Self-Organized Pulse Generator in a Reaction-Diffusion System

本文研究了Bonhoeffer-van der Pol （BVP）型反应扩散体系的时空格局。这项研究的动机来自于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会湮灭，而是构成一个持续向外发射脉冲的域。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BVP方程已被研究了二十多年，但上述性质尚未见报道。在我们的研究中，我们得到了以下事实。脉冲发生器的根源是均匀解和极限环共存的双稳性。此外，系统的兴奋性和双稳性导致了多种时空模式。通过改变振荡强度和激振性参数得到相图。在二维空间中，脉冲发生器形成目标图形。如果增大扩散常数，目标图案就会扩展、局部化，最后静止不动。作为一个相关问题，我们也研究了具有一个活化剂和两个抑制剂的bvp型系统的区域运动。与计算机模拟相结合的界面动力学为我们提供了域的呼吸、摆动和混沌运动的条件。我们已经证明，摆动运动出现在比呼吸运动更宽的参数区域。

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.