Emergent rhythms in coupled nonlinear oscillators due to dynamic interactions
Shiva Dixit, Sayantan Nag Chowdhury, Awadhesh Prasad, Dibakar Ghosh, Manish Dev Shrimali
EEmergent rhythms in coupled nonlinear oscillators due to dynamicinteractions
Shiva Dixit, Sayantan Nag Chowdhury, Awadhesh Prasad, Dibakar Ghosh, a) and Manish Dev Shrimali b) Department of Physics, Central University of Rajasthan, NH-8,Bandar Sindri, Ajmer 305 817,India Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata-700108,India Department of Physics and Astrophysics, University of Delhi, Delhi 110007, India (Dated: 12 January 2021)
The role of a new form of dynamic interaction is explored in a network of generic identical oscillators. The proposed de-sign of dynamic coupling facilitates the onset of a plethora of asymptotic states including synchronous states, amplitudedeath states, oscillation death states, a mixed state (complete synchronized cluster and small amplitude desynchronizeddomain), and bistable states (coexistence of two attractors). The dynamical transitions from the oscillatory to deathstate are characterized using an average temporal interaction approximation, which agrees with the numerical resultsin temporal interaction. A first order phase transition behavior may change into a second order transition in spatialdynamic interaction solely depending on the choice of initial conditions in the bistable regime. However, this possibleabrupt first order like transition is completely non-existent in the case of temporal dynamic interaction. Besides thestudy on periodic Stuart-Landau systems, we present results for paradigmatic chaotic model of Rössler oscillators andMac-arthur ecological model.
Population biology of ecological networks, person to per-son communication networks, brain functional networks,possibility of outbreaks and spreading of disease throughhuman contact networks, to name but a few exampleswhich attest to the importance of researches based on tem-poral interaction approach. Studies based on representat-ing several complex systems as time-varying networks ofdynamical units have been shown to be extremely benefi-cial in understanding real life processes. Surprisingly, inall the previous studies on time-varying interaction, deathstate receives little attention in a network of coupled os-cillators. In addition, only a few studies on dynamic in-teraction have considered the proximity of the individualsystems’ trajectories in the context of their interaction. Inthis paper, we propose a simple yet effective dynamic inter-action scheme among nonlinear oscillators, which is capa-ble of relaxing the collective oscillatory dynamics towardsthe dynamical equilibrium under appropriate choices ofparameters. The dynamics of coupled oscillators can showfascinating complex behaviors including various dynami-cal phenomena. A qualitative explanation of the numeri-cal observation is validated through linear stability analy-sis and interestingly, a linear stability analysis is persuedeven when the system is time-dependent. An elaboratestudy is contemplated to reveal the influences of our pro-posed dynamic interaction in terms of all the network pa-rameters. a) Electronic mail: [email protected] b) Electronic mail: [email protected]
I. INTRODUCTION
Time-varying interaction enjoys a widespread recognitionamong researchers due to its numerous practical applications.Various interdisciplinary research approaches , from boththeoretical and experimental points of view, offer fresh newinsights about the collective phenomena due to time-varyinginteraction. Recently, dynamical systems are found to be anefficient and prominent tool, which open the door to study therole of dynamic interactions in a broad variety of complex sys-tems. The interactions among dynamical systems can give riseto fascinating collective behavior ranging from synchroniza-tion , extreme events , chimera states , suppression ofoscillations to revival of oscillations and many more.Interestingly, most of the previous investigations among dy-namical units are confined within the regime of static inter-actions. Contrary to this, in the present article, we bring thenotion of dynamic interaction on collective behavior of cou-pled nonlinear dynamical systems.The relevance of dynamic interaction has been recognizedalready by considering few general frameworks on the inter-acting nonlinear oscillators, where either the interacting func-tion is changing over time , or the interaction dependson the states of the individual oscillators . In this arti-cle, we consider a new form of dynamic mean-field interac-tion with two distinct possible variations. One of these im-plemented coupling configuration is that individual oscillatorsare interacting with mean-field coupling form for a pre-specified certain time period and they remain isolated for theremaining time window. Another possibility is to introducethe dynamic interaction through the scenario, where individ-ual oscillators are interacting only when the mean state of theoscillators lies within a certain vicinity of the phase space.This type of modulated interaction is quite common in roboticcommunication as well as in wireless communication sys-tems, where transmission is only activated within a particu- a r X i v : . [ n li n . AO ] J a n lar region of the physical space or for a particular specifiedtime . Instead of static (time-independent) coupling for-malism, most of the realistic systems including