Constraint on periodic orbits of chaotic systems given by Random Matrix Theory

TL;DR

This paper establishes a link between spectral fluctuations in chaotic systems and constraints on their classical periodic orbits, suggesting that spectral statistics impose lower bounds on orbit instability, which can test the BGS conjecture.

Contribution

It provides a novel theoretical constraint on the Lyapunov exponents of periodic orbits in chaotic systems based on Random Matrix Theory and spectral fluctuation analysis.

Findings

Lyapunov exponent of periodic orbits must exceed a minimum value of approximately 0.85.

Spectral fluctuation analysis constrains classical orbit instability in chaotic systems.

Potential to identify deviations from the BGS conjecture in specific systems.

Abstract

Considering the fluctuations of spectral functions, we prove that if chaotic systems fulfill the Bohigas-Gianonni-Schmit (BGS) conjecture, which relates their spectral statistics to that of random matrices, therefore by virtue of Gutzwiller trace formula, the instability of classical periodic orbits is constrained. In particular for two-dimensional chaotic systems, the Lyapunov exponent $\lambda_p$ of each periodic orbit $p$ should be bigger than a minimum value $\lambda_{\text{min}} \geq 0.850738$. This opens the possibility of new constraints for a system to be fully chaotic, or the failure of the BGS conjecture.