Dynamics of the local map for a discrete brusselator model

by 1974- Kang, Hunseok

The goal of the thesis is to describe the dynamics of the local map of coupled map lattice(CML) – the discrete Brusselator model. Following Kaneko I view CMLs as phenomenological models of the medium (which is assumed to be homogeneous and unbounded) and I present the dynamical system approach to the analysis of the global behavior of solutions of CML developed in works of V. Afraimovich, M. Brin, D. Orendovici, and Y. Pesin. This analysis is aimed at establishing spatio-temporal chaos associated with the set of traveling wave solutions of CML and describing the dynamics of the evolution operator on this set. The main results claim that the dynamics of the evolution operator on the set of traveling wave solutions is completely determined by the dynamics of the local map thus making the study of the latter as the primary goal of my research. In the case of the Brusselator model, the dynamics of the corresponding local maps is quite complicated, has many interesting properties and displays chaotic behaviors. The model depends on a number of parameters and the dynamics of the corresponding local map varies substantially when these parameters vary. In particular, the local map associated to the Brusselator model has the following properties: (1) it has an open domain of trajectories that escape to infinity; (2) it possesses the Julia set, i.e., an invariant domain that consists of bounded trajectories; (3) it has eventually trapping regions; (4) it has visiting regions; and (5) strange attractors inside the eventually trapping regions. Finally, I carried out the numerical study of strange attractors as well as various aspects of chaotic behaviors for this model. iii