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Edited by: Anna Zakharova, Technische Universität Berlin, Germany

Reviewed by: Jakub Sawicki, Technische Universität Berlin, Germany; Johanne Hizanidis, University of Crete, Greece; Vadim S. Anishchenko, Saratov State University, Russia

This article was submitted to Dynamical Systems, a section of the journal Frontiers in Applied Mathematics and Statistics

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

A new phenomenon of the chimera states cloning in a large two-layer multiplex network with short-term couplings has been discovered and studied. For certain values of strength and time of multiplex interaction, in the initially disordered layer, a state of chimera is formed with the same characteristics (the same average frequency and amplitude distributions in coherent and incoherent parts, as well as an identical phase distribution in coherent part), as in the chimera which was set in the other layer. The mechanism of the chimera states cloning is examined. It is shown that the cloning is not related with synchronization, but arises from the competition of oscillations in pairs of oscillators from different layers.

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. To date, the chimera states have been discovered not only in a variety of theoretical papers [

We consider a two-layer multiplex network with the topology illustrated in

where ^{2} − ^{2})(^{2} − ^{2})(^{2} − ^{2}); the parameters controlling the dynamics of the layers are for definiteness fixed as _{r} = 0.006; _{c} > 0 are the parameters controlling the strength and the time of inter-layer (multiplex) interaction.

If the oscillators do not interact with each other, i.e., _{r} = 0; _{m}(

In our previous paper [

Then, the oscillation phase of the

and

and

where

Cloning of chimera state in system (1). Distribution of instant phases φ, average amplitudes < _{r} = 0.006, _{c} = 1, 000, and

Notice that in addition to the coherent and incoherent parts, the chimera state also contains two isolated oscillators at

Let the chimera state exist in the first layer at the initial instant of time when there is no interaction between the layers (black crosses in _{c} = 1, 000 is shown in

Next we show that the cloning effect is structurally stable. To do this, we introduce a reference layer with the index “0." Let us set in the reference layer a chimera state with the same characteristics as the original chimera that we set in the first layer. Then after the interaction of the first and the second layers we compare states formed in those layers with one in the reference layer using the following characteristics:

where < ω_{i,j} > and < _{i,j} > are averaged frequencies and amplitudes of the oscillators with the number _{0,j} > and < _{0,j} > are those in the reference layer.

The results of such calculations for _{c} = 1, 000 are presented in

Dependence of the maximum errors between the average frequencies _{r} = 0.006, and _{c} = 1, 000.

We showed above that cloning of chimera states takes place when strength of coupling between the elements of the same layer is much smaller than that between the elements of different layers (_{r} < <

where

Notice also that to realize the cloning, initial conditions in non-interacting layers must be formed in a special way. In particular, a chimera state is set in the first layer with coherent part formed by the oscillators, demonstrating high-amplitude oscillations, and incoherent part formed by the oscillators demonstrating alternately low- and high-amplitude oscillations. In the second layer all oscillators are set in the regime of low-amplitude oscillations whose phases are randomly distributed. Moreover, after interaction, the elements of the second layer should switch to the regimes the corresponding elements of the first layer were in initial moment. Thus, we need to consider the evolution of a pair only for two types of initial conditions:

(I.C.)_{1} An oscillator of the first layer is in the regime of high-amplitude oscillations, while that of the second layer is in the regime of low-amplitude oscillations;

(I.C.)_{2} The oscillators of both layers are in the regime of low-amplitude oscillations.

First, we study dynamics of a pair for the case when interaction between its oscillators is not limited by time. Then we obtain the following system of equations:

Since 0 < ε ≪ 1, the system (4) belongs to the class of fast-slow systems. Such systems are characterized by the presence of two timescales (or speeds), namely, fast and slow ones. In the result, the trajectories of the systems have epochs of a slow and a fast movements. In our system _{1} and _{2} are fast variables, while _{1} and _{2} are slow variables. Next to study the dynamics of the system we use GPST theory. According to the GPST, the partition of phase space ℝ^{4} of system (4) into trajectories can be established by studying two subsystems. As ε → 0, the trajectories of system (4) converge during fast epochs to the trajectories of the fast subsystem (or layer equations)

where

The goal of GPST is to use the fast and slow subsystems (5) and (6) to understand the dynamics of the full system (4) for 0 < ε ≪ 1.

