Amplitude chimera and chimera death induced by external dynamical agents
aa r X i v : . [ n li n . AO ] J un Amplitude chimera and chimera death induced by externalagents in two-layer networks
Umesh Kumar Verma and G. Ambika Indian Institute of Science Education and Research(IISER) Tirupati, Tirupati, 517507,India (Dated: 15 June 2020)
We report the emergence of stable amplitude chimeras and chimera death in a two-layer network where one layer has an ensemble of identical nonlinear oscillators in-teracting directly through local coupling and indirectly through dynamic agents thatform the second layer. The nonlocality in the interaction among the dynamic agentsin the second layer induces different types of chimera related dynamical states inthe first layer. The amplitude chimeras developed in them are found to be extremelystable, while chimera death states are prevalent for increased coupling strengths.The results presented are for a system of coupled Stuart-Landau oscillators and canin general represent systems with short-range interactions coupled to another set ofsystems with long range interactions. In this case, by tuning the range of interactionsamong the oscillators or the coupling strength between the two types of systems, wecan control the nature of chimera states and the system can also be restored to homo-geneous steady states. The dynamic agents interacting nonlocally with long-rangeinteractions can be considered as a dynamic environment or medium interacting withthe system. We indicate how the second layer can act as a reinforcement mechanismon the first layer under various possible interactions for desirable effects.
Chimera states are emergent dynamical patterns in a network of coupled identical oscillatorswhere coherent and incoherent domains coexist. There is growing evidence that the study ofchimera states can help to understand the behavior of many real-world systems. Most of thestudies on chimera states are in single networks. Recently interactions of chimera states acrosscoupled layers in multilayer networks are reported. Such studies on multilayer networks,deal with systems where each layer has the same type of dynamics at its nodes. In this work,we study the dynamics of a two layer network where first layer has an ensemble of identicalnonlinear oscillators with local or short-range interactions, and the second has systems with adifferent nodal dynamics and nonlocal or long-range interactions among them. We considerthe second layer to be dynamic agents that can also function as a dynamic environment ininteraction with the network of systems in the first layer. We show how the nonlocality inthe interactions of the second layer can induce chimeras and control related dynamics in thefirst layer. We observe stable amplitude chimera (AC) for weak interlayer coupling, and asinterlayer strength increases, we observe chimera death (CD) and other different types ofsteady-states such homogeneous steady-state (HSS), inhomogeneous steady-state (IHSS), 2-cluster steady-state (2-CSS), and multi-cluster steady-state (MCSS).
I. INTRODUCTION
The study of complex systems using the framework of complex networks has attracted a lot ofattention in recent research in many areas . The emergent behavior in such systems due to inter-action among the dynamical units reveals a variety of interesting cooperative phenomena, such assynchronization , suppression of oscillations , chimera , amplitude chimera , chimera death ,etc. Among these, synchronization is the most widely studied one, and it broadly deals with thetransition from incoherence to coherence among coupled dynamical systems. The suppression ofoscillations observed in such systems is another emergent phenomenon, which can be classified intotwo, namely, amplitude death(AD) and oscillation death(OD) . In AD, coupled oscillators settle ata common stable steady-state, which is the fixed point of the uncoupled system, while OD, refers tothe situation where the final state is a new coupling-dependent steady state(s). In this case, coupledoscillators may settle to different steady states [termed inhomogeneous steady states (IHSS)], or toa homogeneous steady-state (HSS).The chimera state is an interesting spatiotemporal behavior where spatially coherent and in-coherent behavior of oscillators coexist in a network of coupled identical oscillators. Kuramotoand Battogtokh first observed this peculiar spatiotemporal pattern in a network of phase oscilla-tors with a simple symmetric nonlocal coupling scheme , and later this was mathematically es-tablished by Strogatz et al. . Subsequently chimeras were found in periodic oscillators , chaoticoscillators , chaotic maps , time-delay systems and neuronal systems which exhibit burstingdynamics . Initially, chimera states were reported in nonlocally coupled systems, but