Slowing down of so-called chaotic states: ``Freezing'' the initial state
Résumé
The so-called chaotic states that emerge on the model of $XY$ interacting
on regular critical range networks are analyzed. Typical time scales
are extracted from the time series analysis of the global magnetization.
The large spectrum confirms the chaotic nature of the observable,
anyhow different peaks in the spectrum allows for typical characteristic
time-scales to emerge. We find that these time scales {$\tau(N)$}
display a critical slowing down, i.e they diverge as $N\rightarrow\infty$.
The scaling law is analyzed for different energy densities and the
behavior $\tau(N)\sim\sqrt{N}$ is exhibited. This behavior is furthermore
explained analytically using the formalism of thermodynamic-equations
of the motion and analyzing the eigenvalues of the adjacency matrix.