Isochronal synchrony and bidirectional communication with delay-coupled nonlinear oscillators

We propose a basic mechanism for isochronal synchrony and communication with mutually delay-coupled chaotic systems. We show that two Ikeda ring oscillators, mutually coupled with a propagation delay, synchronize isochronally when both are symmetrically driven by a third Ikeda oscillator. This synchronous operation, unstable in the two delay-coupled oscillators alone, facilitates simultaneous, bidirectional communication of messages with chaotic carrier wave forms. This approach to combine both bidirectional and unidirectional coupling represents an application of generalized synchronization using a mediating drive signal for a spatially distributed and internally synchronized multicomponent system.

Figure 1

Scheme for isochronal synchrony. IROs 1 and 2 are bidirectionally coupled, while IRO3 drives them both. In bidirectional communication, messages 1 $\left({\mathrm{m}}_1\right)$ and 2 $\left({\mathrm{m}}_2\right)$ are injected at IROs 1 and 2, respectively. ${\tau }_R$ denotes the ring round-trip time of each IRO.

Figure 2

Synchronization errors (a) $\mid E_1-E_2\mid$ between the mutually coupled oscillators and (b) $\mid E_1-E_3\mid$ between IRO1 and drive system IRO3. When only mutual coupling is initiated at time $T_c=100$, no synchronization between IROs 1 and 2 is observed. After injection of IRO3’s signal at $T_d=200$, IROs 1 and 2 synchronize isochronally, but remain different from the drive. (c) A close-up of the three wave forms after synchronization. The mean values of $\mid E_1\mid$, $\mid E_2\mid$, and $\mid E_3\mid$ are offset to 2, 0, and $-2$, respectively.

Figure 3

(a) The cross correlation ${\rho }_{12}$ between IROs 1 and 2, and (b) ${\rho }_3=({\rho }_{31}+{\rho }_{32})∕2$, the average cross correlation of IROs 1 and 2 to drive system IRO3, as a function of the drive coupling strength ${\kappa }_3$, with ${\kappa }_{12}=0.3$ constant. Isochronal synchrony occurs above the critical coupling strength ${\kappa }_3^{*}\approx 0.27$.

Figure 4

Time traces when the drive system IRO3 is (a) a parameter mismatched Ikeda oscillator (${\tau }_R=1.62$, $E_I=1.05$, $B=0.85$, $\beta =0.01$, $\alpha =5.5$, $G=0.02$, $\gamma =0.8$) and (b) a Rössler oscillator. In both cases, IROs 1 and 2 maintain isochronal synchrony while adopting qualitatively the character of the drive signal. Mean values are offset for display.

Figure 5

Ten round-trip excerpts of the recovered messages (a) $m_1\left(t\right)$ at IRO2 and (b) $m_2\left(t\right)$ at IRO1. The top trace in each figure assumes perfect parameter match between the communicating IROs, while in the middle trace parameters are mismatched by 2% ($E_I=1.02$, $B=0.82$, $\alpha =5.85$, $\gamma =0.98$ for IRO2). The bottom trace depicts message recovery when injection of the transmitted message into the transmitting cavity is not delayed.