physical, bio-logical and social networks possess time-varying connectivity.Our imposed restricted interaction produces unanticipated dy-namical states, that could not be expected if the interactionamong those oscillators is possible in the entire time-domain,or in the whole state space. In fact, there are some real in-stances, where it is not possible to have a continuous interac-tion for all the time and in the entire state-space due to thepractical limitations .Motivated by these facts, we try to capture the essence ofrealistic cases through the paradigm of dynamic interaction incoupled nonlinear oscillators. The remaining part of this pa-per is organized as follows. In Sec. II, we discuss the proposedmechanism of dynamic mean-field interaction in the couplednonlinear oscillators in detail, where the oscillator’s motionaffects the network topology. This is followed by the de-tailed numerical investigations that are carried out for severalsystems including Stuart–Landau system (limit-cycle oscil-lator), Rössler system (chaotic oscillator) and Mac-Arthursystem (ecological system). The generic transitions fromthe oscillatory to steady state are also validated using linearstability analysis in Sec. III. Lastly, we conclude and summa-rize our results in Sec. IV. II. MATHEMATICAL MODEL
The time-evolution of each i -th oscillator ( i = , , · · · , N ) can be described by the following set of equations ,˙ X i = F ( X i ) + εβ ( H X − X i ) , (1)where X i represents state variables of the m -dimensional i -th oscillator and F : R m → R m reflects the intrinsic dynamicsof each node of the network (1). The term X = N ∑ Nj = X j gives the arithmetic mean of the state variables and N ( ≥ ) is the number of independent non-linear dynamical systems.The control parameter ε , reckoning as the strength of the in-teraction among those oscillators, is taken to be identical forall oscillators. β is an m × m diagonal matrix with diagonalentries β kk = , if the k -th component of the oscillatortakes part in the mean-field coupling β kk = , otherwise.Our proposed dynamic interaction is determined by the stepfunction H , which is considered to be a function of mean-fieldterm X and time t . The range of H consists only two values 0and 1. This function H helps to treat our dynamic interactionpolicy as an on–off type of dynamic interaction , where H = H = , the degreeof each node is either 0 (when H =
0) or N − H = H in twodifferent ways. A. Spatial Dynamic Interaction (SDI)
The state-space dependent interaction function H = H ( X , t ) is defined as H ( X , t ) = (cid:40) , if X ∈ R (cid:48) , if X / ∈ R (cid:48) (2)where R (cid:48) ⊆ R m is a subset of the state-space R m , where in-teraction is active. Here, the subset R (cid:48) can be defined in termof ∆ (cid:48) , which is written in the normalized form as ∆ = ∆ (cid:48) / ∆ a ,where ∆ a is the width of the attractor along the clipping direc-tion .Control parameter ∆ plays a decisive role by turning on theinteraction among the dynamical units whenever they are in-side a pre-specified subspace R (cid:48) of the phase space. Specifi-cally, where time-independent and time-varying diffusive in-teractions do not lead to stabilize the unstable stationary pointof the uncoupled system in general, clipping in an inter-val through mean-field diffusive coupling is found to be ben-eficial, which can stabilize the unstable stationary points ofthe isolated systems. Here, for ∆ → + , there is no sufficientinteraction between the units and as a result of our proposednetwork (1), only self-negative feedback is activated. Whilefor ∆ → − , all oscillators are globally coupled with mean-field interaction. B. Temporal Dynamic Interaction (TDI)
On the other hand, we also consider the time-varying func-tion H = H ( X , t ) as a periodic step function of period T ,which is defined as H ( X , t ) = (cid:40) , if 0 < t ≤ τ (cid:48) , if τ (cid:48) < t ≤ T (3)where, τ = τ (cid:48) / T ∈ [ , ] , τ (cid:48) is an active interaction time win-dow and T (= π / ω ) is an average time period of the oscilla-tions of uncoupled system . Here, interaction is active for τ (cid:48) period of time, while it is inactive during the time T − τ (cid:48) asdefined in Eq. (3).Here, τ is an active interaction time-period, where the inter-action switches periodically between the mean-field diffusiveinteraction and self-negative feedback. If the time period ofthe system is T , then the mean-field interaction is activatedfor fraction τ of the cycle, which is followed by self neg-ative feedback for the remaining time window ( T − τ ) . Sowhen τ =
0, the oscillators are always under the effect of thenegative self-feedback and when τ =
1, the oscillators are al-ways coupled through mean-field diffusive interaction for allthe time. For 0 < τ <
1, the oscillators experience interactionthrough mean-field diffusive interaction for time τ T and neg-ative self-feedback for time T ( − τ ) in each cycle of period T .Hence, ∆ and τ are two crucial control parameters of ourmodel. In the following sections, our key interest will be toidentify the emergent collective phenomena due to the inter-play of different parameters ∆ , τ and the coupling strength ε for fixed N number of non-linear oscillators. All these pa-rameters play an important role in order to obtain differentdynamical states. The numerical simulations are done usingthe Runge-Kutta fourth-order (RK4) method for a time of 10 units with a fixed integration time dt = .
