反应扩散系统有着强烈的实际背景，对其斑图动力学的已有研究大多建立在数值分析与模拟的基础上，缺乏深刻的理论分析。分支理论是深入认识反应扩散系统的斑图动力学的重要工具。然而，目前对于反应扩散系统分支问题的研究大多局限于余维一的Hopf分支的情形。本项目针对有界区域上的反应扩散系统，通过对（1）Turing不稳定性与余维二分支；（2）时滞和扩散的联合作用对反应扩散方程动力学的影响；（3）振荡斑图以及时滞、区域边界对斑图动力学的影响这三个科学问题的研究，深入探讨“扩散”、“反应”和“时滞”的联合作用对系统时空动力学的影响，重点研究导致空间结构的Turing分支之间、诱发时间方向周期振动的Hopf分支之间以及它们的相互作用而导致的余维二分支点附近的动力学分类，深刻认识导致振荡班图模式的动力学机制。

Reaction-diffusion system has a strong background in applications. Most of studies on the pattern dynamics of reaction-diffusion system are based on the numerical analysis and numerical simulations, lack of deeply theoretical analysis. Bifurcation theory is a fundamental tool to investigate deeply the pattern dynamics. However, so far, the bifurcation theory on the reaction-diffusion system mostly focus on the codimension-1 Hopf bifurcation. Based on the reaction-diffusion system on the bounded region, this project investigates the following three scientific problems:(1) Turing instability and codimension-2 bifurcation in the bounded region; (2) the effects of diffusion and delay on the dynamics of reaction–diffusion system; (3) oscillatory patterns and the influence of delay, boundary of region on the pattern dynamics. The major objectives of this project are to gain a better understanding of the effects of the interaction of “diffusion”, “delay” and “reaction” on the spatio-temporal dynamics of the reaction-diffusion system, to investigate deeply the dynamical classification near the codimension-2 bifurcation point due to the interaction of Turing bifurcations, interaction of Hopf bifurcations with different wave number and interaction of Turing bifurcation and Hopf bifurcation, and to understand theoretically the dynamical mechanism leading to oscillatory patterns.