later itwas also found in globally coupled systems , locally coupled systems , indirectly coupledsystems , and modular networks . Besides numerical and theoretical studies, chimera patternshave also been demonstrated in laboratory experiments. In particular, chimera patterns were ob-served in an electro-optical system , mechanical systems , chemical oscillators , electrochem-ical systems , electronic circuits , and optical combs . Depending on the initial conditionsand network topology, various types of chimera states are observed on networks, such as ampli-tude mediated chimera , amplitude chimera , chimera death , globally clustered chimera ,phase-flip chimera , imperfect chimera , imperfect traveling chimera , breathing chimera etc.In addition to its established wide prevalence, chimera states are found to play an important rolein the various dynamical behaviors of many real-world systems. For example, in the case of aquaticanimals like dolphins and migratory birds, unihemispheric slow-wave sleep is a phenomenon whereonly one hemisphere of the brain shows sleep activity. The sleeping part of the brain exhibitshighly synchronized activity while awake part of the brain shows desynchronized activity . Dur-ing epileptic seizures, a part of the brain remains highly synchronized, while the remaining part isdesynchronized . Chimera states have also been linked to the various types of brain diseases suchas Alzheimer’s disease, Parkinson’s disease, schizophrenia, and brain tumors . The interplay ofsynchrony and asynchrony as displayed by chimera states plays an important role in brain func-tion and disease as reported in recent studies. Stationary moving chimeras are seen in network ofFitzHugh-Nagumo neurons with empirical structural brain network topology and simulated modularfractal topology .Most of the real world systems are not isolated but interact among themselves as well as withtheir environment or external systems. Such an environment can be modelled as a system of coupledelements where all the elements communicate to each other through dynamical agents or signallingmolecules, that can freely diffuse in the surrounding medium. Examples of such systems includegenetic oscillators , chemical oscillators , and ensemble of cold atoms , etc. There are severalstudies that are focused on the various collective dynamics possible in oscillators that interact witheach other through a dynamic environment .The study of multilayer networks is a recent topic of research that has relevance in understandingthe dynamics of several complex systems, like multilayer structures in neural networks . Wepresent the framework of multilayer networks to study the interaction between two ensembles ofsystems, of which one layer has nonlinear oscillators with local or short-range interactions, andthe other has systems of a different dynamics with nonlocal or long-range interactions. We takethe second layer to be dynamic agents that can together function as a dynamic environment ininteraction with the network of systems in the first layer. We study how the nonlocality in theinteractions of the second layer can induce chimeras and control related dynamics in the first layer,when both layers are connected in a feed back loop. Thus our model is different from most of therecently studied models where both layers have identical dynamical systems .We report how the network of locally connected identical oscillators splits into coexisting coher-ent and incoherent domains due to the influence of the environment having nonlocal interactions.For weak interlayer coupling strength, we see stable amplitude chimera (AC) and as this couplingstrength increases, chimera death (CD) and other different types of steady-states occur. Interest-ingly, emergent dynamics of the oscillators can be controlled by tuning the range of interactions inthe environment layer and we report a variety states like stable amplitude chimera, chimera death,HSS, IHSS, 2-cluster steady state(2-CSS), multi cluster steady state(MCSS), 1-state chimera death,2-state chimera death and travelling waves as possible emergent dynamical states. we note am-plitude chimera state is found to stabilize through nonlocal repulsive coupling in the presence ofattractive coupling in a system of oscillators on a regular network, even for random initial condi- FIG. 1. Schematic diagram of the two layer network where nodes in upper layer L1 (blue) represent thedynamics of oscillators and that the lower layer L2(green) describe the dynamic agents. Each oscillator in L1is connected to the corresponding dynamic agent in L2. tions . Our results are in two layer regular networks, each layer with different nodal dynamicsand the mechanism of creation of chimeras in the present study is thus due to feedback from anotherlayer that is nonlocal in connectivity.
