Laminar Chaos in Experiments: Nonlinear Systems with Time-Varying Delays and Noise

A new type of dynamics called laminar chaos was recently discovered through a theoretical analysis of a scalar delay differential equation with time-varying delay. Laminar chaos is a low-dimensional dynamics characterized by laminar phases of nearly constant intensity with periodic durations and a chaotic variation of the intensity from one laminar phase to the next laminar phase. This is in stark contrast to the typically observed higher-dimensional turbulent chaos, which is characterized by strong fluctuations. In this Letter we provide the first experimental observation of laminar chaos by studying an optoelectronic feedback loop with time-varying delay. The noise inherent in the experiment requires the development of a nonlinear Langevin equation with variable delay. The results show that laminar chaos can be observed in higher-order systems, and that the phenomenon is robust to noise and a digital implementation of the variable time delay.

Figure 1

An illustration of the optoelectronic oscillator we used to observe laminar chaos. Red lines indicate the optical path, black lines indicate the electronic path, and green lines indicate signal processing on the FPGA.

Figure 2

Experimental time series for different values of the mean delay ${\tau }_0$ and for different values of the strength $\zeta$ of the external noise. The trajectories (a)–(c) correspond to a dissipative delay (${\tau }_0=15.0\mathrm{ms}$) and show laminar chaos, whereas the trajectories (d) and (e) correspond to a conservative delay (${\tau }_0=15.4\mathrm{ms}$) and show turbulent chaos.

Figure 3

Trajectories generated from Eq. (4) with parameters from Table S1 for different mean delays ${\tau }_0$ and noise strengths $\sigma$. The trajectories (a)–(c) correspond to a dissipative delay (${\tau }_0=1.50$) and show laminar chaos, whereas the trajectories (d) and (e) correspond to a conservative delay (${\tau }_0=1.54$) and show turbulent chaos.

Figure 4

Detection of the laminar phases. Temporal distribution of the variance ${\sigma }_d^2$ (in units of $T/h$) of the approximate derivatives ($h\approx 0.0033$) of (a) laminar chaotic trajectories of Eq. (4) and (b) experimental trajectories (the same parameters as in Figs. 2 and 3). The laminar phases (low ${\sigma }_d^2$) and the burstlike transitions between them (high ${\sigma }_d^2$) are located around the attractive and repulsive fixed points of the reduced access map [dashed and solid lines in (a)], respectively. In the experimental data the location of the attractive fixed point is not known, and approximated via the local minima of ${\sigma }_d^2$ [dashed lines in (b)]. The rotation number $\rho$ of the access map is $\rho =-3/2$ leading to $q=2$ laminar phases per period.

Figure 5

Connection between the laminar phases for the laminar chaotic trajectories from (a) Fig. 3 (simulation) and (b) Fig. 2 (experiment). The intensity $z_{n+p^{\prime }}$ of the $(n+p^{\prime })$th laminar phase is plotted vs the intensity $z_n$ of the $n$th phase for $p^{\prime }=p=3$. The reconstruction of the nonlinearity $\mathbf{(}z,\mu F(z)\mathbf{)}$ (solid line) is possible even for high noise strengths. For the experimental data the black line represents the fit of the data for $\zeta =0$ to the nonlinearity $\widehat{\mu }\widehat{F}(z)=\widehat{\mu }{\sin }^2(z+{\widehat{\varphi }}_0)+{\widehat{c}}_0$, where $\widehat{\mu }\approx 2.229\pm 0.002$, ${\widehat{\varphi }}_0\approx 0.8074\pm 0.0004$, and ${\widehat{c}}_0\approx -0.103\pm 0.002$, where the $\widehat{·}$ is used to emphasize that the these parameter values are fits to the data shown in Fig. 5, not directly measured. The clipping below $z=0$ is due to the fact that the light intensity cannot be lower than zero.