Transition from clustered state to spatiotemporal chaos in a small-world networks

We study the spatiotemporal patterns in coupled circle maps on a small-world network. This system shows a rich phase diagram with several interesting phases. In particular, we make a detailed study of transition from clustered phase to spatiotemporal chaos. In the clustered state, observed at smaller coupling values, some sites stay close to the fixed point forever while others explore a larger part of the phase space. For stronger coupling, there is a transition to spatiotemporal chaos where no site stays close to fixed point forever. We study this transition as a dynamic phase transition. Persistence acts as a good order parameter for this transition. We find that this transition is continuous. We also briefly discuss other phases observed in this system.

Figure 1

Phase diagram of circle map in two parameter space of coupling strength $ϵ$ and rewiring probability $p$. The abbreviations SFP, FP/PO, FP/Chaos, and STC denote synchronized fixed point, fixed point with periodic orbit, fixed point with chaos, and spatiotemporal chaos, respectively.

Figure 2

Bifurcation diagram in which state variable $x_n\left(i\right)$ ($i=1,2,\dots ,100$) has been plotted as a function of coupling strength $ϵ$ for $p=0.8$.

Figure 3

Space-time plots of the different values of $ϵ$ observed for $p=0.1$. Lattice of size 100 with parameters $\omega =0.068$ and $k=1.0$. The space-time plots show (a) in which for $ϵ=0.01$ it shows the SFP phase, (b) is for $ϵ=0.15$ (FP/PO phase), (c) $ϵ=0.27$ shows a FP/Chaos phase, and (d) $ϵ=0.35$ is in the region of spatiotemporal chaos. Laminar regions are represented by black color.

Figure 4

Fraction of persistent sites $P\left(t\right)$ is plotted as a function of time $t$ at the critical point which clearly displays power law. Lattice size is $L=2\times {10}^4$. Data are averaged over ${10}^3$ initial conditions and critical exponents obtained are (a) 1.291 for $p=0.1$ and (b) 0.239 for $p=0.8$.

Figure 5

Off-critical simulations near critical point. The graph demonstrates data collapse according to the scaling form (5). Lattice size $L=2\times {10}^4$. We averaged over 500 initial conditions. (a) is for $p=0.1$ and (b) for $p=0.8$.

Figure 6

Finite-size scaling for different lattice sizes. (a) $L=100$, 150, 300, and 400 for $p=0.1$ and (b) $L=100$, 300, and 400 for $p=0.8$. The graph demonstrates data collapse according to the scaling form in Eq. (5).

Figure 7

The plot shows $P_l\left(t\right)$ multiplied by $t^{-{\theta }_l}$ with ${\theta }_l=0.239$ for $p=0.8$. Inset shows the raw data of the simulation.

Figure 8

Persistence $P\left(t\right)$ is plotted as a function of time $t$ for $p=0.1$. Lattice size $L=2\times {10}^4$. We averaged over 500 initial conditions. (a) The exponent is 1.155 for $k=0.8$. (b) The exponent is between 2.7 and 3.3 for $k=1.3$.