Theory And Applications Of Center Manifolds And Normal Forms Of Partial Functional Differential Equations

This dissertation develops a simple and effective approach to compute the equivariant normal forms of symmetrical partial functional differential equations (PFDEs) with time lags and parameters. As far as we know, there are so many studies on the calculation of normal forms and their applications to bifurcations of functional differential equations (FDEs), but there are no many theoretical results on the normal forms of PFDEs, especially symmetrical PFDEs. Since more and more PFDEs are used to describe the dynamical behaviors in chemical process, population ecology, cell biology, and control theory etc. It is necessary to investigate the normal form theory of PFDEs in the presence of symmetry, which is one of the most significant tools in investigating the dynamics behavior of PFDEs. The purpose of this dissertation is to establish the normal forms theory of PFDEs based on the normal forms theory previously developed for autonomous retarded FDEs and on the existence of center (or other invariant) manifolds for PFDEs. We first decompose the phase space in such a way that the linearized equations of PFDEs at the stationary solution are equivalent to a series of FDEs, and further transfer the PFDEs to abstract ODEs. Then by computing the normal forms of the abstract ODEs, we can obtain the normal forms of PFDEs. We observe that the form of the reduced vector field relies not only on the information of the linearized system at the critical point but also on the inherent symmetry, and the normal forms can not only present the existence of bifurcations, but also can disclose the spatio-temporal patterns, stability and bifurcation direction of bifurcated solutions. As the illustration of our general results, Hopf bifurcations of several kinds of PFDEs with symmetries and delays are studied. This thesis is organized as follows:Firstly, the method of calculating the normal form of PFDEs is presented. At the beginning, we decompose the phase space according to the characteristic functions of the linear partial differential operators involved in PFDEs. Then the linearized equations of PFDEs at trivial equilibrium are considered. And by enlarging the phase space, we can obtain the abstract ODEs associated with the original PFDEs. Then the process of calculating the equivariant normal form of PFDEs, which preserves the symmetry of the original PFDEs, are presented.Secondly, we illustrate our theoretical results by studying the Hopf bifurca-tions in three classes of PFDEs. The symmetry of the first equation is about the spatial variable, the symmetry of the second equation is about the state variable, and the third equation has both two kinds of symmetries. We show that the method we present are applicable to all kinds symmetrical PFDEs.Thirdly, we present a detailed analysis on the dynamics of a ring network with small world connection and diffusion by employing our theoretical results and symmetry-breaking theory. We analyze the bifurcation of the perfect ring net-work and the small world ring network, including primary steady-state bifurcation, Hopf bifurcation and secondary bifurcation of equilibria, where all the bifurcations mean the local bifurcation. Based on the equivariant branching lemma, we not only obtain the existence of the primary steady-state bifurcation but also analyze the patterns and stability of the bifurcated nontrivial equilibria. Moreover, by means of the equivariant Hopf theorem, we investigate the effect of connection strength on the spatio-temporal patterns of periodic solutions emanating from the trivial equilibrium. By means of the normal forms method we obtain previously, we derive the formula to determine the direction and stability of Hopf bifurcation. Furthermore, we investigate the secondary bifurcation of nontrivial equilibria. Fi-nally, the difference in the dynamics between the perfect system and small world system, shows that small world connection may be used as a simple but efficient switch to control the dynamics of a system.Finally, we employ the center manifold reduction and symmetry-breaking the-ory to study the primary steady state bifurcation, Hopf bifurcation and secondary bifurcation of the ring network with small world connection but without diffusion. Through the comparison of the dynamics of the diffusion systems and diffusion-less systems, we investigate the effect of diffusion on the bifurcation directions, stabilities and the spatio-temporal patterns of bifurcated solutions.