II. INTERACTING TWO LAYER NETWORKS
The multilayer network under study consists of two layers, as shown schematically in Fig. 1. Thetop layer consists of an ensemble of N Stuart-Landau oscillators (SL), with local intralayer diffusivecoupling, called system layer, L1. They have interlayer feedback coupling with the dynamic agentsin the second layer, called L2, with multiplex like i to i coupling. The dynamic agents are 1-d overdamped oscillators with intralayer diffusive couplings that can model the presence of aninteracting environment or medium. Their dynamics is sustained due to feedback from L1 but canin turn influence the dynamics on L1 through the feedback coupling. The dynamics of the two-layernetwork thus modelled is given by˙ x i = ( − x i − y i ) x i − ω y i + σ P i + P ∑ j = i − P ( x j − x i ) + ε s i ˙ y i = ( − x i − y i ) y i + ω x i ˙ s i = − γ s i − ε x i + η P i + P ∑ j = i − P ( s j − s i ) (1)where i = , , . . . , N . x i and y i are the state variables of the i th Stuart-Landau(SL) oscillator. Theindividual SL oscillator exhibits limit cycle oscillations with natural frequency ω . The i th oscillatorinteract with other oscillators directly and indirectly through dynamic agent s i on th other layer withfeedback coupling of strength ε . The dynamics of the dynamic agents s i is considered to be one-dimensional over-damped oscillators with damping coefficient γ >
0. The interaction between theoscillators in first layer is controlled by σ and P , whereas the interaction between dynamic agents insecond layer is controlled by η and P . P and P , ∈ { , N / } , correspond to the number of nearestneighbors in each direction on each layer respectively. They thus represent the range of interactionwith the coupling radius defined by R = PN . For local coupling P =
1, for global coupling P = N and for nonlocal coupling value of P is in the range 1 < P < N /
2. By varying P and P , we canstudy the influence of nonlocality in coupling on the dynamics of first layer. Specifically we study T i m e -101 -101 -101 (b1)1 25 50 75 100-101 y cm (b2)1 25 50 75 100-101 y cm (b3)1 25 50 75 100-101 y cm T i m e -101 -101 -101 (b4)1 25 50 75 100 index(i) -101 y cm (b5)1 25 50 75 100index(i)-101 y cm index(i) -101 y cm (b6) (a1) (a2) (a3)(a4) (a5) (a6) FIG. 2. Space-time dynamics for variable y i and corresponding center of mass averaged over one period ofeach oscillator at different value of coupling strength. (a1, b1) at ε = .
4: synchronized oscillation, (a2, b2) at ε =
2: amplitude chimera, (a3, b3) at ε =
3: inhomogeneous steady state, (a4, b4) at ε = .
5: homogeneoussteady state,(a5, b5) at ε = .
15: one state chimera death(1-CD) and (a6, b6) at ε = .
5: 2-state chimera death.The other parameters are set at ω = R = . R = . σ = η = γ = N = cases where the coupling in system layer L1 is local with nonlocality in environment layer L2 andvice versa.In the model of two layer network introduced above, the dynamic agents s i can be interpreted,based on context, in many different ways. They can be particle species that can freely diffuse in thesurrounding medium and allow individual oscillators to communicate with each other. In the con-text of synthetic bacteria this dynamical agents s i can represent signalling molecules (called auto-inducers) which can freely diffuse in the local medium and in turn effect the collective dynamics ofthe cells . In the case of Belousov-Zhabotinsky (BZ) reaction, s i represents the chemical speciesthat diffuse between autocatalytic beads . Similarly, for metabolic oscillations, s i represents thecommon metabolites that diffuse between cells .In our study, we choose the initial conditions as follows. With random initial conditions and valueof the parameters of the coupled SL oscillators chosen from the inhomogeneous steady-state (IHSS)regime, we find that the coupled system is divided into two domains, one located on the upper branch ( x i , y i ) ≈ ( . , − . ) and other on the lower branch ( x i , y i ) ≈ ( − . , . ) . So we distribute theinitial states of SL oscillators around these two fixed points. The initial conditions of the first halfof the oscillators are distributed around upper branch i.e. ( x i , y i ) = ( . + ∆ξ , − . + ∆ξ ) , where i = , ..... N / ( x i , y i ) = ( − . + ∆ξ , . + ∆ξ ) , where i = N + , ..... N . The initial conditions for s i are 0 . + ∆ξ ,where i = , ..... N . The value of ∆ is 0 .
1, and ξ is a function that gives uniformly distributedrandom numbers between 0 and 1 with zero mean. Throughout the study, the number of oscillators,N is taken as 100, and the dynamics of coupled oscillators, is studied by solving Eq. 1, using fourth–order Runge–Kutta method with a time step 0 .
01. The first 10 values are discarded as transients inthe study. A. Amplitude chimeras and chimera death: L1 with local and L2 with nonlocal interactions
We first consider a case where all the SL oscillators on L1 are coupled to each other locally (i.e., R = P N = .
01) and dynamic agents on L2 are coupled to each other nonlocally with coupling -1 0 1 -1 0 1y i x i (a) -1 0 1-1 0 1y i x i (b) FIG. 3. Phase portraits of coupled SL oscillators (a) at ε = . ε = ω = R = . R = . σ = η = γ =
1, and N = radius R = P N = .
25. We study how the nonlocality or long range interactions in L2 can induceand control chimera states in L1. We fix the value of σ =
10, and η =
10, and vary the strength ofinterlayer coupling, ε .In Fig. 2, we present the space-time plots for variable y i , for the different values of ε . For a valueof ε = .
