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Register a new userPatterns of Chaos Synchronization
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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 can not 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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We extend the concept of generalized synchronization of chaos, a phenomenon that occurs in driven dynamical systems, to the context of autonomous spatiotemporal systems. It means a situation where the chaotic state variables in an autonomous system can be synchronized to each other but not to a coupling function defined from them. The form of the coupling function is not crucial; it may not depend on all the state variables nor it needs to be active for all times for achieving generalized synchronization. The procedure is based on the analogy between a response map subject to an external drive acting with a probability p and an autonomous system of coupled maps where a global interaction between the maps takes place with this same probability. It is shown that, under some circumstances, the conditions for stability of generalized synchronized states are equivalent in both types of systems. Our results reveal the existence of similar minimal conditions for the emergence of generalized synchronization of chaos in driven and in autonomous spatiotemporal systems.
Chaos synchronization may arise in networks of nonlinear units with delayed couplings. We study complete and sublattice synchronization generated by resonance of two large time delays with a specific ratio. As it is known for single delay networks, the number of synchronized sublattices is determined by the Greatest Common Divisor (GCD) of the network loops lengths. We demonstrate analytically the GCD condition in networks of iterated Bernouilli maps with multiple delay times and complement our analytic results by numerical phase diagrams, providing parameter regions showing complete and sublattice synchronization by resonance for Tent and Bernouilli maps. We compare networks with the same GCD with single and multiple delays, and we investigate the sensitivity of the correlation to a detuning between the delays in a network of coupled Stuart-Landau oscillators. Moreover, the GCD condition also allows to detect time delay resonances leading to high correlations in non-synchronizable networks. Specifically, GCD-induced resonances are observed both in a chaotic asymmetric network and in doubly connected rings of delay-coupled noisy linear oscillators.
The properties of functional relation between a non-invertible chaotic drive and a response map in the regime of generalized synchronization of chaos are studied. It is shown that despite a very fuzzy image of the relation between the current states of the maps, the functional relation becomes apparent when a sufficient interval of driving trajectory is taken into account. This paper develops a theoretical framework of such functional relation and illustrates the main theoretical conclusions using numerical simulations.
We show that two coupled map lattices that are mutually coupled to one another with a delay can display zero delay synchronization if they are driven by a third coupled map lattice. We analytically estimate the parametric regimes that lead to synchronization and show that the presence of mutual delays enhances synchronization to some extent. The zero delay or isochronal synchronization is reasonably robust against mismatches in the internal parameters of the coupled map lattices and we analytically estimate the synchronization error bounds.
In this article we synchronize by active control method all 19 identical Sprott systems provided in paper [10]. Particularly, we find the corresponding active controllers as well as we perform (as an example) the numerical synchronization of two Sprott-A models.
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