Revealing the Building Blocks of Spatiotemporal Chaos: Deviations from Extensivity

We have performed high-precision computational studies of the fractal dimension as a function of system length for spatiotemporal chaotic states of the one-dimensional complex Ginzburg-Landau equation. Our data show deviations from extensivity on a length scale consistent with the chaotic length scale, indicating that this spatiotemporal chaotic system is composed of weakly interacting building blocks, each containing about 2 degrees of freedom. Our results also suggest an explanation of some of the “windows of periodicity” found in spatiotemporal systems of moderate size.

Figure 1

(a) Dimension $D$ as a function of system length $L$ for Eq. (2) with $(c_1,c_3)=(3.5,0.95)$. (b) Relative deviations $\Delta D=(D-D_{\mathrm{ext}})/D_{\mathrm{ext}}$ from microextensive behavior as a function of $L$, with $D_{\mathrm{ext}}(L)=\delta L$ and $\delta$ determined at large $L$. Oscillations of length scale ${\xi }_o\approx 8.5$ signal the presence of chaotic building blocks. The arrows indicate missing values of $L$ for which nonchaotic behavior is found. Error bars for $D$, determined from the standard deviation of 200 independent measurements, are typically smaller than 0.002. Each of the 200 measurements is an average of many samples during a time period of at least 75 000 time units following a transient of 80 000 time units, with time step $dt=0.05$ and at least 2 spatial modes per spatial unit.

Figure 2

(a) Oscillation length scale ${\xi }_o$ as a function of $c_3$ for fixed $c_1=3.5$ in Eq. (2). Values of ${\xi }_o$ were determined by fitting a cosine with decaying amplitude to $\Delta D(L)$. A span of at least 3 wavelengths was used for $c_3\ge 0.90$. (b) Ratio ${\xi }_o/{\xi }_{\delta }$ for the same values of $c_3$. ${\xi }_{\delta }$ is determined using Eq. (1) at large $L$. The data indicate that each building block contains about 2 degrees of freedom. (c) Relative deviation $\Delta D$ as a function of building block number, $L/{\xi }_o$, for 6 values of $c_3$. The collapse shows that the deviations depend only on the number of building blocks. Minima occur for integer numbers of building blocks. The data near the minimum at $L/{\xi }_o=5$ for $c_3=0.95$ and $c_3=1.00$ are influenced by a window of periodicity.