Bifurcation Analysis For Several Kinds Of Reaction-diffusion Systems

Bifurcation phenomenon is an important nonlinear phenomenon of nonlinear partial dif-ferential equations. It reflects, when the parameters cross through certain critical values, some structural properties are changed. The study of the bifurcation has a very important significance in theory and practical application. Partial differential problems, which are concerned in this paper, are semi-linear reaction-diffusion systems. The bifurcation problems, which are studied in this paper, are the steady-state bifurcation problems and Hopf bifurcation problems.The main contents of the paper are as follows:Firstly, we consider the Hopf bifurcation problem of a kind of the homogeneous reaction-diffusion model modeling hair growth. We derive the precise conditions for the system to have local Hopf bifurcation, give the analytical expression of the bifurcation points, and the bifurcation periodic solutions near the bifurcation points, and we also study the stability of the bifurcating periodic solutions. In particular, we prove that the system exists two bifurcation points where system can bifurcate the spatially homogeneous periodic solutions. Our results show that: when the system parameter p is sufficiently large, the spatially homogeneous pe-riodic solutions of this system are always stable. The innovation in this part is that, the Hopf bifurcations of this system have been firstly studied, and the existence of spatially homogeneous and non-homogeneous periodic pattern of this system is revealed.Secondly, we study the steady state bifurcations of the afore-mentioned the spatial homo-geneous reaction-diffusion model describing hair growth. We prove the existence conditions of local and global steady state bifurcation of this system. For the local bifurcations, we give the conditions determining the accurate number of the steady state bifurcation points of this system. The innovation in this part is that, we prove the global steady state bifurcation theorem and the conditions determining the accurate number of the steady state bifurcation points for this system.Finally, we study the Hopf bifurcation of a kind of reaction diffusion Maginu model de-scribing the Turing patterns. We study the existence conditions of Hopf bifurcation, the analyt-ical expressions of the Hopf bifurcation points and the analytical expressions of the bifurcation periodic solutions near the bifurcation points. In particular, we prove that all spatially homoge-neous and non-homogeneous periodic solutions of this system are unstable. The innovation in this part is that, the Hopf bifurcations of this system have been firstly studied, and the existence of spatially homogeneous and non-homogeneous periodic pattern of this system is revealed.