复有向图上弱耦合自持振子的Ginzburg-Landau逼近
Ginzburg-Landau approximation for self-sustained oscillators weakly coupled on complex directed graphs
摘要：
在超临界Hopf分支附近，导出了一个基于有向图的反应扩散系统演化的规范形近似.弱扩散耦合被假定为保持相邻节点之间。在此工作假设下，得到复Ginzburg-Landau方程（CGLE），其系数取决于模型的参数和底层网络的拓扑特征。CGLE使人们能够探测同步振荡解的稳定性，如反应扩散系统在Hopf分支之上所显示的。更具体地说，条件可以制定出的对称性破缺不稳定性，最终破坏了均匀的振荡状态的发病。数值实验验证了所提出的理论方案的有效性。记录的CGLE模式类似于恢复后的整合，原来的微扰动态。
英文摘要：
A normal form approximation for the evolution of a reaction-diffusion system hosted on a directed graph is derived, in the vicinity of a supercritical Hopf bifurcation. Weak diffusive couplings are assumed to hold between adjacent nodes. Under this working assumption, a Complex Ginzburg-Landau equation (CGLE) is obtained, whose coefficients depend on the parameters of the model and the topological characteristics of the underlying network. The CGLE enables one to probe the stability of the synchronous oscillating solution, as displayed by the reaction-diffusion system above Hopf bifurcation. More specifically, conditions can be worked out for the onset of the symmetry breaking instability that eventually destroys the uniform oscillatory state. Numerical tests performed for the Brusselator model confirm the validity of the proposed theoretical scheme. Patterns recorded for the CGLE resemble closely those recovered upon integration of the original Brussellator dynamics.
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