New type of chimera and mutual synchronization of spatiotemporal structures in two coupled ensembles of nonlocally interacting chaotic maps

Abstract

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.

Spiral Wave Patterns in Two-Layer 2D Lattices of Nonlocally Coupled Discrete Oscillators. Synchronization of Spiral Wave Chimeras

“…Of particular interest is the synchronization of chimera states in multicomponent systems and networks. Recently, various synchronization scenarios in multiplex and multilayer [23][24][25][26][27][28][29][30][31] networks were studied, such as generalized synchronization [32], inter-layer (external and mutual) synchronization [33][34][35][36][37][38], relay (remote) synchronization [39,40], explosive synchronization [41][42][43][44]. Among a large variety of chimera states [35,, here we concentrate on so-called "spiral wave chimera structures" which are observed in 2D ensembles of coupled nonlinear oscillators [10,22,[73][74][75][76][77][78][79][80][81][82][83][84][85][86].…”

Mean field phase synchronization between chimera states

“…Chimeras were studied in multilayer networks of phase oscillators, 7,[11][12][13]16,19 Hindmarsh-Rose, 7,9,10,14,18 and FitzHugh-Nagumo 27 model neurons, as well as chaotic time-discrete maps. 8,15,17 It was shown that couplings between network layers can suppress or induce [7][8][9][10][11][14][15][16][17] chimera states in individual layers and that chimeras can be identical, almost identical, or distinct across different layers. 7,8,[11][12][13][14][15][16]18 Coupling delays [8][9][10] as well as parameter mismatches or structural differences across layers 7,9,11,[15][16][17] were found to play an important role in this multilayer setting.…”

“…6 Therefore, while chimera states were traditionally studied in one-layer networks, recent work deals with interactions of chimera states across coupled layers in multilayer networks. [7][8][9][10][11][12][13][14][15][16][17][18][19] In particular, it was shown in Ref. 12 that driver-response couplings between layers can induce socalled generalized synchronization, [20][21][22] where the state of the response layer becomes a unique function of the state of the driver layer.…”

“…While these findings on generalized synchronization between chimera states 12 are appealing from a theoretical point of view, their applicability might remain limited because this particular type of synchronization is difficult to detect. Moreover, previous work [7][8][9][10][11][12][13][14][15][16][17] focused on multiplex networks, for which the layers must have the same number of oscillators and the inter-layer coupling connects individual oscillators in a pairwise and ordered topology. These constraints exclude many types of real-world multilayer networks, which in general can have different numbers of nodes in each layer, thereby already ruling out a multiplex connection topology across layers.…”

“…Furthermore, the nodes in the network of one layer might not be directly connected to the nodes in the network of the other layer but instead to some macroscopic variable of the other network. We, therefore, use a more general approach than in previous work [7][8][9][10][11][12][13][14][15][16][17][18][19] and study two-layer networks with a different number of oscillators in each layer. For each layer, we use the classical setting of a ring network of non-locally coupled identical phase oscillators in a chimera state.…”