4, the dynamics on system layer L1, shows synchronized oscillations, which is shown inFig. 2 (a1). By increasing the value of ε ( ε = time steps). We also calculate the center of mass for these twodifferent values of ε using y cm = R T y i ( t ) dt / T , where T = π / ω is the oscillation period for the j th oscillator. The center of mass values are plotted corresponding to ε = . ε =
2, in Fig. 2(b1)and Fig. 2(b2), respectively. From Fig. 2(b2), it is clear that when all the oscillators are coherentin oscillations, y cm =
0, that is zero shift for the center of mass from the origin, while the systemsoscillating with the incoherent region show shift in the values of center of the mass from the origin.When the interlayer coupling strength increased, we find the dynamics in L1, settles to differentsteady states. Thus at ε =
3, L1 exhibits inhomogeneous steady-state (IHSS), as shown in Fig. 2(a3,c3), homogeneous steady-state (HSS) at ε = . ε , L1 stabilises to one state chimera death (1-CD) at ε = .
15 and two-state chimera death (2-CD)at ε = .
5, as shown in Fig. 2(a5, b5) and Fig. 2(a6, b6) respectively.The phase portraits of coupled SL oscillators for the synchronized regime at ε = .
4, and stableamplitude chimera regime at ε = B. Characterization of chimera states and their transitions
In order to characterize the nature of chimera states, we calculate the strength of incoherence (S),as introduced by Gopal, et. al. . This index will help us to distinguish chimera state from variousother collective dynamical states such as the coherent state and incoherent state and can thus beused to study dynamical transitions in the system. We start by calculating w l , i = x l , i − x l , i + , where l = , ... d represents the dimension of individual units in the ensemble, i = , , , ..., N . We dividethe oscillators into M bins of equal size n = N / M , and the local standard deviation σ ( m ) is definedas σ l ( m ) = *vuut n mn ∑ j = n ( m − )+ [ w l , j − ¯ w l , j ] + t , m = , ..., M (2)where ¯ w = n ∑ mnj = n ( m − )+ w l , j ( t ) and h· · · i t represents average over time. Now S is defined as, S = − ∑ Mm = s m M , s m = Θ ( δ − σ l ( m )) (3)where Θ ( · ) is the Heaviside step function. δ is a predefined threshold value, which is taken tobe very small, usually fixed as a certain percentage of difference between x l , i max and x l , i min . In thepresent study we take M =
20 and δ = .
2. In the incoherent domains σ l ( m ) has some finite value ε (a) 0 1 2 3 4 0 1 2 3 4 5 6 7 8 9 10 ρ ε (b) FIG. 4. (a) Strength of incoherence S plotted against interlayer coupling strength ε . S = S =
1, completely desynchronized state and 0 < S < ρ as a function of ε . ρ = ρ is 1,in figure indicates one-cluster chimera death, and ρ =
2, two-states chimera death. Here ω = R = . R = . σ = η = γ =
1, and N = (a)NSCS AC IHSS1-CD 2-CDHSS 2-CSS0 2 4 6 8 100.010.10.20.30.40.5R (b)CS AC IHSS 1-CD2-CDHSS 2-CSS NS FIG. 5. Dynamical domains of N coupled SL oscillators in the parameter plane (a) ε − R for η =
10 and σ =
10, (b) ε − η for R = .
25 and σ =
10 and (c) σ − ε for R = .
25 and η =
10. Here CS, NS, AC, IHSS,HSS 1-CD, 2-CD, and 2-CSS represent complete synchronization, no-synchronization, amplitude chimera,inhomogeneous steady state, homogeneous steady state, one-state chimera death, two-state chimera death, andtwo clusters steady-state respectively. The other parameters are ω = R = . γ =
1, and N = greater than δ , hence the value of s m =
0, while in the coherent domains σ l ( m ) is always zero,and hence s m =
1. Consequently, S takes the value S = S = < S <
1) for chimera state or cluster state. The strength of incoherence (S) is shown in Fig. 4(a) as afunction of the interlayer coupling strength ε which indicates regions of spatial synchronization,non-synchronization and chimera states in L1.Also we characterize different types of multi-chimera states, using a discontinuity measure, whichis based on the distribution of s m . It is defined as , ρ = ∑ Mi = | s i − s i + | , ( s M + = s ) (4)The value of ρ is zero for coherent or incoherent state and unity for chimera state. It takes positiveinteger value between ( < ρ ≤ M / ) for multi-chimera states. Thus for one-cluster chimera death,the value of ρ is 1, and for two-states chimera death ρ = ρ isplotted as a function of ε in Fig. 4(b). From this the region of coherent or incoherent states, onecluster chimera and two cluster chimera states can be identified clearly. µ i | i FIG. 6. Floquet multipliers | µ i | of N =
100 coupled SL oscillators indicating stability of amplitude chimerastate with ε = . R = . R = . γ = ω = η =
10, and σ =
10. Here i = , ... N . C. Phase diagram on parameter planes
We repeat the computation of the two measures, strength of incoherence, S and discontinuitymeasure ρ , for a range of values of the strength and range of nonlocal coupling in L2 and present T i m e -101 -101 -101 (b1)1 25 50 75 100-101 y cm (b2)1 25 50 75 100-101 y cm (b3)1 25 50 75 100-101 y cm T i m e -101 -101 -101 (b4)1 25 50 75 100 index(i) -101 y cm (b5)1 25 50 75 100index(i)-101 y cm index(i) -101 y cm (b6) (a1) (a2) (a3)(a4) (a5) (a6) FIG. 7. Space-time plots for variable y i and corresponding center of mass averaged over one period of eachoscillator at different value of coupling strength. (a1, b1) at ε = .
