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arXiv 2006.02159nlin.AO
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非相干态附近全对全强耦合$λ-ω$振子的准相位约化

Quasi phase reduction of all-to-all strongly coupled $λ-ω$ oscillators near incoherent states

Iván León, Diego Pazó

备注：
11 pages, 5 figures
发表信息：
Phys. Rev. E 102, 042203 (2020)
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摘要：

$N$弱耦合极限环振子系综的动力学可以通过它们的$N$相位使用标准相位缩减技术来捕获。然而，这是一个现象学的事实，所有对所有的强耦合极限环振荡器可能表现为“准相位振荡器”，证明了新的减少策略的需要。我们在这里介绍准相位减少（QPR），一个计划适用于同极对称的振荡器（$λ-ω$系统）。通过应用QPR，我们实现了减少到$N+2$自由度：$N$相位振荡器通过一个独立的复变量相互作用。这种“准相位模型”在非相干态附近是渐近有效的，与耦合强度无关。QPR的有效性示出在一个特定的情况下，一个合奏的Stuart-Landau振荡器，获得精确的稳定性边界的均匀和非均匀的非相干态的各种耦合。QPR超出邻域的不相干的扩展也进行了探讨。最后，一个一般的QPR模型与$N+2M$自由度的耦合，通过一阶$M$谐波。

英文摘要：

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.

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