  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.
