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Discrete dynamical systems with scaling and inversion symmetries.
Vaguiner Rodrigues Dos Santos1, Enrique Chipicoski Gabrick1, Edson Denis Leonel2
1Institute of Physics, University of São Paulo, 05508-090 São Paulo, SP, Brazil.
Chaos (Woodbury, N.Y.)
|June 1, 2026
Summary
This study introduces a novel method to analyze scale invariance in discrete dynamical systems by linking scaling and inversion symmetries. This approach efficiently determines fractal dimensions and Lyapunov exponents, matching standard results with fewer iterations.
Area of Science:
- Dynamical Systems and Chaos Theory
- Fractal Geometry
- Symmetry in Physics
Background:
- Discrete dynamical systems exhibit complex temporal evolution and chaotic behavior.
- Scale invariance is a key property in understanding these systems.
- Traditional methods for analyzing scale invariance can be computationally intensive.
Purpose of the Study:
- To investigate scale invariance in discrete dynamical systems.
- To formulate scale symmetry using inversion symmetry.
- To develop an efficient method for calculating fractal dimensions and Lyapunov exponents.
Main Methods:
- Exploiting the relationship between scaling and inversion transformations.
- Formulating scale symmetry as inversion symmetry.
- Applying the method to paradigmatic discrete dynamical systems.
Main Results:
- Achieved identical numerical values for Lyapunov exponents compared to standard methods.
- Required significantly fewer iterations for computation.
- Naturally provided access to fractal dimensions through a geometric framework.
Conclusions:
- The proposed method is efficient and effective for studying dynamical systems.
- The geometric-based framework offers a new perspective on scale invariance.
- This approach simplifies the computation of key dynamical system properties.
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