Prediction and measurement of transient responses of first difference based chaos control for 1-dimensional maps

TL;DR

This paper investigates the transient responses of a first difference based chaos control method applied to 1D maps, combining theoretical predictions with experimental validation using low-cost analog circuits.

Contribution

It provides the first detailed analysis and experimental validation of transient behaviors in first difference based chaos control for 1D maps.

Findings

Transient responses vary widely, including erratic non-steady convergence.

Experimental results align well with theoretical predictions.

Different transient behaviors are observed depending on system parameters.

Abstract

Chaotic behavior can be produced from difference equations with unstable fixed points. Difference equations can be used for algorithms to control the chaotic behavior by perturbing a system parameter using feedback based on the first difference of the system value. This results in a system of nonlinear first order difference equations whose stable fixed point is the controlled chaotic behavior. Basing the feedback on the first difference produces distinctly different transient responses than when basing feedback on the error from the fixed point. Analog electronic circuits provide the experimental system for testing the chaos control algorithm. The circuits are low-cost, relatively easy to construct, and therefore provide a useful transition towards more specialized real-world applications. Here we present predictions and experimental results for the transient responses of a first difference based feedback control method applied to a chaotic finite difference 1-dimensional map. The experimental results are in good agreement with predictions, showing a variety of behaviors for the transient response, including erratic appearing non-steady convergence.