Synchronization universality classes and stability of smooth coupled map lattices

We study two problems related to spatially extended systems: the dynamical stability and the universality classes of the replica synchronization transition. We use a simple model of one-dimensional coupled map lattices and show that chaotic behavior implies that the synchronization transition belongs to the multiplicative noise universality class, while stable chaos implies that the synchronization transition belongs to the directed percolation universality class.

Figure 1

(Color online) The graph of $f\left(x\right)$ for three values of $a$.

Figure 2

(Color online) Space time pattern of the CML of Eq. (2) with $a=1.9$ and $N=256$ drawn horizontally for a total of $T=300$ time steps drawn vertically from top to bottom. The initial configuration ${\mathit{x}}^0$ is chosen randomly. The color code assigns white (black) whenever $x_i^t=0\left(1\right)$ and a rainbow color scale for other values of $x_i^t$ starting with red for values near zero. Patches of CA behavior (rule 150) appear after a short transient and will eventually fill the whole pattern.

Figure 3

(Color online) Fraction of chaotic orbits as a function of time for different values of $a$ for a total of $M=100$ random initial states, $N=1024$ and $T=100N$.

Figure 4

(Color online) Dependence of the order parameter $⟨u⟩$ on $p$ near the critical value $a_c$. In (a) $T=t_{\mathit{rel}}=5\times {10}^4$, $M=100$, and $N=1024$. In (b) the standard deviation is rather large for the same values of $T$, $t_{\mathit{rel}}$, $M$, and $N$ making it difficult to assign a value to $p_s$. However, by making $t_{\mathit{rel}}={10}^5$ there is a well-defined transition at $p_s=0.303$ as shown in the inset.

Figure 5

(Color online) Synchronization threshold $p_s$ vs $a$. Labels refer to patterns in Fig. 6.

Figure 6

(Color online) Synchronization space time patterns near $p_s$ for different values of $a$. See Fig. 5 for reference. (a) $a=1.8<a_c(\lambda >0),p=0.48<p_s\left(1.8\right)$: the difference field is expanding but low in value, and the synchronized state is not locally absorbing (b) $a=1.9>a_c(\lambda <0),p=0.48⪢p_s\left(1.9\right)$: the difference field is contracting, and the synchronized state is absorbing, even though occasionally patches of nonsynchronized sites appear. (c) $a=1.9>a_c,p=0.36⩾p_s\left(1.9\right)$: the difference field is sustained by the nonlinear term, the unsynchronized sites form a connected cluster. (d) $a=3.5⪢a_c,p=0.36⩾p_s\left(3.5\right)$: the linear contribution to the difference field is (almost) absent, the nonsynchronized cluster is connected and has more “holes” than in case (c). $N=256,T=256$

Figure 7

(Color online) The values of the critical exponents $\delta$, $\beta$, and $z$ for different values of $a$. The vertical line is drawn at $a_c=1.8142$.