Modeling for Spatio-Temporal Chaos

为了更好地理解时空混沌，我们研究了简单的模型方程。特征空间结构的产生和湮灭过程是典型的时空混沌。轧辊结构的重新连接是缺陷混沌中的一个基本过程。我们已经研究了带有附加不稳定性项的Burgers方程。该系统的一个特征空间结构是冲击结构。在时空混沌中，会出现许多冲击，并相互碰撞。由于碰撞，两个激波合并为一个激波。在两个相邻激波之间的较长间隔内会自发地产生一个新的激波。这些产生和合并过程是该模型方程的时空混沌特征。我们建立了一个简单的激波位置模型，得到了一个类似的混沌时间演化，包括激波的产生和合并。对于另一个例子，我们研究了一个耦合的金兹堡-朗道方程，并找到了一个脉冲解作为吸引子。脉冲溶液可以表现出呼吸运动和分裂分叉。结果，时空混沌从单个脉冲解中自发出现。

To understand spatio-temporal chaos better, we have studied simple model eqations. The creation and annihilation processes of characteristics spatial structures is typical of the spatio-temporal chaos. The reconnection of roll structures is a fundamental process in defect chaos.We have studied the Burgers equation with an additional instability term. A characteristic spatial structure in this system is the shock structure. In the spatio-temporal chaos, there appear many shocks and they collide with each other. Two shocks are merged into one shock as a resut of the collision. A new shock is spontaneously created in a long interval between two neighboring shocks. These creation and merging processes are characteristic of the spatio-temporal chaos in this model equation. We have constructed a simple model for the positions of the shocks and obtained a similar chaotic time evolution including the creation and merging of shocks.We have studied a coupled Ginzburg-Landau equation for another example and found a pulse solution as an attractor. The pulse solution can exhibit breathing motion and splitting bifurcation.As a result, spatio-temporal chaos appears spontaneously from a single pulse solution.