Synchronization of chaotic units coupled by their time delayed variables are investigated analytically. A new 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.

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.

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Two neurons coupled by unreliable synapses are modeled by leaky integrate-and-fire neurons and stochastic on-off synapses. The dynamics is mapped to an iterated function system. Numerical calculations yield a multifractal distribution of interspike intervals. The Haussdorf, entropy and correlation dimensions are calculated as a function of synaptic strength and transmission probability.

Two chaotic systems which interact by mutually exchanging a signal built from their delayed internal variables, can synchronize. A third unit may be able to record and to manipulate the exchanged signal. Can the third unit synchronize to the common chaotic trajectory, as well? If all parameters of the system are public, a proof is given that the recording system can synchronize as well. However, if the two interacting systems use private commutative filters to generate the exchanged signal, a driven system cannot synchronize. It is shown that with dynamic private filters the chaotic trajectory even cannot be calculated. Hence two way (interaction) is more than one way (drive). The implication of this general result to secret communication with chaos synchronization is discussed.