5: oscillatory state and (a2, b2) at ε = . ε = .
6: Amplitude chimera. (a4, b4) at ε =
3: multi-cluster steady state, (a5,b5) at ε =
6: homogeneous steady state and (a6, b6) ε =
8: 2-cluster steady state. We set ω = R = . R = . σ = η = γ =
1, and N = the various dynamical states possible on the two parameter phase diagram. We first fix R = . η = σ =
10 and plot the phase diagram ( ε − R ) , which is shown in Fig. 5(a). In this fig-ure, CS, NS, AC, IHSS, HSS 1-CD, 2-CD, and 2-CSS represent the complete synchronization,no-synchronization, amplitude chimera, inhomogeneous steady state, homogeneous steady state,one-state chimera death, two-state chimera death, and two clusters steady-state respectively. Fromthis figure, we see that for non-local coupling radius in layer L2, R < .
1, the layer L1 shows syn-chronized oscillations. As the strength of coupling between layers ε increases, we see transitionsto inhomogeneous steady-state (IHSS), one state chimera death (1-CD), two-state chimera Death(2-CD), and for very strong coupling strength ε there is suppression of chimera giving two-clustersteady-state (2-CSS). In the two cluster steady state, dynamics on L1 is equally divided into twodomains, one located on the upper branch, and the other is located at the lower branch. For alarger range of coupling in L2, with R > .
1, L1 shows synchronized oscillations for weak cou-pling strength ε . Increasing ε , induces in L1 a series of interesting dynamics like amplitude chimerastate (AC), inhomogeneous steady-state (IHSS), homogeneous steady-state (HSS, one state chimeradeath (1-CD), two-state chimera death and in the end, suppression of chimera to 2-CSS.We study the possible emergent states on the parameter plane ( ε − η ) for the fixed values of theparameters R = . σ =
10, and R = .
25. The corresponding phase diagram ( ε − η ) is shownin Fig. 5(b). Here we observe that a stable amplitude chimera regime arises when η >
6. Further,an increase of ε leads to an increase in the chimera death region. Thus nonlocal interactions in L2induce chimera states in L1, but higher strength of coupling or increase in range of nonlocality cansuppress chimera. We also plot the parameter plane ( σ − ε ) for the fixed values of the parameters R = . η =
10, and R = .
25 in Fig. 5(c). Here we observe stable amplitude chimera only forhiger value of σ . We can also see that 1-CD state arises when σ > D. Stability of the amplitude chimera states
We apply the Floquet theory to check the stability of amplitude chimera state. For this,wederive equations for perturbations from the chimera state starting from Eqn 1 as:˙ ξ i = a ξ i − ( ω + x ∗ i y ∗ i ) λ i + σ P i + P ∑ j = i − P ( ξ j − ξ i ) + εκ i ˙ λ i = a λ i + ( ω − x ∗ i y ∗ i ) ξ i ˙ κ i = − γκ i − εξ i + η P i + P ∑ j = i − P ( κ j − κ i ) (5)where a = ( − x ∗ i − y ∗ i ) and a = ( − x ∗ i − y ∗ i ) . x ∗ i , y ∗ i and s ∗ i are the solutions of theamplitude chimera and ξ i , λ i and κ i are the perturbations. Integrating the above equation for onetime period T = π / ω , we can construct the monodromy matrix. Then we calculate the eigenvaluesof the monodromy matrix, to get the Floquet multipliers ( µ i ) , that characterize the stability ofa periodic orbit. If all | µ i | are less then one (except for the Goldstone mode i.e. | µ | =
1) thecorresponding periodic orbit is stable. In Fig 6 we plot the values of all Floquet multipliers foramplitude chimera state. Since all values of | µ i | is less than one except | µ | =
1, it is clear that theperiodic orbits constituting the amplitude chimera are stable.
