论文标题
不连贯状态附近的全部强烈耦合$λ-ω$振荡器的准阶段减少
Quasi phase reduction of all-to-all strongly coupled $λ-ω$ oscillators near incoherent states
论文作者
论文摘要
$ n $弱耦合极限循环振荡器的合奏的动力可以使用标准减少技术来捕获其$ n $阶段。然而，这是一个现象学事实，即全部强烈耦合的极限周期振荡器可能以“准方案振荡器”的形式行为，这表明需要新的还原策略。我们在这里介绍了准相减少（QPR），该方案适用于具有极性对称性的相同振荡器（$λ-ω$系统）。通过应用QPR，我们将$ n+2 $自由度的自由度减少：$ n $相位振荡器通过一个独立的复杂变量进行交互。这种“准相模型”在不连贯状态的附近渐近有效，而与耦合强度无关。在特定情况下说明了QPR的有效性，即Stuart-Landau振荡器的集合，获得了各种耦合的均匀和非均匀不连贯状态的确切稳定性边界。还探讨了QPR在不一致之外的扩展。最后，通过第一个$ m $谐波的耦合获得了具有$ n+200万美元自由度的一般QPR模型。
The dynamics of an ensemble of $N$ weakly coupled limit-cycle oscillators can be captured by their $N$ phases using standard phase reduction techniques. However, it is a phenomenological fact that all-to-all strongly coupled limit-cycle oscillators may behave as "quasiphase oscillators", evidencing the need of novel reduction strategies. We introduce here quasi phase reduction (QPR), a scheme suited for identical oscillators with polar symmetry ($λ-ω$ systems). By applying QPR we achieve a reduction to $N+2$ degrees of freedom: $N$ phase oscillators interacting through one independent complex variable. This "quasi phase model" is asymptotically valid in the neighborhood of incoherent states, irrespective of the coupling strength. The effectiveness of QPR is illustrated in a particular case, an ensemble of Stuart-Landau oscillators, obtaining exact stability boundaries of uniform and nonuniform incoherent states for a variety of couplings. An extension of QPR beyond the neighborhood of incoherence is also explored. Finally, a general QPR model with $N+2M$ degrees of freedom is obtained for coupling through the first $M$ harmonics.
