Self-Organized Pulse Generator in a Reaction-Diffusion System

本文研究了一类Bonhoven-van der Pol（BVP）型反应扩散系统的时空斑图。这项研究的动机是在计算机模拟脉冲碰撞中意外观察到的特殊行为。也就是说，一对传播脉冲在碰撞时不湮灭，而是构成持续发射出射脉冲的域。这为我们提供了一个反例，即耗散系统中的脉冲在碰撞时简单地消失。我们称这种现象为自组织脉冲发生器。虽然边值问题方程的研究已有二十多年的历史，但上述性质还未见报道。脉冲发生器的起源是双稳态，即一个均匀解和一个极限环共存。此外，系统的可激发性和双稳性导致了多种时空模式。通过改变振荡强度和激发度参数，得到了相图。在两个维度中，目标图案由脉冲发生器形成。作为相关问题，我们还研究了具有一个激活剂和两个抑制剂的BVP型系统中的畴运动。计算机模拟的界面动力学为我们提供了畴的呼吸、摆动和混沌运动的条件。我们已经表明，摆动运动出现在一个更广泛的参数区域比呼吸运动。

We have investigated the spatio-temporal pattern in a Bonhoeffer-van der Pol (BVP) type reaction- diffusion system. This study was motivated from the peculiar behavior observed unexpectedly in computer simulations of pulse collision. That is, a pair propagating pulses do not annihilate upon collision but constitute a domain which emits outgoing pulses persistently. This provides us with anti-example that pulses in a dissipative system simply disappear upon collision. We call this phenomenon a self-organized pulse generator. Although BVP equation had been studied for more than twenty years, the above property had not been reported.In our study, we have obtained the following facts. The origin of the pulse generator is the bistability such that a uniform solution and a limit cycle coexist. Furthermore, the excitability of the system and the bistability cause a variety of spatio-temporal patterns. The phase diagram is obtained by changing the parameters of oscillatory strength and excitability. In two dimensions, a target pattern is formed by the pulse generator. If one increases the diffusion constant, the target pattern becomes extended, localized and finally motionless.As a related problem, we have also studied domain motion in a BVP-type system with one activator and two inhibitors. The interface dynamics associated with computer simulations provides us with the condition of breathing, wiggling and chaotic motion of domains. We have shown that the wiggling motion appears in a wider parameter region than the breathing motion.