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submittedInternational audienceThe so-called chaotic states that emerge on the model of $XY$ interactingon regular critical range networks are analyzed. Typical time scalesare 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 characteristictime-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 thebehavior $\tau(N)\sim\sqrt{N}$ is exhibited. This behavior is furthermoreexplained analytically using the formalism of thermodynamic-equationsof the motion and analyzing the eigenvalues of the adjacency matrix

Slowing Down of So-called Chaotic States: “Freezing” the Initial State