Bifurcation Analysis In Several Classes Of Dynamical Systems With Delay

Delayed dynamical systems have attracted much attention in the past few decadessince many process in natural science can be described by such systems. So it’s of greatsignificance to study the dynamics of delayed dynamical systems. Bifurcation analysis isan important and effective way to obtain dynamics of delayed systems. By bifurcation,we mean that the topological structure of a flow with parameters is changed as parame-ters is cross a critical value. Generally speaking, bifurcations consists of local, semi-localand global bifurcations. Hopf bifurcation is a common and important local bifurcation.It studies the change of stability of equilibria and the consequent occurrence of periodicsolutions with small amplitude near equilibria. Therefore, Hopf bifurcation is an effectiveway to find periodic solutions. As we know, the phase space of delayed systems is usuallyof infinite dimension, and the bifurcation analysis for such systems needs not only classi-cal theory of dynamical systems, but also knowledge of other mathematics branches, suchas topology, algebra and functional analysis. In other words, delayed dynamical systemsis an interdiscipline. In this thesis, we mainly consider several kinds of delayed dynamicalsystems with strongly practical background. The organization is as follows:1. Dynamics of a food-limited population model.For the spatially homogeneous case, we first establish a theorem on the stabilityof the positive equilibrium and the existence of Hopf bifurcation via analyzing the dis-tribution of roots to the corresponding characteristic equation. Then we get a sufficientcondition for global Hopf bifurcation to exist by Wu’s global Hopf bifurcation theorem.For the spatially inhomogeneous case (more precisely, a partial functional differentialequation with Dirichlet boundary condition), we obtain the existence for bifurcation ofpositive steady states employed the idea of phase plane analysis, and further prove theexistence of Hopf bifurcation at positive steady states as well as the formula for deter-mining the stability of bifurcated periodic solutions and the direction of Hopf bifurcation.Finally, we present a discussion on the effect of diffusion on stability.2. Hopf bifurcation of a haematopoietic stem cell model with a single delay.We first study the distribution of roots to the characteristic equation, which is a expo- nential polynomial equation with delay-dependent coefficients. Consequently, we obtainthe stability threshold and the existence of Hopf bifurcation. Then we derive an explicitformula for determining the stability and direction of periodic solutions bifurcated fromHopf bifurcations using the normal form theory combined with center manifold argu-ments. The persistence of periodic solution induced by Hopf bifurcation as parametersvary in a large range is confirmed.3. Dynamics of a coupled Mackey-Glass electronic circuits model with multipledelays.The distribution of roots to the characteristic equation is complicated because ofmultiple delays. To analyze it, we first figure out the stability interval of one delay, andthen discuss the effects of other delays. Consequently, we find a sequence of Hopf bifur-cation points and derive an explicit algorithm for determining the direction and stability ofbifurcated periodic solutions by appealing to the normal form theory and center manifoldargument.4. Hopf bifurcation in a genetic regulatory model with delay.The existence and uniqueness of positive equilibrium are established. Choosing timedelay τ as the parameter, we obtain a sufficient condition for Hopf bifurcation to happen.5. Dynamics of an age-structured population model of a single species living in twoidentical patches.This is a delayed system arose in population biology. It has rich dynamical behavior,which is still largely opened. We here perform some bifurcation to this model. Someproperties of Hopf bifurcation are obtained by the normal form theory coupled with thecenter manifold theorem.