Strong and Weak Chaos in Nonlinear Networks with Time-Delayed Couplings

We study chaotic synchronization in networks with time-delayed coupling. We introduce the notion of strong and weak chaos, distinguished by the scaling properties of the maximum Lyapunov exponent within the synchronization manifold for large delay times, and relate this to the condition for stable or unstable chaotic synchronization, respectively. In simulations of laser models and experiments with electronic circuits, we identify transitions from weak to strong and back to weak chaos upon monotonically increasing the coupling strength.

Figure 1

Time evolution of the complex electric field’s amplitude $|\mathcal{E}(t)|$ in the synchronization manifold (SM) vs time in units of the delay time $\tau =10\mathrm{ns}$ in the regime of weak chaos ($\sigma =21{\mathrm{ns}}^{-1}$).

Figure 2

(a) Maximum Lyapunov exponents ${\lambda }_m$ (solid line) and ${\lambda }_0$ (dashed line) of the SM for $\tau =10\mathrm{ns}$ vs coupling strength $\sigma$. (b) Enlarged view for small coupling strengths $\sigma$.

Figure 3

(a) ${\lambda }_m\tau$ for the SM vs delay time $\tau$ in the regime of weak chaos ($\sigma =21{\mathrm{ns}}^{-1}$). (b) $\ln (|{\lambda }_m-{\lambda }_0|/\sigma )$ for the SM vs delay time $\tau$ in the regime of strong chaos ($\sigma =12{\mathrm{ns}}^{-1}$). Parameters as in Fig. 2.

Figure 4

(a) ${\lambda }_m\tau$ for the SM vs coupling strength $\sigma$ for $\tau =100\mathrm{ns}$. (b) ${\lambda }_m\tau$ for the SM vs $\ln (\sigma /|{\lambda }_0|)$ in the regime of weak chaos near the left (gray line) and right (black line) divergences.

Figure 5

(a) Experimental setup to measure the difference between strong and weak chaos. (b) Simulated ${\lambda }_m$ (solid line) and ${\lambda }_0$ (dashed line) of the SM of the two electronic circuits and experimentally measured cross-correlation $C$ [red (gray) line] between the maxima of the time series of the two electronic circuits vs coupling strength $\sigma$.