Determining the sub-Lyapunov exponent of delay systems from time series

For delay systems the sign of the sub-Lyapunov exponent (sub-LE) determines key dynamical properties. This includes the properties of strong and weak chaos and of consistency. Here we present a robust algorithm based on reconstruction of the local linearized equations of motion, which allows for calculating the sub-LE from time series. The algorithm is inspired by a method introduced by Pyragas for a nondelayed drive-response scheme [K. Pyragas, Phys. Rev. E 56, 5183 (1997)]. In the presented extension to delay systems, the delayed feedback takes over the role of the drive, whereas the response of the low-dimensional node leads to the sub-Lyapunov exponent. Our method is based on a low-dimensional representation of the delay system. We introduce the basic algorithm for a discrete scalar map, extend the concept to scalar continuous delay systems, and give an outlook to the case of a full vector-state system, from which only a scalar observable is recorded.

Figure 1

Sub-LE of the logistic map with delayed feedback $\tau =100$ as a function of the feedback strength $\kappa$. The blue (thick gray) line shows the true exponent from linearization and the narrow black line with markers shows the approximation to the sub-LE obtained from nearest-neighbor reconstructions perfectly coinciding with true sub-LE. There are three transitions between strong and weak chaos.

Figure 2

Sub-LE of the continuous Ikeda delay system for $\tau =10$. By construction ${\lambda }_0=-1$ for all $\kappa$ as indicated by the dashed black line. The reconstruction algorithm is applied with three different sampling methods: black, $\delta t=0.4$ and one sampling point at $\theta =0.2$; blue (gray), $\delta t=0.2$ and one sampling point at $\theta =0.1$; red (dark gray) with dots, $\delta t=0.4$ and two sampling points located at ${\theta }_1=0.1,{\theta }_2=0.3$. In the steady-state regime at $\kappa <2.3$, the reconstruction is not possible. At $2.3<\kappa <2.9$ the dynamics is periodic. The subsequent chaotic dynamics is characterized by amplitudes and bandwidth increasing with $\kappa$. At low feedback the sub-LE is recovered correctly even for the coarse sampling. With increasing feedback the reconstruction error increases as well, so a proper calculation of the sub-LE requires smaller $\delta t$ or higher $K$.

Figure 3

Spectrum of sub-LEs from the Lang-Kobayashi equations as a function of the feedback strength $\kappa$ consisting of the maximum (blue), second (green), and minimum (red) exponents. Dashed lines indicate the true values calculated by a linearization. Solid lines denote the reconstructed values using nearest-neighbor reconstruction with a time step of $\delta t=10\mathrm{ps}$, one support point in the center of the delay term integral $\left({\theta }_1=5\mathrm{ps}\right)$, and 200 nearest neighbors for the regression at each point. The full vector state of a trajectory of $1\mu \mathrm{s}$ was used and another $1\text{−}\mu \mathrm{s}$ trajectory served as pool for nearest neighbors.

Figure 4

Spectrum of sub-LEs from the Lang-Kobayashi equations as a function of the feedback strength $\kappa$ consisting of the maximum (blue), second (green), and third (red) exponents. Dashed lines indicate the true values calculated by a linearization. Solid lines (both with and without markers) denote the reconstructed values using nearest-neighbor reconstruction with a time step of $\delta t=10\mathrm{ps}$. The total length of the time series was $20\mu \mathrm{s}$ or $N=2\times {10}^6$ points with splitting in $5\%$ reference and $95\%$ pool; see Appendix pp1. Only the intensity of the simulated semiconductor laser was used as a scalar observable of the trajectory underlying the calculation of the spectrum. Solid lines without markers denote the result obtained by simple embedding using three support points at $(t_n,t_{n-1},t_{n-2})$ and three support points for the first delay at $(t_n-\tau ,t_{n-1}-\tau ,t_{n-2}-\tau )$. Solid lines with markers denote the result of the extended embedding using also the past state two delay times before at $(t_n-2\tau ,t_{n-1}-2\tau ,t_{n-2}-2\tau )$.

Figure 5

Reconstruction of the sub-LE of the delayed logistic map at feedback $\kappa =0.25$ with additive Gaussian measurement noise. The true exponent ${\lambda }_0=0.1588$ is indicated by the dashed horizontal line. The horizontal axis is the effective noise strength $r_{\mathrm{eff}}$. The reconstructed exponent is shown for $J=4$ (black), $J=40$ (red with dots), and $J=400$ (blue with circles), with pool sizes $2\times {10}^3,2\times {10}^4$, and $2\times {10}^5$, respectively.

Figure 6

Convergence of the reconstructed sub-LE of the numerical Ikeda system (solid line with markers) to the true sub-LE ${\lambda }_0=-1$ (dashed line) with increasing number of nearest neighbors $J$ at feedback strength $\kappa =10$.