Exploring Transition from Stability to Chaos through Random Matrices

TL;DR

This paper demonstrates how random matrices with Wishart-like properties can effectively detect the transition from stability to chaos in the Chirikov standard map, using eigenvalue density analysis to identify the chaos boundary.

Contribution

It adapts a phase transition detection technique from spin systems to chaos theory, accurately locating the chaos boundary in the standard map.

Findings

Successfully identified the chaos boundary at K≈2.43

Validated the use of eigenvalue density for chaos monitoring

Extended phase transition methods to dynamical systems

Abstract

This study explores the application of random matrices to track chaotic dynamics within the Chirikov standard map. Our findings highlight the potential of matrices exhibiting Wishart-like characteristics, combined with statistical insights from their eigenvalue density, as a promising avenue for chaos monitoring. Inspired by a technique originally designed for detecting phase transitions in spin systems, we successfully adapt and apply it to identify analogous transformative patterns in the context of the Chirikov standard map. Leveraging the precision previously demonstrated in localizing critical points within magnetic systems in our prior research, our method accurately pinpoints the Chirikov resonance-overlap criterion for the chaos boundary at $K\approx 2.43$, reinforcing its effectiveness.