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We study chaotic systems with multiple time delays that range over several orders of magnitude. We show that the spectrum of Lyapunov exponents (LE) in such systems possesses a hierarchical structure, with different parts scaling with the different delays. This leads to different types of chaos, depending on the scaling of the maximal LE. Our results are relevant, in particular, for the synchronization properties of hierarchical networks (networks of networks) where the nodes of subnetworks are coupled with shorter delays and couplings between different subnetworks are realized with longer delay times. Units within a subnetwork can synchronize if the maximal exponent scales with the shorter delay, long range synchronization between different subnetworks is only possible if the maximal exponent scales with the long delay. The results are illustrated analytically for Bernoulli maps and numerically for tent maps.
We use symbolic dynamics to study discrete-time dynamical systems with multiple time delays. We exploit the concept of avoiding sets, which arise from specific non-generating partitions of the phase space and restrict the occurrence of certain symbol
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 oth
We study the synchronization of chaotic units connected through time-delayed fluctuating interactions. We focus on small-world networks of Bernoulli and Logistic units with a fixed chiral backbone. Comparing the synchronization properties of static a
Networks of nonlinear units with time-delayed couplings can synchronize to a common chaotic trajectory. Although the delay time may be very large, the units can synchronize completely without time shift. For networks of coupled Bernoulli maps, analyt
We study systems of identical coupled oscillators introducing a distribution of delay times in the coupling. For arbitrary network topologies, we show that the frequency and stability of the fully synchronized states depend only on the mean of the de