Pattern Formation in Systems with Multiple Delayed Feedbacks

Dynamical systems with complex delayed interactions arise commonly when propagation times are significant, yielding complicated oscillatory instabilities. In this Letter, we introduce a class of systems with multiple, hierarchically long time delays, and using a suitable space-time representation we uncover features otherwise hidden in their temporal dynamics. The behavior in the case of two delays is shown to “encode” two-dimensional spiral defects and defects turbulence. A multiple scale analysis sets the equivalence to a complex Ginzburg-Landau equation, and a novel criterium for the attainment of the long-delay regime is introduced. We also demonstrate this phenomenon for a semiconductor laser with two delayed optical feedbacks.

Figure 1

Spiral defects in a system with delays, Eq (1). (a) Typical time series of the absolute value $|z(t)|$. Spatiotemporal representation of the time series using pseudospace coordinates, Eq. (3), reveals the spiral defects: (b) Snapshot of the spatial profile in the pseudospace coordinates ($x$, $y$) for ${\theta }_0=0.4$. (c) Constant level lines for the phase of $z$. Circles denote the positions of defects. Parameters: $a=-0.985$, $b=0.4$, $c=0.6$ (corresponding to $P=0.015$), $d=-0.75+i$, ${\tau }_1=100$, and ${\tau }_2=10000$. Initial conditions are chosen randomly. See the Supplemental Material [19] for the dynamics of the patterns.

Figure 2

Defects turbulence in delayed system, Eq. (1). Same as in Fig. 1 for a different value of $d=-0.1+i$. Spatiotemporal representation in (b) and (c) reveals defects turbulence.

Figure 3

“Small” delays effect. (a) One-parameter curve $x(t)$, $y(t)$ in the pseudospace determined by Eq. (3) for $ϵ=0.05$. For larger distances between the branches (smaller delays), the line does not resolve the cores of the spiral of the corresponding Ginzburg-Landau model. (b) Numerical histograms of $|z|$ for the DT regime (parameters as in Fig. 2) for increasing the delays values. Histograms for smaller delays (here, ${\tau }_1=25$, ${\tau }_2=25{\tau }_1$) correspond to bounded, periodic solutions with no defects, reached after a transient. A tail in the distribution appears for higher delays (here, ${\tau }_1=50$, ${\tau }_2=50{\tau }_1$). The dashed line is a reference curve $P(|z|)\sim |z{|}^1$.

Figure 4

Dynamics of the solution $E$ of the system in Eq. (5), represented as snapshot in the pseudospace for the parameter values: ${\tau }_1=10^2$, ${\tau }_2=10^4$, ${\eta }_1={\eta }_2=0.1$, $T=10^2$, and $J=-0.17$. (a), (b) Amplitude and phase of $E$ for $\alpha =2$, showing the occurrence of spiral defects. (c) Amplitude of $E$, defects turbulence regime for $\alpha =4$. The temporal behavior is presented as movies in the Supplemental Material [19]. (d) Statistics of the field amplitude in the case (c); the line is a power-law fit of the tail with exponent 0.98.