Sublattice synchronization of chaotic networks with delayed couplings

Synchronization of chaotic units coupled by their time-delayed variables is investigated analytically. A type of cooperative behavior is found: Sublattice synchronization. Although the units of one sublattice are not directly coupled to each other, they completely synchronize without time delay. The chaotic trajectories of different sublattices are only weakly correlated but not related by generalized synchronization. Nevertheless, the trajectory of one sublattice is predictable from the complete trajectory of the other one. The spectra of Lyapunov exponents are calculated analytically in the limit of infinite delay times, and phase diagrams are derived for different topologies.

Figure 1

Small networks of chaotic units. Double lines signify bidirectional couplings whereas arrows show unidirectional couplings. It turns out that each ring with an even number $N$ of units has Lyapunov exponents which are identical to a chain with $N∕2+1$ units.

Figure 2

Lyapunov spectra for the different modes $q$ for $N=4$, $a=1.5$, $\tau =40$, and $\kappa =1∕4$. At $\epsilon =4∕9$ the $(q=1)$ mode becomes stable ($\to$ sublattice synchronization). At $\epsilon =2∕3$ the $(q=2)$ mode $(+1,-1,+1,-1)$ becomes stable, too ($\to$ complete synchronization). See also Fig. 3.

Figure 3

Phase diagram for $a=1.5$. See text for assignment of the different regions to complete and sublattice synchronization.

Figure 4

Cross correlation $C$ between $A_t$ and $B_{t+\Delta }$ for $a=1.5$, $\tau =40$, and $\epsilon =0.7$. (a) Complete synchronization. (b), (c) Sublattice synchronization.

Figure 5

Two trajectories of the sublattice synchronization phase for the networks of Fig. 1 with $\tau =1$, $\epsilon =0.8$, $\kappa =0$, and $a=3.0$. Obviously, there is no functional relation between the two trajectories.