Length scale of interaction in spatiotemporal chaos

Extensive systems have no long scale correlations and behave as a sum of their parts. Various techniques are introduced to determine a characteristic length scale of interaction beyond which spatiotemporal chaos is extensive in reaction-diffusion networks. Information about network size, boundary condition, or abnormalities in network topology gets scrambled in spatiotemporal chaos, and the attenuation of information provides such characteristic length scales. Space-time information flow associated with the recovery of spatiotemporal chaos from finite perturbations, a concept somewhat opposite to the paradigm of Lyapunov exponents, defines another characteristic length scale. High-precision computational studies of asymptotic spatiotemporal chaos in the complex Ginzburg-Landau system and transient spatiotemporal chaos in the Gray-Scott network show that these different length scales are comparable and thus suitable to define a length scale of interaction. Preliminary studies demonstrate the relevance of these length scales for stable chaos.

Figure 1

Spatiotemporal chaos of the variable $a$ in the (a) complex Ginzburg-Landau (cGL) ring network and in the (b) transient phase of the Gray-Scott (GS) ring network. The cGL system is plotted for $300$ time units, $N=500$ nodes, and for the parameters $c_1=3.5$, $c_3=0.95$, and $D=4$. The GS system is plotted for $300$ time units, $N=500$ nodes, and for the parameters $\mu =33.7$, $\Phi =2.8$, and $D=16$. A fourth-order Runge-Kutta integration method was used; large values of $a$ are color coded in white.

Figure 2

Deviation of space-time averages [$\mathrm{X̄}$, Eq. (6)] from those of a large ($N=1000$) ring network [${\mathrm{X̄}}_0$, Eq. (7)] as a function of network size $N$: (a) $\ln |\mathrm{r̄}-{\mathrm{r̄}}_0|$ for the cGL ring network ($c_1=3.5$, $c_3\in \{0.85,0.95,1.2\}$) with $r=\left|z\right|$ the distance to the unstable focus; (b) $\ln |\mathrm{ā}-{\mathrm{ā}}_0|$ for the GS ring network ($\mu =33.7$, $\Phi =2.8$); (c) $\ln |\mathrm{X̄}-{\mathrm{X̄}}_0|$ for the same GS ring network as in (b) and different variables $X$ ($X=a^2$, $X=(\mathit{da}/\mathit{dn}){}^2$, and $X=r$ with $r$ the distance to the unstable focus). The envelopes of the curves in (a) and (b) are fitted with a subjectively chosen linear function. The linear fit from (b) is copied into (c) to guide the eye. The deviations of space-time averages in (a) do not match the deviations from extensivity of the Lyapunov dimension in [25].

Figure 3

Deviation of time averages [$\langle X_n\rangle$, Eq. (5)] from the space-time average of a large ring network [${\mathrm{X̄}}_0$, Eq. (7)] for varying nodes $n$ near a no-flux boundary ($N=1000$): (a) $\ln |\langle r_n\rangle -{\mathrm{r̄}}_0|$ for the cGL ring network ($c_1=3.5$, $c_3\in \{0.85,0.95,1.2\}$) with $r=\left|z\right|$ the distance to the unstable focus; (b) $\ln |\langle a_n\rangle -{\mathrm{ā}}_0|$ for the GS ring network ($\mu =33.7$, $\Phi =2.8$); (c) $\ln (s_X|\langle X_n\rangle -{\mathrm{X̄}}_0|)$ [35] for the same GS ring network as in (b) and different variables $X$ ($X=a^2$, $X=\mathit{da}/\mathit{dn}$, and $X=r$ with $r$ the distance to the unstable focus). $s_X$ is a scale factor chosen to align all three traces vertically. The top trace corresponds to $\ln [{max}_X(s_X|\langle X_n\rangle -{\mathrm{X̄}}_0|)]$; it is raised above the other traces for clarity. Envelopes of curves in (a) and (c) are fitted with a subjectively chosen linear function. The slope of the linear fit in (c) is shown in (b) for reference.

Figure 4

Shortcut-induced deviations from microextensivity $E$ [Eq. (9)] that are based on time averages $\langle X\rangle$ for varying shortcut lengths $k_2-k_1$ in a ring network ($N=1000$) with a single shortcut between node $k_1=1$ and $k_2$, for (a) the cGL system ($c_1=3.5$, $c_3=0.95$) and for (b) the GS system ($\mu =33.7$, $\Phi =2.8$). In each of the figures a subjectively chosen linear fit is determined and plotted twice (with the same slope) as a visual reference.