E. Suppression of chimera: L1 and L2 with nonlocal interactions
We now consider the case when R > .
01 i.e. the ensemble of SL oscillators interact directlythrough nonlocal coupling in layer L1 while the dynamic agents also interact nonlocally in the layer2. We first fix R = . R = . σ =
10 and η =
10, and plot space-time plot for differentvalues of ε (Fig. 7). For weak interlayer coupling strength at ε = . ε = .
3, we observe travelingwave(TW) dynamics in L1 (Fig. 7(a2,b2)). However further increase to ε = .
6, results in stable (a)TWCS AC MCSS HSS 2-CSS0 2 4 6 8 100.010.10.20.30.40.5R (b)MCSSIHSSCS AC2-CSS0.01 0.1 0.2 0.3 0.4 0.5R FIG. 8. Dynamical states of coupled SL oscillators in the parameter plane (a) ( ε − R ) for R = .
05 and (b) ( R − R ) for ε = ω = σ = η = γ =
1, and N = CS MCSSIHSS 2-CSS0 2 4 6 8 100.010.10.20.30.40.5R FIG. 9. Dynamical states of coupled SL oscillators in the parameter plane (a) ( ε − R ) for R = .
01 with ω = σ = η = γ =
1, and N = amplitude chimera as is clear from Fig. 7(a3,b3) and at ε = ε = ε =
8, which are shown in Fig. 7(a5,b5) and Fig. 7(a6,b6) respectively.For this coupling scenario, we plot phase diagram ( ε − R ) keeping other parameter values as ω = σ =
10 and η =
10. In Fig. 8(a) the phase diagram in the parameter space ( ε − R ) are shownfor R = .
05. It shows the regions of traveling wave (TW), stable amplitude chimera(AC) and HSSstates that arise for higher value of coupling radius R . In this case we also see multi-cluster steadystate (MCSS) and two cluster steady state (2-CSS) with increase of R . We also present a phasediagram in the parameter space ( R − R ) for ε = R . In the parameter space we also have MCSS when R > .
04. Wealso observe two clusters steady state (2-CSS) for higher value of R . From this parameter space itis clear that amplitude chimera state occurs only for small value of R . F. L1 with nonlocal and L2 with local interactions
We also consider the case where oscillators in L1 are coupled to each other nonlocally (i.e., R > .
01) while dynamical agents in L2 are coupled locally (i.e., R = . ( ε − R ) , is shown in Fig. 9. In this case, we do not see chimera states even thoughL1 has nonlocal couplings. When the value of R is small L1 shows a transition from completesynchronized state to IHSS state as the coupling strength increases, and further transition fromIHSS to 2-CSS. For an increase in the range of coupling, L1 mostly shows only 2-CSS.0 III. CONCLUSION
In summary, we present emergent behavior in a two-layer network, in which layer L1 is formedby an ensemble of identical oscillators interacting through a local coupling, and layer L2 formsanother network of dynamic agents with nonlocal coupling among them. The two layers are put ina feedback loop so that they can mutually influence their dynamics. For the specific case of coupledStuart-Landau oscillators with the limit cycle dynamics, we show how the layer L2 functioning as adynamic environment can be tuned to control the dynamics in L1.Our study indicates that the long-range interactions in L2, can induce stable amplitude chimeraand chimera death in L1, even when L1 has only short-range or local interactions. With an in-crease in coupling strength between the layers and range of interaction in L2, different types ofsteady-states such as homogeneous steady state, inhomogeneous steady-state, and two- clusters-steady-state are found to occur. In the chimera death regime, we find one state chimera death andtwo-state chimera death. We compute two quantifiers, strength of incoherence to identify occur-rence of chimera and discontinuity