Waveform proportionality and Taylor's law in coupled Lorenz systems

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 Rössler 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.