Figure 5

Logarithm of same-time mutual information $M_{\mathit{mn}}$ vs. node separation $|n-m|$ for (a) the variable $a$ of the cGL ring network ($c_1=3.5$) and for (b) the variable $a$ of the GS ring network ($\mu =33.7$, $\Phi =2.8$). For variable $r$ the deviations from the linear trend were more pronounced. Each curve was approximated with a least-squares linear fit for the largest range of node separations $|n-m|$ with an approximately linear relation. The probabilities were computed from histograms with five bins for each variable, so $25$ bins for the joint probability $p(X_m,X_n)$.

Figure 6

Logarithm of transfer entropy $T_{\Delta n\Delta t}$ vs. time delay $\Delta t$ for a variety of node spacings $\Delta n$ ($\Delta n=0,5,10,15,\text{etc.}$ from top to bottom) for (a) the cGL ring network ($c_1=3.5$, $c_3=0.95$) and for (b) the GS ring network ($\mu =33.7$, $\Phi =2.8$). Each curve is shifted in $\Delta t$ using the estimated information transfer velocity $v$ ($v=4.2$ for cGL system, and $v=3.3$ for GS system) to allow the peaks to align. The break in the graphs at $\Delta n=0$, $\Delta t=u$ occurs because $T_{\Delta n\Delta t}=0$ at that point [Eqs. (11) and (12)]. The location $m$ of the additional entropy source is at $|n-m|=N/2$ with $N=1000$ the network size.

Figure 7

Spatiotemporal deviations in the average nodal characteristics for perturbed [Eq. (14)] and unperturbed [Eq. (1)] network dynamics at node $n$ and at a time $\tau$ after the onset of the perturbation. (a) $\ln |\langle r_{n\tau }\rangle -{\mathrm{r̄}}_0|$ for the cGL ring network [$c_1=3.5$, $c_3=0.95$, $N=500$, $P=50$, ${\mathbf{q}}_0=(5,0)$, and $K=7.2\times {10}^6$], and (b) $\ln |\langle a_{n\tau }\rangle -{\mathrm{ā}}_0|$ for the GS ring network [$\mu =33.7$, $\Phi =2.8$, $N=500$, $P=300$, ${\mathbf{q}}_0=(2,0)$, and $K=6.0\times {10}^5$]. The other parameters for the spatiotemporally localized finite perturbation are $n_0=N/2$, $w_n=10$, and $w_t=1$. The dotted lines have slope equal to the information transfer velocity [Eq. (13)] and are plotted for comparison.

Figure 8

(a) The data from Fig. 7, collapsed along the temporal ($\tau$) axis. (b) The data from Fig. 7, collapsed along the spatial ($n$) axis. Note that the time axis is scaled differently for the cGL and GS traces.

Figure 9

Length scales of interaction for stable chaos in the coupled map lattice [Eq. (19) with $a=0.07$, $b=2.7$, $c=0.1$, and $\epsilon =2/3$]. (a) Deviation of time averages [$\langle x_n\rangle$, Eq. (5)] from the space-time average of a large ring network [${\mathrm{x̄}}_0$, Eq. (7)] for varying nodes $n$ near a no-flux boundary ($N=1000$), in analogy to Fig. 3. (b) Logarithm of transfer entropy $T_{\Delta n\Delta t}$ vs. time delay $\Delta t$ for a variety of node spacings $\Delta n$ ($\Delta n=0,1,2,3,\text{etc.,}$ from top to bottom), in analogy to Fig. 6. Each curve is shifted in $\Delta t$ using the estimated information transfer velocity $v$ ($v=0.75\text{nodes}/\text{timestep}$) to allow the peaks to align. (c) Spatiotemporal deviations in the average nodal characteristics for perturbed [Eq. (14)] and unperturbed [Eq. (19)] network dynamics at node $n$ and at a time $\tau$ after the onset of the perturbation, $\ln |\langle x_{n\tau }\rangle -{\mathrm{x̄}}_0|$, (with parameters $N=200$, $P=100$, $n_0=N/2$, $q_0=1$, $K=1.3\times {10}^8$, $w_n=3$, and $w_t=2$ [43]), in analogy to Fig. 7. The dotted lines have slope equal to the information transfer velocity $v$ [Eq. (13)], and the double dotted lines have slope equal to the disturbance propagation velocity $v_d$ in Eq. (20).

Figure 10

A transformation that turns two ring networks each having $N$ nodes into one network with $2N$ nodes, without changing the local network structure or the total number of nodes.