approach that integrates both mathematical and molecular biology. From the recent discoveries in molecular biology and through a mathematical approach, we propose that the mechanism of circadian rhythm is based upon the combination of both negative and positive feedback. / Master of Science

Este trabalho é concernente a periodicidade na equação de Wright. Provaremos a existência de soluções periódicas não constantes, explorando o conceito de ejetividade de um teorema de ponto fixo. Além disso, provamos a existência de uma seqüência infinita de Bifurcação de Hopf. / This work is concerned with periodicity in the Wright's equation. We prove the existence of nonconstant periodic solutions by exploiting the ejectivity concept in a theorem of fixed point. Furthemore, we prove the existence of an infinite sequence of Hopf Bifurcations.