Laminar chaos in experiments and nonlinear delayed Langevin equations: A time series analysis toolbox for the detection of laminar chaos

Recently, it was shown that certain systems with large time-varying delay exhibit different types of chaos, which are related to two types of time-varying delay: conservative and dissipative delays. The known high-dimensional turbulent chaos is characterized by strong fluctuations. In contrast, the recently discovered low-dimensional laminar chaos is characterized by nearly constant laminar phases with periodic durations and a chaotic variation of the intensity from phase to phase. In this paper we extend our results from our preceding publication [Hart, Roy, Müller-Bender, Otto, and Radons, Phys. Rev. Lett. 123, 154101 (2019)], where it is demonstrated that laminar chaos is a robust phenomenon, which can be observed in experimental systems. We provide a time series analysis toolbox for the detection of robust features of laminar chaos. We benchmark our toolbox by experimental time series and time series of a model system which is described by a nonlinear Langevin equation with time-varying delay. The benchmark is done for different noise strengths for both the experimental system and the model system, where laminar chaos can be detected, even if it is hard to distinguish from turbulent chaos by a visual analysis of the trajectory.

Figure 1

Visualization of the criterion (5) for laminar chaos by a heat map of the Lyapunov exponent of the access map, which corresponds to the sinusoidal delay $\tau \left(t\right)={\tau }_0+Asin\left(2\pi t\right)/\left(2\pi \right)$. The contour lines correspond to a constant Lyapunov exponent (from bottom to top $\lambda \left[R\right]=-0.1,-0.3,-0.5,\cdots$). For Eq. (1) with $\mu =2.2$, $f\left(z\right)={sin}^2(z+{\varphi }_0)$, and ${\varphi }_0=\frac{\pi }{4}$ the criterion (5) is fulfilled above the dashed red line, which corresponds to the contour line $\lambda \left[R\right]=-\lambda \left[F\right]\approx -0.31$. The vertical lines represent regions of ${\tau }_0$ where the criterion (5) is fulfilled for a fixed delay amplitude $A=0.9$.

Figure 2

(a) Block diagram of Eq. (1). This block diagram shows the essential ingredients for laminar chaos: a feedback loop with a band-limiting element (low-pass filter), a nonlinearity, and a time-varying time delay. An additional condition is given by Eq. (5), which is visualized in Fig. 1. (b) 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 3

(a) Bifurcation diagram and (b) Lyapunov exponent $\lambda \left[F\right]$ of the map $z^{\prime }=\mu f\left(z\right)=\mu {sin}^2(z+{\varphi }_0)$, where ${\varphi }_0=\frac{\pi }{4}$, versus the bifurcation parameter $\mu$. A positive and negative Lyapunov exponent $\lambda \left[F\right]$ for a given $\mu$ is indicated by the usage of red and black color, respectively. The observations of laminar chaos presented in this paper were done with $\mu =2.2$, which is indicated by the vertical dashed lines. In this case $\lambda \left[F\right]\approx 0.31$ [horizontal dashed line in (b)].

Figure 4

Experimental time series, where $T=200$, $f\left(z\right)={sin}^2(t+{\varphi }_0)$, $\mu =2.2$, ${\varphi }_0=\frac{\pi }{4}$, and $\tau \left(t\right)={\tau }_0+0.9sin\left(2\pi t\right)/\left(2\pi \right)$ 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 (${\tau }_0=1.5$) and show laminar chaos, whereas the trajectories (d) and (e) correspond to a conservative delay (${\tau }_0=1.54$) and show turbulent chaos.

Figure 5

Trajectories generated from Eq. (11), where $T=200$, $f\left(z\right)={sin}^2(t+{\varphi }_0)$, $\mu =2.2$, ${\varphi }_0=\frac{\pi }{4}$, and $\tau \left(t\right)={\tau }_0+0.9sin\left(2\pi t\right)/\left(2\pi \right)$ for different values of the mean delay ${\tau }_0$ and the noise strength $\sigma$. The trajectories (a)–(c) correspond to a dissipative delay (${\tau }_0=1.5$) and show laminar chaos, whereas the trajectories (d) and (e) correspond to a conservative delay (${\tau }_0=1.54$) and show turbulent chaos.

