Chaotic versus stochastic behavior in active-dissipative nonlinear systems

We study the dynamical state of the one-dimensional noisy generalized Kuramoto-Sivashinsky (gKS) equation by making use of time-series techniques based on symbolic dynamics and complex networks. We focus on analyzing temporal signals of global measure in the spatiotemporal patterns as the dispersion parameter of the gKS equation and the strength of the noise are varied, observing that a rich variety of different regimes, from high-dimensional chaos to pure stochastic behavior, emerge. Permutation entropy, permutation spectrum, and network entropy allow us to fully classify the dynamical state exposed to additive noise.

Figure 1

Spatiotemporal patterns of the noisy gKS solution for different values of $\sigma$ and $\delta$. (a) $\delta =0,\sigma =0.05,$ (b) $\delta =0,\sigma =0.5,$ (c) $\delta =0,\sigma =1.0,$ (d) $\delta =0.2,\sigma =0.05,$ (e) $\delta =0.2,\sigma =0.5,$ (f) $\delta =0.2,\sigma =1.0,$ (g) $\delta =0.4,\sigma =0.05,$ (h) $\delta =0.4,\sigma =0.5,$ and (i) $\delta =0.4,\sigma =1.0,$

Figure 2

Time variation of the signal $u_G$ for $\delta =0$ and different values of $\sigma$.

Figure 3

Power spectrum density $S_u\left(f\right)$ for different values of $\sigma$ and for (a) $\delta =0$, (b) $\delta =0.2$, and (c) $\delta =0.4$. The red dashed line corresponds to a power law with exponent $-1.56\pm 0.08$.

Figure 4

(a) Permutation entropy $h_p$ as function of $\sigma$ for $\delta =0$, 0.2, and 0.4. The solid line corresponds to the permutation entropy obtained from the EW equation. (b) Surface plot of $h_p$ as function of $\delta$ and $\sigma$.

Figure 5

Frequency distribution of permutation patterns and their standard deviation at different $\sigma$ for (a) the noisy gKS solution with $\delta =0.2$, and (b) the EW solution. Forbidden patterns are shown as red dots.

Figure 6

(a) Number of forbidden patterns $N_f$ of the noisy gKS solution as functions of $\delta$ and $\sigma$. (b) Critical value ${\sigma }_c$ as function of $\delta$ above which the noisy gKS becomes fully stochastic. Solid lines correspond to a data fit to the function given by Eq. (9).

Figure 7

(a) Network entropy $h_n$ as function of $\sigma$ for $\delta =0$, 0.2, and 0.4. The solid line corresponds to the network entropy obtained from the EW equation. (b) Surface plot of $h_n$ as function of $\delta$ and $\sigma$.

Figure 8

Correlation coefficient $C$ against the predicted time $t_P$ for increment $\mathrm{\Delta }{u}_G$ with different values of $\sigma$ for (a) the noisy gKS solution for $\delta =0.2$ and (b) the EW solution. Panel (c) shows the correlation coefficient $C(t_P=1)$ as functions of $\sigma$ and $\delta$ for $\mathrm{\Delta }{u}_G$ of the noisy gKS solution.