Dynamics of oscillatory and excitable fields with nanlocal coupling

在本研究项目中，从理论上研究了具有有效非局域耦合的自振荡介质所表现出的各种新型图案动力学。对一种特殊的三组分反应扩散模型及其在Hopf分叉附近的简化形式进行了数学和数值分析。证明了无相位奇异性的旋转螺旋波的存在。这发生在螺旋核心附近的一组振荡器未能与该场同步时，这些振荡器经历了相对较弱的内场。我们的反应扩散模型及其约化被证实表现出这一行为2。证实了软模湍流的存在。软模湍流是一种时空混沌，它是在系统破坏一定的连续对称性的条件下，空间均匀的状态变得图灵不稳定而产生的。对我们的约化方程(非局部耦合复GL方程)进行了线性稳定性分析，结果表明在适当的条件下出现了软模湍流特有的特征值谱类型。方程的直接数值模拟证实了预期湍流行为的发生

In this research project, a variety of new types of pattern dynamics exhibited by self-oscillatory media with effective non-local coupling were studied theoretically. A specific 3-component reaction-diffusion model as well as its reduced form near the Hopf bifurcation were analyzed mathematically and numerically1. Existence of rotating spiral waves without phase singularity was demonstrated. This occurs when a group of oscillators near the spiral core, which experience relatively weak internal field, fails to synchronize with this field. Our reaction-diffusion model and its reduction were confirmed to exhibit this behavior2. Existence of soft-mode turbulence was confirmed. Soft-mode turbulence is a type of spatio-temporal chaos which arises when a spatially uniform state becomes Turing unstable under the condition that the system already breaks a certain continuous symmetry. A linear stability analysis for our reduced equation (non-locally coupled complex GL equation) was carried out, which revealed that the type of eigenvalue spectrum peculiar to soft-mode turbulence appear under suitable condition. Direct numerical simulation of the equation confirmed the occurrence of the expected turbulent behavior