Electronically--implemented coupled logistic maps

TL;DR

This paper presents a simple, low-cost electronic implementation of the logistic map and its coupling, demonstrating experimental and simulated agreement in chaotic behavior analysis.

Contribution

It introduces a versatile electronic circuit for the logistic map and its coupling, enabling easy modifications and potential extension to complex systems.

Findings

Experimental results closely match simulations.

Behavior consistent across various parameters and coupling strengths.

Lyapunov exponents and phase portraits confirm chaotic dynamics.

Abstract

The logistic map is a paradigmatic dynamical system originally conceived to model the discrete-time demographic growth of a population, which shockingly, shows that discrete chaos can emerge from trivial low-dimensional non-linear dynamics. In this work, we design and characterize a simple, low-cost, easy-to-handle, electronic implementation of the logistic map. In particular, our implementation allows for straightforward circuit-modifications to behave as different one-dimensional discrete-time systems. Also, we design a coupling block in order to address the behavior of two coupled maps, although, our design is unrestricted to the discrete-time system implementation and it can be generalized to handle coupling between many dynamical systems, as in a complex system. Our findings show that the isolated and coupled maps' behavior has a remarkable agreement between the experiments and the simulations, even when fine-tuning the parameters with a resolution of $\sim 10^{-3}$. We support these conclusions by comparing the Lyapunov exponents, periodicity of the orbits, and phase portraits of the numerical and experimental data for a wide range of coupling strengths and map's parameters.