For definiteness, we consider the jth oscillator in the ith layer. The oscillation frequencies and phases of oscillators were calculated as follows. Coherent part of the chimera state is formed by first 99 ( j = 1 − 99) oscillators with high-amplitude oscillations, and incoherent part is formed by oscillators with j = 101 − 199 that demonstrate alternately the low- and high-amplitude oscillations. The example of chimera state is presented in Figure 2A by black crosses (distribution of instant phases φ, average amplitudes, and average frequencies ). In our previous paper we showed that various chimera states exist in a separate layer of system (1) at the chosen parameters and obtained chimera states at the experimental system consisting of seven bistable self-exciting oscillators with linear couplings. Thus, an oscillator can be either in the regime of low-amplitude oscillations with a dimensionless frequency of 0.0039 or, in the case of initial conditions “outside” the unstable cycle, in the regime of high-amplitude oscillations with a dimensionless frequency of 0.0021. An unstable equilibrium state is located at the coordinate origin ( u = 0, v = 0). The regions of attraction of these cycles are separated by an unstable limit cycle. Two stable limit cycles with “low” and “high” amplitudes exist on the ( u, v) phase plane (see Figure 1B). If the oscillators do not interact with each other, i.e., d r = 0 d m( t) ≡ 0, then the dynamics of each oscillator is described by second-order equation. The dynamics of the network is described by the following system: Each layer of the network is a ring of locally and linearly coupled relaxational oscillators with phase portrait shown in Figure 1B. We consider a two-layer multiplex network with the topology illustrated in Figure 1A. for the case of a multiplex network with an arbitrary dimension of the layer and give a theoretical justification for the cloning effect based on the study of the fast-slow dynamics of the model using the methods of Geometric Singular Perturbation Theory (GPST). In this article, we generalize the results of Dmitrichev et al. We called this effect the chimera states cloning. Recently we presented an example of a two-layer multiplex network with seven oscillators in each layer where due to short-term interaction, one more chimera is emerged which is identical to the initial one (excluding the phase distribution of the incoherent part). Note that in all these works the interaction of chimera states led to the formation of new chimera states (in some cases with synchronous coherent parts) that are different from the pre-existing chimera state. The effects of generalized synchronization of chimera states, synchronization of chimera states in ensembles with asymmetrical connections, synchronization of chimera states in multiplex networks with delays, synchronization of chimera states in a two-layer multiplex network with adaptive connections in each layer, synchronization of chimera states in modular networks, interaction of chimera states with fully coherent or fully incoherent states, etc. At present, great attention is paid to studying of interaction of chimera states. Similar states have also been registered in the neural activity of animal brain networks. To date, the chimera states have been discovered not only in a variety of theoretical papers, but also in experimental systems of various natures, for example, mechanical, optical, chemical, and radiotechnical ones. Study of the formation of chimera states, i.e., peculiar types of hybrid states consisting of oscillators with coherent and incoherent behavior is one of the hot problems of the modern non-linear dynamics.
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