标题： 波形比例性和耦合洛伦兹系统中的泰勒定律 显示英文标题

标题： Waveform proportionality and Taylor's law in coupled Lorenz systems

摘要： 泰勒定律（TL），即一个量的均值和方差之间的幂律关系，在多个科学领域中都被观察到。 尽管其普遍存在，但导致TL的潜在机制尚未完全阐明。 特别是，TL指数为2的频繁经验观察需要进一步研究。 在之前的一项研究[Phys. Rev. Lett. 134, 167202 (2025)]中，我们假设同步有助于产生指数为2的TL。 为了验证这一假设，我们采用了耦合振子模型，每个振子由不同的动力系统描述：一个食物链模型，Rössler系统，Brusselator和Lorenz系统。 我们的分析和数值结果表明，强耦合会导致一种同步形式，其中时间序列彼此成比例，从而导致指数为2的TL。 在这里，我们将之前的关于耦合Lorenz系统的发现扩展，并提供详细的计算。 我们的分析和数值结果表明，在强耦合下，波形比例性和指数为2的泰勒定律不仅出现在原始Lorenz系统中，也出现在广义和超混沌Lorenz系统中。 显示英文摘要

摘要： Taylor's law (TL), a power-law relationship between the mean and variance of a quantity, has been observed across diverse scientific disciplines. Despite its ubiquity, the underlying mechanisms responsible for TL are not yet fully elucidated. In particular, the frequent empirical observation of TL with an exponent 2 warrants further investigation. In a previous study [Phys. Rev. Lett. 134, 167202 (2025)], we hypothesized that synchronization contributes to the emergence of TL with an exponent 2. To validate this hypothesis, we employed coupled oscillator models, with each oscillator described by a distinct dynamical system: a food chain model, the R\"ossler system, the Brusselator, and the Lorenz system. Our analytical and numerical results demonstrated that strong coupling leads to a form of synchronization wherein time series become proportional to each other, consequently resulting in TL with an exponent 2. Here, we extend our previous findings for the coupled Lorenz system and provide detailed calculations. Our analytical and numerical results demonstrate that, under strong coupling, waveform proportionality and Taylor's law with an exponent 2 emerge not only in the original Lorenz system but also in the generalized and hyperchaotic Lorenz systems.