From system (5) one can see that _{1} = _{2} =

where

Since

the trajectories of system (7) [and so system (5)], except for equilibrium states, relax to one of the stable equilibrium states. Moreover, since the system is gradient their trajectories relax to the equilibrium states by the fastest way. The number and type of equilibrium states depend on the parameters and may change due to saddle-node bifurcations. For example, for

Qualitative phase portrait of fast subsystem (7) for

The first two algebraic equations in system (6) define a critical manifold _{1} and _{2} to obtain the slow flow on

System (8) is a system of linear inhomogeneous algebraic equations for derivatives

If Δ ≠ 0, then system (8) has the only solution

Note that in the four-dimensional phase space ℝ^{4} of system (4), algebraic equation Δ ^{4} of system (4) corresponds to a submanifold, whose stability with respect to the trajectories of the fast subsystem coincides with the stability of the corresponding equilibrium state. Since the coordinates of the equilibrium states of system (5) depend on two parameters

where _{1}, _{2}) depicts submanifolds

_{1}, _{2}). The boundaries of stable submanifolds (

Since the system (4) on submanifolds

Let us study the dynamics of system (4) for initial conditions (I.C.)_{1}. Note, the dynamics of system (4) is formed by the alternating dynamics of fast and slow epochs, which results in a “stitched” trajectory.

The initial conditions (I.C.)_{1} corresponds to one of the stable submanifolds of slow motions

where _{n}. System (6), and hence system (10), have the only equilibrium point at the origin. Therefore, the trajectories starting on ^{sn} is also the equilibrium state of the fast system (5). From this condition we find

By using Equation (11) fast system (5) can be rewritten in the form

The eigenvalues of the Jacobian matrix of (12) in the point ^{sn} are given by

Because of Equation (13), the point ^{sn} is a saddle-node with an unstable separatrix and a stable nodal branch. Further we show that there are such values of the parameter ^{u}(^{sn}) tends to the stable node

Consider the level curves of the function _{1}, _{2}) = ^{sn}, where ^{sn}. In ^{sn}) are marked with different colors. Each color corresponds to the same value of

At the points of these lines the following conditions are satisfied:

Consider the asymptotic behavior of the separatrix of the saddle-node ^{sn}. Taking into account Equation (8), the location of the level curves of the function ^{u}(^{sn}) asymptotically tends to the equilibrium state ^{sn}, we increase the value of the parameter ^{*} appear (see ^{*} disappears, and following the arrangement of lines (14) and the level curves of the function ^{u}(^{sn}) tends to the node ^{*}). This corresponds to the merging of the node ^{sad} (^{*}, this equilibrium state disappears, and the separatrix ^{u}(^{sn}) tends to the equilibrium state ^{*} depends on the coordinates of the point ^{sn}.

A part of level map of the function _{1}, _{2}) taken at the saddle-node equilibrium state ^{sn} (marked by red color). White color marks region with higher levels of _{2} (respectively variable _{1}). ^{*}, ^{sad} are additional saddle-node equilibrium states and

To describe such possible transitions, we introduced the distance _{1}, _{2}) of system (9) from the origin to

^{u}(^{sn}) for different values of ^{u}(^{sn}) of any of the saddle-nodes in the fast system (5) asymptotically tends to the stable node ^{4} the equilibrium state corresponds to a stable manifold _{1}, _{2}) → (−_{1}, −_{2}). Thus, transitions from ^{4}, and high-amplitude oscillations are established in system (4). The initial conditions found by us do not exhaust the entire set of initial conditions under which the oscillation amplitude changes from low to high in the second oscillator.

The partition of _{1}, _{2}) to a next stable submanifold. The red color in

Let us study the dynamics of system (4) for initial conditions (I.C.)_{2}. These conditions correspond to one of the stable submanifolds of slow motions _{1} we have analyzed the behavior of the trajectories leaving the submanifolds.

So far, we have considered the dynamics of system (4) without any restrictions on the interaction time. However, in the initial model, the layers interact only during the time _{c}. We numerically investigated the dynamics of system (3). For the initial conditions such as (I.C.)_{1}, _{1} type, interacting during _{c} = 300 and _{2} type. Here, interaction of the pairs does not lead to high-amplitude oscillations, and the regime of low-amplitude oscillations persists.

Temporal snapshots of the pair of short-term coupled elements [system (3)] for 1,000 initial conditions such as _{1}; _{2}. Parameters values: _{c} = 300.

