Fig. 4: Graphical fixed-point analysis.
From: Bistability and time crystals in long-ranged directed percolation
For a power–law exponent α < 1, the dynamics of Eq. (6) is understood from the FP analysis of the equation x = fμ(x). a For a control parameter \(\mu <{\mu }_{{\rm{c}}}^{0}=0.6550(8)\), the system is inactive, corresponding to a single FP x0 = 0: at long times, the system ends up in the empty, absorbing state with state variables si = 0 for all sites i. b At the critical point \(\mu ={\mu }_{{\rm{c}}}^{0}\), a new semi-stable FP emerges at xc = 0.5216(9), that is, unstable from his left and stable on his right. c Increasing μ above \({\mu }_{{\rm{c}}}^{0}\), the semi-stable FP splits into an unstable FP x1 < xc and a stable FP x2 > xc. Depending on whether the initial density p1 is <x1 or >x1, the system flows towards density n = x0 = 0 or n = x2 > 0, respectively, indicating bistability.
