Spatial recurrence plots

We propose an extension of the recurrence plot concept to perform quantitative analyzes of roughness and disorder of spatial patterns at a fixed time. We introduce spatial recurrence plots (SRPs) as a graphical representation of the pointwise correlation matrix, in terms of a two-dimensional spatial return plot. This technique is applied to the study of complex patterns generated by coupled map lattices, which are characterized by measures of complexity based on SRPs. We show that the complexity measures we propose for SRPs provide a systematic way of investigating the distribution of spatially coherent structures, such as synchronization domains, in lattice profiles. This approach has potential for many more applications, e.g., in surface roughness analyzes.

Figure 1

Spatial return maps for a coupled map lattice (1) with $N=1001$ at a fixed time $n=5000$, and (a) $\alpha =2.0$, $ϵ=0.1$; (b) $\alpha =1.0$, $ϵ=0.1$; (c) $\alpha =0.5$, $ϵ=0.1$; (d) $\alpha =2.0$, $ϵ=0.5$; (e) $\alpha =1.0$, $ϵ=0.5$; (f) $\alpha =0.5$, $ϵ=0.5$ (large plot) and $\alpha =0.2$, $ϵ=0.5$ (small plot).

Figure 2

Space-time-amplitude plots for the CML (1), with $N=51$ sites, $ϵ=0.5$ and $\alpha =$ (a) $2.0$, (b) $1.0$, and (c) $0.5$.

Figure 3

Spatial correlation integral versus radius $l$ for the CML (1) with $N=201$, at a fixed time $n=2000$ and (a) $\alpha =2.0$, $ϵ=0.1$; (b) $\alpha =1.0$, $ϵ=0.1$; (c) $\alpha =0.5$, $ϵ=0.1$; (d) $\alpha =2.0$, $ϵ=0.5$; (e) $\alpha =1.0$, $ϵ=0.5$; (f) $\alpha =0.5$, $ϵ=0.5$.

Figure 4

Spatial correlation exponent as a function of the embedding dimension $m$, for the CML (1) with $N=1001$, at a fixed time $n=2000$, $\alpha =5.0$, and $ϵ=0.1$. The straight line is a least squares fit with slope equal to the correlation dimension density $d_2=0.36$.

Figure 5

Spatial recurrence plots for different values of the cutoff radius for the CML (1) with $N=201$, $\alpha =3.73$, $ϵ=1.0$, $n=2000$ and (a) $l=0.1$, (b) $l=0.2$, (c) $l=0.4$, (d) $l=0.6$, (e) $l=0.8$, (f) $l=1.0$.

Figure 6

Spatial recurrence plots for the following $(\alpha ,ϵ)$ values in the coupling parameter space: (a) $(2.0,0.1)$, (b) $(1.0,0.1)$, (c) $(0.5,0.1)$, (d) $(2.0,0.5)$, (e) $(1.0,0.5)$, (f) $(0.5,0.5)$. We used a cutoff radius $l=0.1$ and $N=201$.

Figure 7

Average spatial correlation integral versus the range parameter for different coupling strengths. We used a cutoff radius $l=0.1$ and $N=201$.

Figure 8

Average laminarity versus the range parameter for different coupling strengths. We used a cutoff radius $l=0.1$ and $N=201$.

Figure 9

Average trapping length versus the range parameter for different coupling strengths. We used a cutoff radius $l=0.1$ and $N=201$.

Figure 10

Spatial pattern (at a fixed time $n=2001$) and the corresponding SRP (cutoff radius $l=0.05$) obtained for a lattice with $N=1001$ sites, and coupling parameters $ϵ=1.0$, and (a) $\alpha =2.0$ and (b) $\alpha =2.8$.

Figure 11

Spatial autocorrelation function versus spatial delay (measured in number of lattice sites) for the two patterns considered in Fig. 10.