Patterns of chaos synchronization

Small networks of chaotic units which are coupled by their time-delayed variables are investigated. In spite of the time delay, the units can synchronize isochronally, i.e., without time shift. Moreover, networks cannot only synchronize completely, but can also split into different synchronized sublattices. These synchronization patterns are stable attractors of the network dynamics. Different networks with their associated behaviors and synchronization patterns are presented. In particular, we investigate sublattice synchronization, symmetry breaking, spreading chaotic motifs, synchronization by restoring symmetry, and cooperative pairwise synchronization of a bipartite tree.

Figure 1

Two mutually coupled units.

Figure 2

Phase diagram for $\alpha =3/2$ (analytical result).

Figure 3

Ring of four units.

Figure 4

Sublattice synchronization in a triangular lattice with periodic boundaries. The double lines signify bidirectional couplings. The self-feedback is not drawn to simplify the illustration.

Figure 5

Ring of four units with two different coupling parameters ${\kappa }_1$ and ${\kappa }_2$. (a) Illustration of the setup. (b) Phase diagram for Bernoulli system with $\alpha =3/2$ and $\epsilon =11/20$ (analytical result). The regions of no synchronization are labeled with a dash $(-)$.

Figure 6

Triangle (three bidirectionally coupled units) with a unidirectionally attached infinitely large lattice. (a) Double lines signify bidirectional couplings whereas arrows show unidirectional couplings. The self-feedback is not drawn to simplify the illustration. The colors indicate the synchronization pattern (sublattice synchronization) of region III. (b) Phase diagram for Bernoulli system, $\alpha =3/2$. Analytical result combining (1) the analytical synchronization region for a ring of three units (—) and (2) the region where identical units which are driven by an identical signal synchronize (– –).

Figure 7

Symmetric (a) and asymmetric (b) chain without self-feedback and with two different time delays.

Figure 8

Setups where each unit on one side is coupled to all units of the other side. (a) There are $N$ different delay times which are pairwise identical for one unit of side $A$ and one unit of side $B$. (b) There are $N$ different shifts in the Bernoulli map which are pairwise identical for one unit of side $A$ and one unit of side $B$. The shaded dots indicate that the signals of each side of the tree are summed up and a single signal is transmitted to the other side and there distributed (second shaded dot) among the units.

Figure 9

Phase diagrams for the Lang-Kobayashi laser equations. Every circle or point represents one simulation. Open circles $(○)$ show a cross correlation $C>0.99$, which can be regarded as synchronization in the case of the laser equations. Figure (a) shows results for two mutually coupled lasers, $A\rightleftarrows B$, before applying transformation (21), whereas (b) shows the same diagram after the parameter transformation. The synchronization region of two mutually coupled units (b) can be mapped [using Eq. (8)] to a region (c) which is similar to the synchronization region of a driver-receiver system (d), $A^{\prime }\leftarrow S\to A$. After the parameter transformation (21), the synchronization regions for the Lang-Kobayashi equations, (b), (c), and (d), look very similar to the ones for the Bernoulli maps, Fig. 2.