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

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.