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.

The dissertation studies about the existence of three different types of attractors of three multi-component reaction-diffusion systems. These reaction-diffusion systems play important role in both chemical kinetics and biological pattern formation in the fast-growing area of mathematical biology. In Chapter 2, we prove the existence of a global attractor and an exponential attractor for the solution semiflow of a reaction-diffusion system called Boissonade equations in the L2 phase space. We show that the global attractor is an (H, E) global attractor with the L∞ and H2 regularity and that the Hausdorff dimension and the fractal dimension of the global attractor are finite. The existence of exponential attractor is also shown. The upper-semicontinuity of the global attractors with respect to the reverse reaction rate coefficient is proved. In Chapter 3, the existence of a pullback attractor for non-autonomous reversible Selkov equations in the product L2 phase space is proved. The method of grouping and rescaling estimation is used to prove that the L4-norm and L6-norm of solution trajectories are asymptotic bounded. The new feature of pinpointing a middle time in the process turns out to be crucial to deal with the challenge in proving pullback asymptotic compactness of this typical non-autonomous reaction-diffusion system. In Chapter 4, asymptotical dynamics of stochastic Brusselator equations with multiplicative noise is investigated. The existence of a random attractor is proved via the exponential transformation of Ornstein-Uhlenbeck process and some challenging estimates. The proof of pullback asymptotic compactness here is more rigorous through the bootstrap pullback estimation than a non-dynamical substitution of Brownian motion by its backward translation. It is also shown that the random attractor has the L2 to H1 attracting regularity by the flattening method.

京都大学 / 新制・課程博士 / 博士(理学) / 甲第23675号 / 理博第4765号 / 新制||理||1683(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 坂上 貴之, 教授 泉 正己, 教授 國府 寛司 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM