SUPA, Institute for Complex Systems and Mathematical Biology, King's College, University of Aberdeen, Aberdeen AB24 3UE, United Kingdom.
Phys Rev E. 2018 Jan;97(1-1):012203. doi: 10.1103/PhysRevE.97.012203.
We discuss the behavior of the largest Lyapunov exponent λ in the incoherent phase of large ensembles of heterogeneous, globally coupled, phase oscillators. We show that the scaling with the system size N depends on the details of the spacing distribution of the oscillator frequencies. For sufficiently regular distributions λ∼1/N, while for strong fluctuations of the frequency spacing λ∼lnN/N (the standard setup of independent identically distributed variables belongs to the latter class). In spite of the coupling being small for large N, the development of a rigorous perturbative theory is not obvious. In fact, our analysis relies on a combination of various types of numerical simulations together with approximate analytical arguments, based on a suitable stochastic approximation for the tangent space evolution. In fact, the very reason for λ being strictly larger than zero is the presence of finite-size fluctuations. We trace back the origin of the logarithmic correction to a weak synchronization between tangent and phase-space dynamics.
我们讨论了在大型异质、全局耦合的相位振荡器集合的非相干相中最大 Lyapunov 指数 λ 的行为。我们表明，与系统大小 N 的标度取决于振荡器频率间隔分布的细节。对于足够规则的分布，λ∼1/N，而对于频率间隔的强烈波动，λ∼lnN/N（独立同分布变量的标准设置属于后者类别）。尽管对于大 N，耦合很小，但严格的微扰理论的发展并不明显。事实上，我们的分析依赖于各种类型的数值模拟与近似分析论点的结合，基于切线空间演化的合适随机逼近。事实上，λ严格大于零的原因是有限大小波动的存在。我们追溯到对数修正的起源是切线和相空间动力学之间的弱同步。
