Statistical performance of local attractor dimension estimators in non-Axiom A dynamical systems

Abstract

We investigate various estimators based on extreme value theory (EVT) for determining the local fractal dimension of chaotic dynamical systems. In the limit of an infinitely long time series of an ergodic system, the average of the local fractal dimension is the system’s global attractor dimension. The latter is an important quantity that relates to the number of effective degrees of freedom of the underlying dynamical system, and its estimation has been a central topic in the dynamical systems literature since the 1980s. In this work, we propose a framework that combines phase space recurrence analysis with EVT to estimate the local fractal dimension around a particular state of interest. While the EVT framework allows for the analysis of high-dimensional complex systems, such as the Earth’s climate, its effectiveness depends on robust statistical parameter estimation for the assumed extreme value distribution. In this study, we conduct a critical review of several EVT-based local fractal dimension estimators, analyzing and comparing their performance across a range of systems. Our results offer valuable insights for researchers employing the EVT-based estimates of the local fractal dimension, aiding in the selection of an appropriate estimator for their specific applications.