Figure 6

Estimation of the delay period: (a) low frequency region of the power spectrum of the approximated derivatives ($h\approx 0.033$) of trajectories of Eq. (11), where $T=200$, $f\left(z\right)={sin}^2(t+{\varphi }_0)$, $\mu =2.2$, ${\varphi }_0=\frac{\pi }{4}$, and $\tau \left(t\right)={\tau }_0+0.9sin\left(2\pi t\right)/\left(2\pi \right)$, which show laminar chaos (${\tau }_0=1.5$, $\sigma =0.02$) and turbulent chaos (${\tau }_0=1.54$, $\sigma =0.1$). The arrow points to the peak, which corresponds to the delay period and can be easily identified. The spectra are computed by averaging 10 spectra, which are in turn computed from distinct windows of the length of ${10}^5$ delay periods. (b) Power spectra of the approximated derivatives ($h=0.001T_{\tau }$) of the experimental trajectories that are partially shown in Fig. 4, where the frequency is given in multiples of the delay frequency ${\nu }_{\tau }=T_{\tau }^{-1}$. Some of the peaks at the multiples of the delay frequency ${\nu }_{\tau }$ (vertical lines) are clearly visible. However, if the delay frequency ${\nu }_{\tau }$ is not known, longer time series are needed to distinguish the peaks at the multiples of ${\nu }_{\tau }$ from the background as it is possible for the spectra in (a).

Figure 7

Detection of the laminar phases: temporal distribution of the variance ${\sigma }_d^2$ (in units of $T$) of the approximated derivatives ($h\approx 0.0033$) of (a) laminar chaotic trajectories of Eq. (11) with the same parameters as in Fig. 5 and (b) experimental trajectories for different external noise strengths $\zeta$. ${\sigma }_d^2$ is defined by Eq. (13) and is a measure for the fluctuation strength of the trajectory at a specific time relative to the internal clock induced by the time-varying delay. The laminar phases are located in the neighborhood of the attractive fixed points of the reduced access map [dashed lines in (a)] and the burstlike transitions between the laminar phases are located at the repulsive fixed points of the access map (solid lines). In this case, the denominator of the rotation number of the access map is given by $q=2$, which leads to two laminar phases per period. The temporal distribution of the variance ${\sigma }_d^2$ in (b) for the experimental time series was shifted, such that the numerically computed local minimum that corresponds to the longest laminar phase is located at $t=0$. The dashed lines indicate the local minima, which where determined numerically.

Figure 8

(a) Reconstruction of the nonlinearity (solid black line) of Eq. (11) from a trajectory that shows laminar chaos by plotting the intensity level of the $(n+p^{\prime })\mathrm{th}$ laminar phase $z_{n+p^{\prime }}$ against the intensity level of the $n\mathrm{th}$ laminar phase $z_n$. In the case of laminar chaos, there are infinitely many positive $p^{\prime }\in \mathbb{N}$ such that the points $(z_n,z_{n+p^{\prime }})$ resemble a line. For the smallest of these numbers $p^{\prime }=p$ this line is the nonlinearity $\mu f\left(z\right)$ (plotted here). We used the same parameters as in Fig. 5. (b) Reconstruction of the nonlinearity for experimental trajectories, which where realized with external noise of different strength $\zeta$. The black line represents the fit $\widehat{\mu }\widehat{f}\left(y\right)$ of the reconstructed nonlinearity for $\zeta =0$, where $\widehat{\mu }\approx 2.229\pm 0.002$ and $\widehat{f}\left(y\right)={sin}^2(y+{\widehat{\varphi }}_0)+{\widehat{c}}_0$, where ${\widehat{\varphi }}_0\approx 0.8074\pm 0.0004$ and ${\widehat{c}}_0\approx -0.103\pm 0.002$. The error estimates represent the confidence intervals returned by NonlinearModelFit in Mathematica 11.3 (twice the standard error). The clipping below $y_n=0$ and $y_{n+p}=0$ is due to the fact that the system is an optical system, where the amplitude of the trajectories corresponds to the light intensity, which can not be lower than zero.

Figure 9

(a) Autocorrelation $C\left(\theta \right)$ of trajectories of Eq. (11) with the same parameters as in Fig. 5. The time average in Eq. (15) was taken over $10000$ delay periods. As a possible quantity for the distinction between laminar and turbulent chaos, the half of the full width at half-maximum is indicated by the vertical dashed lines. For $\sigma =0$ the autocorrelation of the laminar chaotic trajectory is approximately piecewise linear as indicated by Eq. (17) (unevenly dashed black line). Here we have $q=2$, leading to an autocorrelation function with two linear segments with nonzero slope. The values of $\theta$ where the slope changes correspond to the widths of the two laminar phases per delay period, which are obtained for the chosen set of parameters. (b) Autocorrelation $C\left(\theta \right)$ of experimental trajectories, which where realized with external noise of different strengths $\zeta$. The time average in Eq. (15) was taken over at least 5000 delay periods. Since noise is always inherent to experimental systems in the absence of external noise ($\zeta =0$), the autocorrelation $C\left(\theta \right)$ in this case also slightly deviates from the theoretical behavior given by Eq. (17) (unevenly dashed black line).

