Spatial chaos of traveling waves has a given velocity

We study the complexity of stable waves in unidirectional bistable coupled map lattices as a test tube to spatial chaos of traveling patterns in open flows. Numerical calculations reveal that, grouping patterns into sets according to their velocity, at most one set of waves has positive topological entropy for fixed parameters. By using symbolic dynamics and shadowing, we analytically determine velocity-dependent parameter domains of existence of pattern families with positive entropy. These arguments provide a method to exhibit chaotic sets of stable waves with arbitrary velocity in extended systems.

Figure 1

(Color online) Entropy-velocity diagram of stable TW in $\left(ϵ,T\right)$ space for $a=0.8$. Painted points code for the numerical TW entropy $\frac{\text{ln}{P}_{19,v}}{19}$ (the period $L=19$ is chosen large enough for robustness). Gray levels (colors ) correspond to velocities $v$, from 0—left (light gray, yellow) to 1—right (dark gray, brown) 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. Intensity is proportional to entropy (maximal values along the axis $T=0.5$). 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

(Color online) Entropy-velocity diagram in the $\left(ϵ,T\right)$-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.