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Subject of this work is the investigation of generic synchronization phenomena in interacting complex systems. These phenomena are observed, among all, in coupled deterministic chaotic systems. At very weak interactions between individual systems a transition to a weakly coherent behavior of the systems can take place. In coupled continuous time chaotic systems this transition manifests itself with the effect of phase synchronization, in coupled chaotic discrete time systems with the effect of non-vanishing macroscopic mean field. Transition to coherence in a chain of locally coupled oscillators described with phase equations is investigated with respect to the symmetries in the system. It is shown that the reversibility of the system caused by these symmetries results to non-trivial topological properties of trajectories so that the system constructed to be dissipative reveals in a whole parameter range quasi-Hamiltonian features, i.e. the phase volume is conserved on average and Lyapunov exponents come in symmetric pairs. Transition to coherence in an ensemble of globally coupled chaotic maps is described with the loss of stability of the disordered state. The method is to break the self-consistensy of the macroscopic field and to characterize the ensemble in analogy to an amplifier circuit with feedback with a complex linear transfer function. This theory is then generalized for several cases of theoretic interest.

Transition to Coherence in Populations of Coupled Chaotic Oscillators: A Linear Response Approach
(2001)

We consider the collective dynamics in an ensemble of globally coupled chaotic maps. The transition to the coherent state with a macroscopic mean field is analyzed in the framework of the linear response theory. The linear response function for the chaotic system is obtained using the perturbation approach to the Frobenius-Perron operator. The transition point is defined from this function by virtue of the self-excitation condition for the feedback loop. Analytical results for the coupled Bernoulli maps are confirmed by the numerics.

We examine the influence of noise on the propagation of harmonic signals with two frequencies through discrete bistable media. We show that random fluctuations enhance propagation of this kind of signals for low coupling strengths, similarly to what happens with purely monochromatic signals. As a more relevant finding, we observe that the frequency being propagated with better efficiency can be selected by tuning the intensity of the noise, in such a way that for large noises the highest frequency is transmitted better than the lower one, whereas for small noises the reverse holds. Such a noise-induced frequency selection can be expected to exist for general multifrequency harmonic signals.