We study numerically the dynamics of a network made of two coupled one-dimensional ensembles of discrete-time systems. The first ensemble is represented by a ring of nonlocally coupled Henon maps, and the second one -by a ring of nonlocally coupled Lozi maps. We find that the network of coupled ensembles can realize all the spatio-temporal structures which are observed both in the Henon map ensemble and in the Lozi map ensemble when uncoupled. Moreover, we reveal a new type of spatiotemporal structure, a solitary state chimera, in the considered network. We also establish and describe the effect of mutual synchronization of various complex spatiotemporal patterns in the system of two coupled ensembles of Henon and Lozi maps.PACS numbers: 05.45.-a, 02.60.-x Keywords: ensemble of nonlocally coupled oscillators, multilayer systems, chimera states, synchronization of spatiotemporal structures, synchronization region, inertial and dissipative couplingRecently studying the formation and evolution of various spatiotemporal patterns in ensembles or networks of coupled oscillators has become one of the most rapidly developing and highly attractive research topics in the nonlinear science and its applications. This exclusive interest is especially related to the discovery of a novel type of spatiotemporal structure -a chimera state. A lot of attention is paid to the dynamics of coupled ensembles of identical elements with various coupling topologies, but of particular interest are coupled ensembles with different types of network elements. In the latter case, the enrichment of regimes as well as the synchronization of spatiotemporal patterns is expected to be observed.In the present paper we analyze the spatiotemporal dynamics of a network made of two coupled rings of Henon and Lozi maps with nonlocal coupling. Our numerical studies have shown that this network can demonstrate both the spatiotemporal regimes, which are observed in separate rings, and a new type of chimera structure, called a solitary state chimera. We have also established the possibility of realizing the mutual synchronization of various complex spatiotemporal structures in the network of two coupled rings. The identity of synchronous patterns is confirmed by calculating the cross-correlation coefficient. The existence of a finite region of synchronization in the coupling parameters plane of the considered system is shown for an exemplary synchronous structure.

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We investigate solitary states and solitary state chimeras in a ring of nonlocally coupled systems represented by FitzHugh-Nagumo neurons in the oscillatory regime. We perform a systematic study of solitary states in this network. In particular, we explore the phase space structure, calculate basins of attraction, analyze the region of existence of solitary states in the system’s parameter space, and investigate how the number of solitary nodes in the network depends on the coupling parameters. We report for the first time the occurrence of solitary state chimera in networks of coupled time-continuous neural systems. Our results disclose distinctive features characteristic of solitary states in the FitzHugh-Nagumo model, such as the flat mean phase velocity profile. On the other hand, we show that the mechanism of solitary states’ formation in the FitzHugh-Nagumo model similar to chaotic maps and the Kuramoto model with inertia is related to the appearance of bistability in the system for certain values of coupling parameters. This indicates a general, probably a universal desynchronization scenario via solitary states in networks of very different nature.

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We study numerically forced synchronization of a heterogeneous multilayer network in the regime of a complex spatiotemporal pattern. Retranslating the master chimera structure in a driving layer along subsequent layers is considered, and the peculiarities of forced synchronization are studied depending on the nature and degree of heterogeneity of the network, as well as on the degree of asymmetry of the inter-layer coupling. We also analyze the possibility of synchronizing all the network layers with a given accuracy when the coupling parameters are varied.

We study relay and complete synchronization in a heterogeneous triplex network of discrete-time chaotic oscillators. A relay layer and two outer layers, which are not directly coupled but interact via the relay layer, represent rings of nonlocally coupled two-dimensional maps. We consider for the first time the case when the spatiotemporal dynamics of the relay layer is completely different from that of the outer layers. Two different configurations of the triplex network are explored: when the relay layer consists of Lozi maps while the outer layers are given by Henon maps and vice versa. Phase and amplitude chimera states are observed in the uncoupled Henon map ring, while solitary state regimes are typical for the isolated Lozi map ring. We show for the first time relay synchronization of amplitude and phase chimeras, a solitary state chimera, and solitary state regimes in the outer layers. We reveal regimes of complete synchronization for the chimera structures and solitary state modes in all the three layers. We also analyze how the synchronization effects depend on the spatiotemporal dynamics of the relay layer and construct phase diagrams in the parameter plane of inter-layer vs intra-layer coupling strength of the relay layer.

We study the dynamics of a two-dimensional lattice of nonlocally coupled-map-based neuron models represented by Rulkov maps. It is firstly shown that this discrete-time neural network can exhibit spiral and target waves and corresponding chimera states when the control parameters (the coupling strength and the coupling radius) are varied. It is demonstrated that one-core, multicore, and ring-shaped core spiral chimeras can be realized in the network. We also reveal a novel type of chimera structure—a target wave chimera. We explore the transition from spiral wave chimeras to target wave structures when varying the coupling parameters. We report for the first time that the spiral wave regime can be suppressed by applying noise excitations, and the subsequent transition to the target wave mode occurs.