The quasi-periodic behavior on different earthquake models. The first row show the analysis of the autocorrelation function for the avalanche time series of (a) the random spatial earthquake model with 〈 k 〉 = 228 , (b) the OFC earthquake model, and (c) the earthquake model on cycle with k = 4. Each type of graph or network is simulated for the system size N=128². Different curves correspond to different degrees of dissipation β ( = 4 α ) or α. The position of the peaks indicates that the system has a quasi-periodic behavior with a period proportional to the degree of dissipation: T = T m ( 1 − β ) = T m ( 1 − 4 α ) . For visual clarity, curves α=0.10 and α=0.15 have been shifted along the x axis in the earthquake model on cycle, x → x − 100 and x → x + 100 , respectively. The second row shows the fast Fourier transform (FFT) for the avalanche time series of (a) the random spatial earthquake model with 〈 k 〉 = 228 , β=0.6 and N=128² ( r c = 0.0325 ), (b) the OFC model with α=0.15 and N=128², and (c) the earthquake model on cycle with α=0.15, N=128², and k = 4. The quasi-periodic behavior is robust T=1/f and the period is consistent with that of the autocorrelation function.