Erratum: Spatial chaos of traveling waves has a given velocity [Phys. Rev. E 80, 025203(R) (2009)]

Figure 1

Entropy-velocity diagram of stable TW in $(\epsilon ,T)$ space for $a=0.8$. Painted points code for the numerical TW entropy $\frac{\mathit{log}{P}_{19,v}}{19}$ (the period $L=19$ is chosen large enough for robustness). Gray levels (colors) correspond to velocities $v$, from 0 – left (light gray, blue) to 1 – right (dark gray, black) and including $\frac{1}{7},\frac{1}{6},\frac{1}{5},\frac{1}{4},\frac{2}{7},\frac{1}{3},\frac{2}{5},\frac{3}{7},\frac{1}{2}$ and symmetric values (from light gray, orange, to dark gray, brown). Intensity is proportional to entropy (maximal values along the axis $T=0.5$). Clearly, some domains with small (respectively, large) velocity overlap with domains with velocity 0 (respectively, 1) indicating that a chaos of fixed points (respectively, chaos of TW with $v=1$) may coexist with a chaos of patterns with intermediate velocity. Curves delimit regions where TW symbolic sequences—with arbitrary pattern in a given chaotic set and given velocity—were analytically proved to solve Eq. (3); see text.

Figure 2

Entropy-velocity diagram in the $(\epsilon ,T)$ square and complementary domains where the entropy is analytically shown to cancel (outer light gray regions). (a) $a=0.4$ and (b) $a=0.8$; the rectangle delimits the region displayed in Fig. 1.