Dynamics, correlation scaling, and synchronization behavior in rings of delay-coupled oscillators

We study the dynamics of unidirectionally delay-coupled nonlinear oscillators. Cascading them within a ring of fixed total propagation delay, we demonstrate simple scaling behavior of correlation properties. In fact, the correlation properties of a ring with $N$ elements can be deduced from the autocorrelation of the single delayed feedback system. Coupling a ring element to a chain of unidirectionally coupled identical oscillators, we achieve complete synchronization between elements of chain and ring, evidencing generalized synchronization among the other elements, even if uncorrelated.

Figure 1

(Color online) (a) Oscillator with delayed feedback. (b) Two bidirectionally delay-coupled oscillators. (c) A ring configuration of $N$ unidirectionally delay-coupled oscillators which can optionally be coupled to a chain of oscillators.

Figure 2

Time traces (left panel), power spectra (middle panel), and normalized intensity autocorrelation functions (AC, right panel) of the emission of one laser for different values of $N$. (a)–(c) correspond to a SL with delayed optical feedback, (d)–(f) to two bidirectionally coupled SLs, (g)–(i) to 4 SLs in a ring configuration, and (j)–(l) to 100 SL in a ring configuration.

Figure 3

AC peak (gray squares) and its envelope (black crosses) around the delay time vs the number of elements in the ring for (a) semiconductor lasers and (b) Ikeda oscillators. An exponential fit, $\alpha \exp [-\beta (N-1\left)\right]$, has been calculated of the envelope (solid black lines). Black circles indicate the maximum of the envelope of the AC peak at $t=M\tau$ $(t=MT)$ of one laser (Ikeda) with delayed feedback. The gray dashed lines are guides to the eyes.

Figure 4

Maximum value of the cross correlation $\left({\mathrm{CC}}_{\max }\right)$ between the sender (SL $N∕2$) and the transmitter (SL 0) (black squares) and between the sender and receiver (SL $N∕2^{\prime }$) (gray circles) vs the number $N$. The solid black line is an exponential fit using $\alpha \exp [-\beta (N-1\left)\right]$. The gray line is a guide to the eyes.