Discrete dynamical systems with scaling and inversion symmetries

TL;DR

This paper introduces a unified framework that treats scale invariance in discrete dynamical systems via inversion symmetry. By defining inverse sets, inversion functions, and their differential equations, the authors derive both fractal measures and Lyapunov exponents from inversion-based scaling relations, achieving results that match standard methods with far fewer iterations. The approach is demonstrated on self-similar fractals and chaotic maps (tent, logistic, Chebyshev), highlighting exponential inversion laws tied to the fractal dimension difference $d_F-d_E$ and revealing a fast, geometry-driven route to quantify chaos. The framework provides practical, conceptually transparent tools for analyzing fractal geometry and chaotic dynamics in discrete systems, with potential broad applicability to nonlinear science.

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.

Figures (14)

• Figure 1: The inversion transformation in the plane, defined by $r_P.r_Q=r^2$, which maps external points $P$ into internal points $Q$ and vice versa.
• Figure 2: Phase portrait of the map $x_{m+1}=\mu x_{m}^3$ for $\mu>0$, where the orbit generated from a non-null initial condition $x_0$ belongs to one of the inverse sets $I^{+}_{1x}$, $I^{+}_{2x}$, $I^{-}_{1x}$, or $I^{-}_{2x}$.
• Figure 3: A geometric figure resulting from the union of the inverse sets formed by equilateral triangles.
• Figure 4: Sierpinski triangle up to iteration $n=4$, illustrating that the perimeter and area measurements are inversely related with respect to iteration $n=2$.
• Figure 5: Illustration of the mappings $F^{m}(x)$, $F^{m+1}(x)$, $F^{m+2}(x)$, and $F^{m+3}(x)$ of a chaotic system, which are related by the scale factor $\rho=e^{\lambda}$ in the asymptotic limit $m\rightarrow\infty$.