Figure 10

How to find delay parameters that correspond to laminar chaos: full width at half-maximum (FWHM) ${\theta }_{1/2}$ of the autocorrelation $C\left(\theta \right)$ of trajectories of Eq. (11), $f\left(z\right)={sin}^2(t+{\varphi }_0)$, $\mu =2.2$, ${\varphi }_0=\frac{\pi }{4}$, and $\tau \left(t\right)={\tau }_0+0.9sin\left(2\pi t\right)/\left(2\pi \right)$ under variation of ${\tau }_0$ for different noise strengths $\sigma$, with $T=200$ (solid) and $T=1000$ (dashed). For laminar chaos the FWHM is large, whereas for turbulent chaos the FWHM is small. The boundaries of the regions of ${\tau }_0$ where the criterion (5) for laminar chaos is fulfilled are represented by the vertical lines. The slight deviation between the latter and the boundaries represented by the jumps of the FWHM for $\sigma =0$ is caused by the finite $T<\infty$ and vanishes for $T\to \infty$.

Figure 11

(a) RMS error between the experimentally measured synchronization error and simulated synchronization errors with different noise strengths $\zeta$. (b) Comparison of experimental and simulated synchronization errors without noise and with $\zeta =0.001$.

Figure 12

Determination of the delay period by computing the inverse participation ratio (IPR) of the temporal distribution of variances ${\sigma }_d^2$. Here, the IPR of ${\sigma }_d^2$ in dependence of the relative deviation of the estimated delay period from the exact delay period is shown. The IPR exhibits a sharp peak at the exact delay period, whereas for larger deviations the IPR is close to one, which indicates a constant temporal distribution of variances ${\sigma }_d^2$. The peak width is bounded by the inverse of the length of the time series (in delay periods) $N^{-1}$, which is indicated by the solid vertical lines that are located at $\pm N^{-1}$, where $N=1000$.

Figure 13

Temporal distribution of the variance ${\sigma }_d^2$ (in units of $T$) of the approximated derivatives ($h\approx 0.0034$) of (a) laminar chaotic trajectories of Eq. (11) with the same parameters as in Fig. 5 and (b) experimental trajectories for different strengths $\zeta$ of the external noise. ${\sigma }_d^2$ is defined by Eq. (13) and is a measure for the fluctuation strength of the trajectory at a specific time relative to the internal clock induced by the time-varying delay. The temporal distribution of the variance ${\sigma }_d^2$ in (b) for the experimental time series was shifted, such that one of the numerically computed local minima is located at $t=0$. The dashed lines in (a) and (b) indicate the local minima, which where determined numerically after denoising ${\sigma }_d^2$ by applying a Gaussian filter with a standard deviation of $50h$.

Figure 14

Attempt to reconstruct the nonlinearity from trajectories that show turbulent chaos for different noise strength, where we considered (a) trajectories of Eq. (11) with the same parameters as in Fig. 5 and (b) experimental trajectories with the same parameters as in Fig. 4. For numerical (experimental) data, in (a) [(b)] the value $z_{n+p^{\prime }}$ ($y_{n+p^{\prime }}$) is plotted against $z_n$ ($y_n$), where in this case $p^{\prime }=3$ and the $z_n=z\left(t_n^{†}\right)$ are the values of the trajectories at the periodic sequence of local minima $t_n^{†}$ of the temporal distribution of the variance ${\sigma }_d^2$, where only the $q=2$ lowest local minima inside one delay period are considered. The points $(z_n,z_{n+p^{\prime }})$ and $(y_n,y_{n+p^{\prime }})$, respectively, do not resemble a line for all positive $p^{\prime }\in \mathbb{N}$, which leads to a clear distinction from laminar chaos in contrast to the temporal distribution of the variance ${\sigma }_d^2$. Nevertheless, for $p^{\prime }=3$ signatures of the nonlinearities $\mu f$ and $\widehat{\mu }\widehat{f}$ (black solid lines) are also visible in (a) and (b), respectively, where $\widehat{\mu }\widehat{f}$ is given by the fit to the experimental data in Fig. 8. This is due to the fact that in our example the conservative delay is close to a dissipative delay, where the access map has the rotation number $\rho =-\frac{p}{q}=-\frac{3}{2}$ (see text). For this example, an infinite number of points $(z_n,z_{n+p^{\prime }})$ or $(y_n,y_{n+p^{\prime }})$ would fill a square-shaped set.