Visualization-system of spatio-temporal patterns in natural sciences

1988年，我们在广岛大学数学系开发了与局域网系统下的小型超级计算机相连的数值模拟可视化系统。通过使用这个系统，我们的第一个研究是可视化的时空图案动力学的燃烧，通过数值求解两个数学模型的反应扩散类型。第一个例子是演示一维燃烧波的传播。我们可以观察到，在波的速度发生分叉现象。也就是说，当我们改变两侧的温度差X =<plus-minus>* 时，以恒定速度传播的波通过霍普夫分叉变得不稳定，并出现一个新的以周期速度传播的波。这种现象在真实的实验中也能观察到。另一个例子是在二维环形区域中形成蜂窝状火焰的图案。1989年，我们对二元合金中的相分离和粗化过程的理论可视化现象感兴趣。我们数值求解了几个反应扩散方程模型，迄今已提出理解这些现象。结果显示出如此复杂和美丽的图案，这是迄今为止没有预料到的。

In 1988, we developed visualization system of numerical simulations which connected with mini-super computer under local network systems in the department of Mathematics at Hiroshima University. By using this system, our first study is to visualize spatio-temporal patterns dynamics in combustions, by numerically solving two mathematical models of reaction-diffusion type. The first example is to demonstrate the propagation of combustion waves in one dimension. We can observe that there occurs a bifurcation phenomenon in the wave velocity. That is, when we change the difference of the temperatures at the both side X = <plus-minus>*, the wave which propagates with constant velocity becomes unstable through Hopf-bifurcation and there appears a new wave which propagates with periodic velocity. This phenomenon can be also observed in real experiments. Another example is the pattern formation of cellular flame in a 2 dimensional annular domain. We find that radially symmetric flame patterns become unstable and then there appear cellular flame patterns through static bifurcatioIn 1989, we are interested in visualizing the phenomena of phase separation and coarsening process in binary alloy from a theoretical view point. We numerically solved several reaction-diffusion equation models which has been proposed so far to understand these phenomena. The results show so complex and beautiful patterns which has not been expected until now.