Discrete dynamical systems with scaling and inversion symmetries

TL;DR

This paper introduces a novel approach leveraging scale and inversion symmetries to analyze discrete dynamical systems, enabling efficient computation of fractal dimensions and Lyapunov exponents with fewer iterations.

Contribution

It formulates scale symmetry through inversion symmetry and applies this framework to determine key dynamical properties more efficiently than standard methods.

Findings

Accurate Lyapunov exponents obtained with fewer iterations

Fractal dimensions computed directly from geometric framework

Method matches standard results, confirming efficiency

Abstract

In this work, we investigate scale invariance in the temporal evolution and chaotic regime of discrete dynamical systems. By exploiting the close interrelation between scaling and inversion transformations, we formulate scale symmetry in terms of inversion symmetry. As applications of our approach, we determine fractal dimensions and compute Lyapunov exponents for paradigmatic dynamical systems using scaling and inversion symmetries. By comparing our method with standard approaches, we obtain identical numerical values for the Lyapunov exponents using only a small number of iterations. Furthermore, our geometric-based framework naturally provides access to the fractal dimension. The agreement with standard results demonstrates that the proposed method is efficient and can be effectively employed in the study of dynamical systems.