01 after removingenough transients ( ∼ units). III. RESULTS FOR LIMIT CYCLE SYSTEM: COUPLEDSTUART-LANDAU OSCILLATOR
We consider N identical Stuart Landau (SL) oscillators z j , j = , , · · · , N coupled through the spatio-temporal dynamicinteraction. The dynamical equations of the coupled systemare given as˙ z j = ( ρ + i ω − | z j | ) z j + ε ( Hz − Re ( z j )) . (4)Here, j = , , · · · , N , where N is the total number of os-cillators in the dynamical network. z j = x j + iy j ∈ C isthe state variable of the j -th oscillator with i = √− z = N ∑ Nk = Re ( z k ) = N ∑ Nk = x k is the mean field term of the cou-pled system. The parameter ρ ∈ R acts like a bifurcation pa-rameter and a Hopf bifurcation occurs at ρ =
0. For ρ ≤ j -th oscillator stabilizes into the triv-ial stationary point z j =
0, and for ρ >
0, single SL oscillatorexhibits a stable periodic attractor with radius √ ρ and eigenfrequency ω . The function H is considered as a function of Re ( z k ) , ( k = , , · · · , N ) and time t . We investigate the col-lective behavior of the coupled SL oscillators with dynamicspatial and temporal interaction by varying control parame-ters such as ρ , ω , ∆ , τ and ε . For the numerical simula-tions, initial conditions are chosen randomly from the inter-val [ − , ] × [ − , ] . Without loss of any generality, ρ = ω = A. Spatial dynamic interaction (SDI):
To illustrate the effect of SDI, two distinct Figs. 1 (a)and 1 (b) are presented for fixed coupling strength ε = . ∆ = . ∆ = .
8, respectively. Here, minima andmaxima of the attractor (with ρ =
1) are − ∆ a = ∆ assigns the active region [ global minima of the attractor , global minima of the attractor + ∆ a × ∆ ] as per our implementation. Hence, ∆ = . ( H = ) is [ − , − . ] .Similarly, ∆ = . [ − , . ] . Clearly, the dynamic couplingis active only within the gray-shaded region and it is turnedoff ( H = ) outside of that region in Figs. 1 (a) and 1 (b).Note that, a key difference is observed between ∆ = . ∆ = .
8. For ∆ = .
4, the coupling-activated subspace does -101-1 -0.5 0 0.5 1 (a) -101-1 -0.5 0 0.5 1
USP USP (d) t (x-xi) (-xi) (x-xi) T (-xi) τ t y x x -1010 1 2 3 4 5 -1010 1 2 3 4 5 T τ x (b)(c) FIG. 1.
Transient trajectories : The dynamics of the first oscillatorof the coupled SL oscillator is shown here without loss of any gen-erality. The interaction is activated in the shaded region (gray) (a) for ∆ = .
4, and (b) for ∆ = . (c) τ = .
4, and (d) τ = . τ period of time while absentfor the rest of the time-period. For all the subfigures, N =
100 and ε = . not contain the origin, the unstable stationary point of theisolated SL oscillator. For this specific choice of ∆ = . ( x , y ) .One is from the interaction active region [ − , − . ] and theother one is from ( − . , ] . The initial choice of ( x , y ) fromthe coupling active region gives rise to a limit cycle (solidorange), whereas the initial condition for ( x , y ) from theinactive region leads to cessation of oscillations (dashed blue)under the influence of only negative self-feedback and thesystem consequently converges to the origin. This bistablebehavior is completely vanished for ∆ = .