The occurrence of high-amplitude oscillations for the initial conditions (I.C.)_{1} depends on the values of the parameters _{c} and _{c}) shows the dependence of the probability of establishing high-amplitude oscillations. The area highlighted in black corresponds to the establishment of high-amplitude oscillations from any initial conditions. The area highlighted in shades of gray corresponds to the establishment of high-amplitude oscillations from only some initial conditions. And finally the area marked in white corresponds to those values of the parameters for which high-amplitude oscillations are not established at all. Note that there are threshold values for both parameters _{c}. The existence of threshold value for _{c} is associated with the motion time _{c} is determined by the dynamics of both oscillators and it is not related to the periods of high and low oscillations of the isolated oscillator.

Dependence of establishing probability of high-amplitude oscillations in the pair of short-term coupled elements [system (3)] for initial conditions such as (_{1} on the parameters (_{c}). Parameters values:

Thus, it has been established that in the case of initial conditions (I.C.)_{1}, there are values of the parameters _{c} corresponding to the emergence of high-amplitude oscillations in the pairs of interacting oscillators belonging to the different layers. On the other hand, it has been shown that in the case of initial conditions (I.C.)_{2}, when the oscillators belonging to the different layers do not change their initial regimes after the interaction and keep demonstrating low-amplitude oscillations.

Now let us consider the dynamics of multiplex network (1) based on the findings of the previous subsections. It can be divided into two main stages.

(a) In the time interval 0 < _{c}, oscillators of different layers interact with each other through inter-layer couplings with strengths, _{c}, greatly exceeding those of the diffusive intra-layer couplings, _{r}. Therefore, in this stage the main contribution to the dynamics of the system due to the dynamics of the interacting pairs. We have established that as a result of this dynamics, the pairs of oscillators with high-amplitude oscillations are formed in (1) from the initial conditions (I.C.)_{1}. On the other hand, the pairs of oscillators with low-amplitude oscillations are formed in (1) from the initial conditions (I.C.)_{2}. This means that the average amplitude distribution in the first layer does not change, while that of the second layer becomes the same as in the first one.

(b) For _{c}, there are no inter-layer couplings, and the oscillators interact only through diffusive intra-layer ones. Under the influence of these couplings, the neighboring oscillators with similar amplitudes become phase-locked with each other at some average frequency and form the coherent part of the chimera state. The neighboring oscillators with different amplitudes do not become phase-locked with other oscillators and form the incoherent part of the chimera state with distinguished bell-shaped distributions of average frequencies and amplitudes. Thus, the same chimera state is formed in the second layer as in the first one. Note that a finite interaction time is required to stop the competition of oscillations of pairs of oscillators. Otherwise, new complex states arise in the layers and they differ from the initial chimera.

In a large two-layer multiplex network with short-term couplings, a new phenomenon of the chimera states cloning, has been discovered and studied. Each layer of the system has a ring topology and consists of relaxation oscillators having two stable limit cycles on their phase planes. The oscillators inside the layers interact through diffusive couplings, while those of different layers interact by means of multiplex couplings. When the chimera state existing in one of the layers interacts for a while with oscillations of the other layer having a random distribution of phases, the same chimera state appears in the latter layer. Note that the time of occurrence of the chimera state in the second layer is less than the minimal partial oscillation period. We have found that the phenomenon is not related with synchronization of oscillations existing in the layers, but instead is determined by the competition of high- and low-amplitude oscillations. Using GPST, we showed that competition of oscillations in each (multiplex) pair of oscillators in the multiplex network is controlled by switching four-dimensional slow-fast dynamics. We have analytically established the initial conditions leading to the trajectories in phase space which start from stable “competitive” submanifolds of slow motions and then transit to stable “winner” submanifolds. The “competitive” submanifold corresponds to the case where oscillations in different layers have different (low and high) oscillation amplitudes. The “winner” submanifold corresponds to the case where oscillations in different layers have high amplitudes. Transitions between stable submanifolds occur along the trajectories of a two-dimensional fast subsystem. The given initial conditions belong to the basin of attraction of both the initial chimera state and the clone. We found that strength, as well as time of multiplex interaction, play a crucial role in the existence of the cloning effect of chimera states. A chimera clone is formed with 100% probability if the strength and time of multiplex interaction exceed certain threshold values. Below these threshold values a chimera clone occurs with a certain probability. Note that the effect of chimera state cloning does not depend on the choice of boundary conditions, since the dynamics of pairs of oscillators plays a crucial role in its existence. We hope also that the cloning effect is not specific to considered model and exists in other models, since the conditions necessary for it to take place are fairly general.

VN and AD conceived the original ideas. VN supervised the project. AD performed the numerical computations. VN, AD, and DS wrote the manuscript.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.