Andrei Bukh

This paper considers the effects of forced and mutual synchronization of complex spatiotemporal structures in a two-layer network of nonlocally coupled logistic maps in the presence of inhomogeneous interlayer coupling. Two different types of coupling topology are considered: the first one is the sparse interlayer coupling with randomly distributed coupling defects, and the second type is the cluster interlayer coupling, providing the coupling via designated finite groups of elements. The latter type of coupling topology is considered for the first time. As a quantitative measure of the synchronization effect on the network, variance averaged over time and variance averaged both over time and network elements are used. We analyze how the synchronization measure changes depending on a degree of the interlayer coupling sparseness. We also identify a cluster of network elements which can provide almost complete synchronization in the network under study when the interlayer coupling is introduced along them.
This paper is dedicated to the memory of our teacher and scientific supervisor Prof. Vadim S. Anishchenko who passed away last November.

The features of external and mutual synchronization of spiral wave structures including chimera states in the interacting two-dimensional lattices of nonlocally coupled Nekorkin maps are investigated. The cases of diffusive and inertial couplings between the lattices are considered. The lattices model a neuronal activity and represent two-dimensional lattices consisting of $N \times N$ elements with $N = 200$. It is shown that the effect of complete synchronization is not achieved in the studied lattices, and only the regime of partial synchronization is realized regardless of the case of coupling between the lattices. It is important to note that the conclusion is applied not only to the regimes of spiral wave chimeras, but also to the regimes of regular spiral waves.

Effects of synchronization of chimera states are studied numerically in a two-layer network of nonlocally coupled nonlinear chaotic discrete-time systems. Each layer represents a ring of nonlocally coupled logistic maps in the chaotic mode. A control parameter mismatch is introduced to realize distinct spatiotemporal structures in isolated ensembles. We consider external synchronization of chimeras for unidirectional intercoupling and mutual synchronization in the case of bidirectional intercoupling. Synchronization is quantified by calculating the crosscorrelation coefficient between the symmetric elements of the interacting networks. The same quantity is used to determine finite regions of synchronization inside which the cross-correlation coefficient is equal to 1. The identity of synchronous structures and the existence of finite synchronization regions are necessary and sufficient conditions for establishing the synchronization effect. It is also shown that our results are qualitatively similar to the synchronization of periodic self-sustained oscillations.