Chaotic Griffiths Phase with Anomalous Lyapunov Spectra in Coupled Map Networks

Dynamics of coupled chaotic oscillators on a network are studied using coupled maps. Within a broad range of parameter values representing the coupling strength or the degree of elements, the system repeats formation and split of coherent clusters. The distribution of the cluster size follows a power law with the exponent $\alpha$, which changes with the parameter values. The number of positive Lyapunov exponents and their spectra are scaled anomalously with the power of the system size with the exponent $\beta$, which also changes with the parameters. The scaling relation $\alpha \sim 2(\beta +1)$ is uncovered, which is universal independent of parameters and among random networks.

Figure 1

Examples of typical time series in the CMN (1): $x_{2n}(i)$ as a function of time step $n$ are overlaid for all elements $i$, after discarding initial transients of $10^6$ steps. Instead of plotting every time step, the variables are plotted every two time steps to make them clearly discernible, since the period-two oscillation is inherent in the logistic map. $a=1.7$ and $N=200$. (a) $\epsilon =0.5$, $k=20$ [phase (ii)]. (b) $\epsilon =0.35$, $k=15$ [phase (iii)]. (c) $\epsilon =0.2$, $k=10$ [phase (iv)]. (d) $\epsilon =0.05$, $k=10$ [phase (v)].

Figure 2

Phase diagram of the CMN (1) with $a=1.7$ and $N=200$. Each phase (i)–(v) (see text) is determined by the Lyapunov exponents as described in the text (see Supplemental Material [37] for the diagrams on ${\lambda }_1$, ${\lambda }_2$, $N_p$, and an index for macroscopic order defined therein). The configuration of the phase diagram is independent of $a$, while the phase boundary is shifted.

Figure 3

Temporal evolution of the maximal cluster size. $a=1.7$, $k=20$, and $N=1000$. The cluster is computed by using the threshold $\delta ={10}^{-3}$, while this intermittent behavior does not vary as long as it is sufficiently small. $\epsilon =0.5$ (blue line), $\epsilon =0.55$ (green line), $\epsilon =0.6$ (red line), in the chaotic Griffiths phase.

Figure 4

The distribution $P(s)$ of cluster size $s$. Log-log plot. $a=1.7$, $k=20$, and $N=16384$. The results from $\epsilon =0.45$, 0.475, 0.5, 0.525, 0.55, 0.575, and 0.6 are plotted with different colors. The distribution is obtained by sampling over $10^3$ steps, with 100 initial conditions, over 100 networks, by using the threshold $\delta ={10}^{-3}$, while the exponents do not vary as long as this threshold is sufficiently small, and also the network sample dependence is negligible.

Figure 5

(a) Dependence of the exponent $\alpha$ (red) and $\beta$ (blue) upon $\epsilon$, from the data of Figs. 4 and 6. (b) Relationship between the exponents $\alpha$ (abscissa) and $\beta$ (ordinate). Apart from the data of Fig. 4 ($\circ$,red), the data with varying $k$ by fixing $\epsilon$ at 0.6 ($△$,green), the data $a=1.9$ ($\square$,blue), and $a=1.7$ on a regular random network ($◊$, yellow), as well as that with increasing the small-world structure ($\star$, purple) and that with increasing community structure ($\times$, black), with fixed $k$ and varying $\epsilon$ are plotted. (see also Supplemental Material [37], Fig. S5). The error bars are computed from the least-square fit of the power law.

Figure 6

(a) The number of positive Lyapunov exponents $N_p$ plotted as a function of the system size $N$, for $\epsilon =0.45$, 0.48, 0,51, 0.54, 0.57, and 0.6 with different colors. $a=1.7$, $k=20$. (b) The scaled Lyapunov spectra $\lambda (z)$ with $z=i/N^{\beta }$. The Lyapunov exponents are computed from 2000 time steps over 12 networks samples and 4 initial conditions.