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\title{DYNAMICS OF MULTISTABILITY STATES AND FORMATION OF CHIMERA IN MULTI-LAYERS NETWORK}

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\author{ \twlsfb Nikita Frolov
    \affiliation{
      Department of\\
      Automation, Control, Mechatronics,\\
      Saratov State Technical University,\\
      Russia\\
      phrolovns@gmail.com\\
    }
    \and \twlsfb Vladimir Maksimenko
    \affiliation{
      Department of\\
      Automation, Control, Mechatronics,\\
      Saratov State Technical University,\\
      Russia\\
      maximenkovl@gmail.com
    }
    \and \twlsfb Vladimir Makarov
    \affiliation{
      Department of\\
      Electrotechnics and Electronics,\\
      Saratov State Technical University,\\
      Russia\\
      vladmak404@gmail.com\\
    }
}

%% another two or three authors, uncomment if necessary
\author{\twlsfb Mikhail Goremyko
    \affiliation{
      Saratov State Technical University,\\
      Russia\\
      gormv67@sstu.ru
    }
    \and \twlsfb Alexei Koronovskii
    \affiliation{
      Faculty of Nonlinear Processes,\\
      Saratov State University,\\
      Russia\\
      alexey.koronovskii@gmail.com
    }
    \and \twlsfb Alexander Hramov
    \affiliation{
      Department of\\
      Automation, Control, Mechatronics,\\
      Saratov State Technical University,\\
      Russia\\
      hramovae@gmail.com
    }
}

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\maketitle


\begin{abstract}
In this work we study the conditions of chimera states excitation in multiplex network of non-locally coupled Kuramoto-Sakaguchi (KS) oscillators. In the framework of current research we analyze the dynamics of the homogeneous network containing identical oscillators. To perform the analysis we have used both numerical simulation and analytical technique, namely the Ott-Antonsen (OA) ansatz, to consider the low-dimensional behavior of KS network. We have shown that the fully identical layers, demonstrated individually different chimera due to the initial mismatch, come to the identical chimera state with the increase of inter-layer coupling.
\end{abstract}


\begin{keywords}
Multistability, complex network, Kuramoto-Sakaguchi network, chimera state, multiplexing
\end{keywords}


Nowadays, the study coupled oscillators collective behavior excites a great interest from the viewpoint of chimera states analysis \cite{1,2,3,4,5}. Such effect occurs in the networks of identical oscillators. Chimera state is a special state of complex oscillator network which represents simultaneous existence of spatial regions of coherent and incoherent subgroups. It has been firstrstly discovered by Kuramoto and Battogtokh in their work \cite{1}. This phenomena has also been found in a network of non-locally coupled nonlinear media described by the complex Ginzburg-Landau system \cite{6} and in a network of Kuaramoto-Sakaguchi phase model \cite{7}. Recent studies show that chimera-like states may arise in networks of oscillators with various coupling: global coupling \cite{8}, nearest neighbor local coupling\cite{9} too. Along with theoretical study chimera-state has been observed in experiment on chemical \cite{10}, electronic \cite{11}, electrochemical \cite{12} and opto-electronic \cite{13} systems.

However, studies of chimera states were conned so far to the domain of single-layer network. Among many other effects associated with the emergence of chimera states, important and less studied topics are the stability of chimera states \cite{14}, how different isolated networks individually be in any of the states, chimera, coherent and even incoherent, may be affected when they interact with each other? Obviously, such a situation can appear in real systems, particularly, relevant to many areas of science (e.g., neuroscience \cite{15,16}) and technology. Its consideration along with a theoretical investigation demands due attention for prospective practical use \cite{17}. In this paper we address this issue of chimera states in a framework of multilayer networks and explore the resultant effect when they interact with other. Each layer consists of nonlocally coupled identical oscillators and, separately, may be in either of the states, chimera states, coherent, incoherent states. Such multilayer network structure exists in real world and has been widely used currently both for the analysis of real data and explaining multilayer character of real-world networks \cite{18,19}. From the dynamical systems' perspective, the multilayer formulation has been applied to networks whose layers coexist or alternate in time \cite{19}. In both the cases, the multilayer formulation allowed synchronization regions that arise as a consequence of the interplay between the layers' topologies and their coupling \cite{20,21,22}, and dened new type of synchronization based on the coordination between the layers \cite{23}.

Generally, the multilayer network model is characterized by nodes that have two types of links. One type establishes intra-coupling interaction between the nodes located in the same layer. The second type determines the inter-coupling of the dynamic elements between the layers. Depending on the specic objectives of the multilayer conguration, the inter-layer relation between the elements of a network may be quite different \cite{20}. We focus on a multilayered network in which inter-layer relations match the model described in a recent work \cite{24}. Using this particular multilayer model, we consider different possible dynamical status: one layer sustains with chimera states and other layers may be in chimera states, coherent and incoherent states. We present examples of two-layered and three-layered networks and demonstrate that the original chimera states in isolated networks can be suppressed and recreated by variation of the inter-layer coupling strength. Most importantly, we reveal, that excitation of chimera state by multiplexing leads to emergence of an unknown kind of inter-layer chimera states where all layers in the network demonstrate the identical chimera pattern. To clarify the presence of the observed phenomena we have studied the continuous model using the Ott-Anttonsen approach \cite{25}, which results showed excellent match with our numerical simulations.

\section*{Acknowledgements}
This work has been supported by the Russian Foundation of Basic Research (grants 15-02-00624, 16-32-00334).

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