Observation of bifurcations and hysteresis in experimentally coupled logistic maps

TL;DR

This paper experimentally investigates bifurcations and hysteresis in coupled logistic maps using electronic circuits, confirming theoretical predictions and revealing robust multi-stable regions despite parameter mismatch and noise.

Contribution

It provides the first precise experimental validation of bifurcation and hysteresis phenomena in coupled logistic maps with electronic implementations.

Findings

Experimental confirmation of bifurcation points matches theoretical predictions.

Multi-stable regions are robust despite parameter mismatch.

Hysteresis observed in coupled logistic maps under real electronic conditions.

Abstract

Initially, the logistic map became popular as a simplified model for population growth. In spite of its apparent simplicity, as the population growth-rate is increased the map exhibits a broad range of dynamics, which include bifurcation cascades going from periodic to chaotic solutions. Studying coupled maps allows to identify other qualitative changes in the collective dynamics, such as pattern formations or hysteresis. Particularly, hysteresis is the appearance of different attracting sets, a set when the control parameter is increased and another set when it is decreased -- a multi-stable region. In this work, we present an experimental study on the bifurcations and hysteresis of nearly identical, coupled, logistic maps. Our logistic maps are an electronic system that has a discrete-time evolution with a high signal-to-noise ratio ($\sim10^6$), resulting in simple, precise, and reliable experimental manipulations, which include the design of a modifiable diffusive coupling configuration circuit. We find that the characterisations of the isolated and coupled logistic-maps' dynamics agrees excellently with the theoretical and numerical predictions (such as the critical bifurcation points and Feigenbaum's bifurcation velocity). Here, we report multi-stable regions appearing robustly across configurations, even though our configurations had parameter mismatch (which we measure directly from the components of the circuit and also infer from the resultant dynamics for each map) and were unavoidably affected by electronic noise.