Waveform proportionality and Taylor's law in networks of Lorenz systems

Abstract

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 have not yet been fully elucidated. In particular, the frequent empirical observation of TL with an exponent 2 warrants further investigation. In a previous study, 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, resulting in TL with an exponent 2. Here, we extend our previous findings for the coupled Lorenz system and provide detailed calculations. We analytically and numerically show that, under strong coupling, waveform proportionality and TL with an exponent 2 emerge for some variables in the globally coupled original, generalized, and hyperchaotic Lorenz systems. Furthermore, we derive a condition under which these properties hold for the original Lorenz system on networks. We then numerically confirm that when heterogeneous oscillators are randomly distributed within the network, waveform proportionality and TL with an exponent 2 emerge across several representative network structures, such as a two-dimensional lattice, random networks, and scale-free networks.