Generalized synchronization in relay systems with instantaneous coupling

We demonstrate the existence of generalized synchronization in systems that act as mediators between two dynamical units that, in turn, show complete synchronization with each other. These are the so-called relay systems. Specifically, we analyze the Lyapunov spectrum of the full system to elucidate when complete and generalized synchronization appear. We show that once a critical coupling strength is achieved, complete synchronization emerges between the systems to be synchronized, and at the same point, generalized synchronization with the relay system also arises. Next, we use two nonlinear measures based on the distance between phase-space neighbors to quantify the generalized synchronization in discretized time series. Finally, we experimentally show the robustness of the phenomenon and of the theoretical tools here proposed to characterize it.

Figure 1

(a) Relay configuration scheme of the system of Eqs. (1). (b) Synchronization error $\langle e\rangle$ (see text for definition) between systems 1 and 2 (blue line), and between systems 2 and 3 (red line) as a function of the coupling strength $\sigma$. The critical coupling ${\sigma }_c$ marks the beginning of the RS regime.

Figure 2

(a) Lyapunov spectrum as a function of $\sigma$ (only the positive, zero, and slightly negative Lyapunov exponents are shown). The gray filled areas are windows where the dynamics is periodic or only slightly chaotic (see the text for further explanations). The critical coupling ${\sigma }_c$ coincides to a very high precision with the coupling strength at which the second largest Lyapunov exponent vanishes. (b) SPP and $N$ index (see text for definition) vs $\sigma$, with system 1 as the domain set and 2 as the co-domain set for the possible mapping (blue line for SPP and green line for $N$ index) and vice versa (red line for SPP and light blue line for $N$ index).

Figure 3

Schematic representation of the experimental setup. The bidirectional coupling is adjusted by means of three digital potentiometers X9C104 (Coupler-XDCP) whose parameters $C_{\mathrm{u}/\mathrm{d}}$ (up/down resistance) and $C_{\mathrm{step}}$ (increment of the resistance at each step) are controlled by a digital signal coming from a DAQ card. See text for the full details of the experimental system.

Figure 4

(a) Synchronization error between systems 1 and 2 (blue line) and between systems 2 and 3 (red line) as a function of $\sigma$. (b) SPP and $N$ index (see text for definitions) vs $\sigma$, with the same color stipulations as in the caption of Fig. 2 (b).