measure to distinguish different types of chimeras and chimeradeath states. We use them to identify regions of different emergent dynamics in phase diagrams onparameter planes. In all types of emergent behaviour, the dynamics in the layer L2 matches that oflayer L1, and both layers exhibit spatially coherence and the temporarily phase-shifted dynamics.On repeating the study for larger N values, we observe qualitatively similar results.We also present two other possibilities, where both layers have nonlocal interactions as well asthe case where oscillators in L1 interact through nonlocal coupling and are coupled to L2 that hasonly local interactions. In the former case, we observe traveling waves and stable amplitude chimerafor weak coupling strength but mostly multi-clusters steady-state (MCSS) and HSS states. In thelatter case, the system settles to 2-cluster steady-state and multi-cluster steady states, even thoughthe layer L1 has nonlocal interactions.In the limiting case of no coupling in layer L1 but nonlocal coupling in L2, we see homogeneoussteady-state and 2-cluster steady states. Similarly, with L1 having nonlocal coupling but L2 hasno coupling, only multi-cluster steady states and 2- cluster steady states are seen to occur. In bothcases, synchronized states occur for low coupling strengths.The model of interacting two layer networks presented is very generic and can be applied toa wide class of systems ranging from chemical oscillators , synthetic genetic and neuronalsystems , systems of bacteria communicate with each other through chemical species . In general,the study illustrates how dynamics in one layer can be controlled by tuning that in the other, evenwhen both have different intrinsic dynamics and different ranges of interactions. V. A. Maksimenko, V. V. Makarov, B. K. Bera, D. Ghosh, S. K. Dana, M. V. Goremyko, N. S. Frolov, A. A. Koronovskii,and A. E. Hramov, Phys. Rev. E , 052205 (2016). S. Majhi, M. Perc and D. Ghosh, Sci. Rep. , 39033 (2016). S. Majhi, M. Perc, and D. Ghosh, Chaos , 073109 (2017). J. Sawicki, I. Omelchenko, A. Zakharova, and E. Sch¨oll, Eur. Phys. J. Special Topics , 1161 (2018). D. J. Watts and S. H. Strogatz, Nature (London) , 440 (1998). A. S. Pikovsky, M. G. Rosenblum, and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences, CambridgeNonlinear Science Series (Cambridge University Press, Cambridge, UK, 2001). G. Saxena, A. Prasad, and R. Ramaswamy, Phys. Rep. , 205 (2012). A. Koseska, E. Volkov, and J. Kurths, Phys. Rev. Lett. , 024103 (2013). Y. Kuramoto and D. Battogtokh, Nonlinear Phenom. Complex Syst., , 3884 (2002). A. Koseska, M. Kapeller, and E. Sch¨oll, Phys. Rev. Lett. , 154101 (2014). T. Banerjee, EPL , 60003 (2015). D. M. Abrams and S. H. Strogatz, Phys. Rev. Lett. , 174102 (2004). S. Ulonska, I. Omelchenko, A. Zakharova, and E. Sch¨oll, Chaos , 094825 (2016). C. Gu, G. St-Yves, and J. Davidsen, Phys. Rev. Lett. , 134101 (2013). I. Omelchenko, Y. Maistrenko, P. H¨ovel, and E. Sch¨oll, Phys. Rev. Lett. , 234102 (2011). R. Gopal, V. K. Chandrasekar, A. Venkatesan, and M. Lakshmanan, Phys. Rev. E , 052914 (2014). B. K. Bera, D. Ghosh, and M. Lakshmanan, Phys. Rev. E , 012205 (2016). A. Yeldesbay, A. Pikovsky, and Michael Rosenblum, Phys. Rev. Lett. , 144103 (2014). V. K. Chandrasekar, R. Gopal, A. Venkatesan, and M. Lakshmanan, Phys. Rev. E , 062913 (2014). C. R. Laing, Phys. Rev. E , 050904(R) (2015). J. Hizanidis, N. Lazarides, and G. P. Tsironis, Phys. Rev. E , 032219 (2016). V. K. Chandrasekar, R. Gopal, D. V. Senthilkumar, and M. Lakshmanan, Phys. Rev. E , 012208 (2016). R.Gopal, V.K.Chandrasekar, D.V.Senthilkumar, A.Venkates and M.Lakshmanan, Communications in Nonlinear Scienceand Numerical Simulation, , 30 (2018). J. Hizanidis, N. E. Kouvaris, G. Zamora-L`opez, A. D´ıaz-Guilera and C. G. Antonopoulos, Sci. Rep. , 19845 (2016). M. Tinsley, S. Nkomo, and K. Showalter, Nature Phys. , 662 (2012). A. M. Hagerstrom, T. E. Murphy, R. Roy, P. H¨ovel, I. Omelchenko and E. Sch¨oll, Nature Phys., , 658 (2012). E. A. Martens, S. Thutupalli, A. Fourri`ere, and O. Hallatschek, Proc. Nat. Acad. Sci. USA , 10563 (2013). S. Nkomo, M.R. Tinsley, and K. Showalter, Phys. Rev. Lett. , 244102 (2013). M. Wickramasinghe, and I.Z. Kiss, PLoS ONE 8, e80586 (2013). M. Wickramasinghe, and I.Z. Kiss, Phys. Chem. Chem. Phys. , 18360 (2014). L.V. Gambuzza, A. Buscarino, S. Chessari, L. Fortuna, R. Meucci, M. Frasca, Phys. Rev. E , 032905 (2014). D.P. Rosin, D. Rontani, N.D. Haynes, E. Sch¨oll, D.J. Gauthier, Phys. Rev. E , 030902(R) (2014). E.A. Viktorov, T. Habruseva, S.P. Hegarty, G. Huyet, B. Kelleher, Phys. Rev. Lett. , 224101 (2014). G. C. Sethia, A. Sen, and G. L. Johnston, Phys. Rev. E , 042917 (2013). J. H. Sheeba, V. K. Chandrasekar, and M. Lakshmanan, Phys. Rev. E , 055203(R) (2009). T. Kapitaniak, P. Kuzma, J. Wojewoda, K. Czolczynski and Y. Maistrenko, Sci. Rep. , 6379 (2014). B. K. Bera, D. Ghosh, and T. Banerjee, Phys. Rev. E , 012215 (2016). T. Chouzouris, I. Omelchenko, A. Zakharova, J. Hlinka, P. Jiruska, and E. Sch¨o, Chaos , 045112 (2018). D. M. Abrams, R. Mirollo, S. H. Strogatz, and D. A. Wiley, Phys. Rev. Lett. N. C. Rottenberg, C. J. Amlaner, and S. L. Lima, Neurosci Biobehav Rev. , 817 (2000). A. Rothkegel and K. Lehnertz, New J. Phys. , 055006 (2014). P. J. Uhlhaas, and W. Singer, Neuron , 155 (2006). A. Kuznetsov, M. Krn, and N. Kopell, SIAM J. Appl. Math. , 392 (2004). R. Toth, A.F. Taylor, and M.R. Tinsley, J. Phys. Chem. B , 10170 (2006). J. Javaloyes, M. Perrin, and A. Politi, Phys. Rev. E , 011108 (2008). G. Katriel, Physica D , 2933 (2008). V. Resmi, G. Ambika, and R. E. Amritkar, Phys. Rev. E , 046212 (2011). V. Resmi, G. Ambika, R. E. Amritkar, and G. Rangarajan, Phys. Rev. E , 046211 (2012). D. Ghosh and T. Banerjee, Phys. Rev. E , 062908 (2014). P. R. Sharma, N. K. Kamal, U. K. Verma, K. Suresh, K. Thamilmaran, and M. D. Shrimali, Phys. Lett. A , 3178 (2016). A. Sharma, U. K. Verma, and M. D. Shrimali, Phys. Rev. E , 062218 (2016). U. K. Verma, N. K. Kamal, and M. D. Shrimali,
Chaos Solitons & Fractals , 55 (2018). U. K. Verma, A. Sharma, N. K. Kamal, and M. D. Shrimali, Physics Letters A , 2122 (2018). U. K. Verma, A. Sharma, N. K. Kamal, and M. D. Shrimali, Chaos , 063127 (2019). U. K.Verma, S. S. Chaurasia, and S. Sinha, Phys. Rev. E , 032203 (2019). K. Sathiyadevi, V. K. Chandrasekar, D. V. Senthilkumar, and M. Lakshmanan Phys. Rev. E , 032207 (2018). K. Sathiyadevi, V. K. Chandrasekar, D. V. Senthilkumar, and M. Lakshmanan Phys. Rev. E , 032207 (2018). J. Garcia-Ojalvo, M. B. Elowitz, and S. H. Strogatz, Proc. Natl. Acad. Sci. USA , 10955 (2004). A. Taylor, M. Tinsley, F. Wang, Z. Huang, and K. Showalter, Science , 614 (2009). M. Tinsley, A. Taylor, Z. Huang, F. Wang, and K. Showalter, Phys. D (Amsterdam, Neth.) , 785 (2010). D. J. Schwab, A. Baetica, P. Mehta, Physica D , 1782 (2012). K. Premalatha, V. K. Chandrasekar, M. Senthilvelan, and M. Lakshmanan, Phys. Rev. E , 052213 (2016). L. Tumash, A. Zakharova, J. Lehnert, W. Just, and E. Sch¨oll, Europhys. Lett. , 20001 (2017). E. Ullner, A. Zaikin, E. I. Volkov, and J. Garc´ıa-Ojalvo, Phys. Rev. Lett. , 148103 (2007). N. M. Dotson and C. M. Gray, Phys. Rev. E94