Instituto de Física de Cantabria (IFCA), CSIC-Universidad de Cantabria, 39005 Santander, Spain.
Phys Rev E. 2019 Jul;100(1-1):012211. doi: 10.1103/PhysRevE.100.012211.
Phase reduction is a powerful technique that makes possible to describe the dynamics of a weakly perturbed limit-cycle oscillator in terms of its phase. For ensembles of oscillators, a classical example of phase reduction is the derivation of the Kuramoto model from the mean-field complex Ginzburg-Landau equation (MF-CGLE). Still, the Kuramoto model is a first-order phase approximation that displays either full synchronization or incoherence, but none of the nontrivial dynamics of the MF-CGLE. This fact calls for an expansion beyond the first order in the coupling constant. We develop an isochron-based scheme to obtain the second-order phase approximation, which reproduces the weak-coupling dynamics of the MF-CGLE. The practicality of our method is evidenced by extending the calculation up to third order. Each new term of the power-series expansion contributes with additional higher-order multibody (i.e., nonpairwise) interactions. This points to intricate multibody phase interactions as the source of pure collective chaos in the MF-CGLE at moderate coupling.
相位降阶是一种强大的技术，它使得我们可以用相位来描述弱扰的极限环振荡器的动力学。对于振荡器的集合，相位降阶的一个经典例子是从平均场复金兹堡-朗道方程（MF-CGLE）推导出的 Kuramoto 模型。然而，Kuramoto 模型是一个一阶相位近似，它显示出完全同步或非相干性，但没有 MF-CGLE 的任何非平凡动力学。这一事实需要在耦合常数的一阶以上进行扩展。我们开发了一种基于等时的方案来获得二阶相位近似，它再现了 MF-CGLE 的弱耦合动力学。我们的方法的实用性通过将计算扩展到三阶来证明。幂级数展开的每个新项都贡献了额外的高阶多体（即非成对）相互作用。这表明，在中等耦合下，MF-CGLE 中的纯集体混沌的来源是复杂的多体相位相互作用。