8. For ∆ = .
8, thetrajectories always settle down to sinusoidal like oscillationsirrespective of the choice of initial conditions belonging to [ − , ] × [ − , ] .To explore the effect of initial conditions, we plot the basinof attraction with respect to the variables x - x in Fig. 2 (a) bykeeping fixed the values of the other variables y ( ) = . y ( ) = .
0. For simplicity, only N = ∆ = . ε = .
1, the choice of ( x ( ) , y ( )) from the inac-tive region ( − . , . ] always leads to AD, but that is not nec-essarily reflect the true story. Definitely, the stabilization ofAD state depends not only on the initial condition of any sin-gle oscillator. Other variable’s initial conditions as well as thevelocity fields near the interaction switching on-off region arealso equally important to stabilize the death state. For the par-ticular choice of y ( ) = . y ( ) = .
0, the basin of thecoexisting attractors reveals that suppression of oscillations toAD is possible if z ( ) = x ( ) + x ( ) ∈ [ − . , . ] . How-ever, this bistable behavior is completely suppressed with in-creasing coupling strength ε . An example is depicted in Fig. FIG. 2. (a)
Effect of initial conditions : Here, ∆ = . ε = . N = y ( ) = . y ( ) = . x i ( ) are varied uniformly from − .
02 for i = ,
2. For z ( ) = x ( ) + x ( ) ∈ [ − . , . ] (approximately), the only stable attractor is the am-plitude death (AD) state (blue). Beyond this region, synchronizedlimit cycles (CS) are found. (b) Time–series near the transitionpoint for two coupled SL oscillators : Here, the initial conditionis ( x ( ) , y ( ) , x ( ) , y ( )) = ( − . , . , − . , . ) and ∆ = . ε from 2 . .
6, boththe oscillators settle down to a common stable death state from thesynchronized oscillatory behavior. N = ε = .
5, but the oscillatory solu-tions lose their stability for ε = . N =
100 coupled SLoscillators, we analyze the interplay of the parameters ε and ∆ in Fig. 3 (a). We find here that there exists a critical value ∆ ∗ ≈ .
49 that designates two major transition scenarios. For ∆ > .
49, the coupled SL oscillators are oscillating coherentlyat very small non-zero coupling strength ε . With increment of ε , a transition takes place from the synchronized state (CS) to oscillation death (OD) for ∆ > .
49. Even after a cer-tain threshold of ε , the coexistence of oscillatory states andOD states are observed in the parameter plane ε − ∆ , which isshown as BS ∆ < .
49, the suppres-sion of oscillation from the oscillatory state is also perceivedfor ε ≥ .
0. The Jacobian matrix of the coupled systems (4)with H = ( O , O , · · · , O (cid:124) (cid:123)(cid:122) (cid:125) N times ) , where O = ( , ) is the unstable stationary point of the SL oscilla-tor, is given by the block diagonal matrix A ⊕ A ⊕ A · · · ⊕ A ( N times). Here, A is the Jacobian matrix of the isolated sys-tem with only negative self-feedback at O = ( , ) . The eigenvalues of A are λ , = ρ − ε ± (cid:112) ( ε − ω )( ε + ω ) . (5)This eigenvalue analysis suggests that amplitude death(AD) is impossible for ω ≤
1. For any value of ω > λ , to zero, we get ε HB = ρ . Set-ting λ , =
0, we find ε PB = ρ + ω ρ , where ρ (cid:54) =
0. Thesebifurcation points ε HB = ε PB = ρ = ω = FIG. 3. (a)
Diverse emergent dynamical behaviors of N = coupled SL oscillators in the parameter plane ε - ∆ : Numericalsimulation of the asymptotic behavior of the system (4) gives riseto several dynamical states including NS (desynchronized oscilla-tory state), CS (complete synchronized oscillatory state), AD (am-plitude death state), OD (oscillation death state), MX (mixed state,where desynchronized small oscillations and a group of completelysynchronized large oscillations co-exist), BS1 (bistable regime withcomplete synchronized state and amplitude death state) and BS2(where complete synchronized state and oscillation death state co-exist). (b) Bifurcation diagram of the system (1) with respect tothe coupling strength ε for ∆ = .
45: Shaded region is the BS1state. Here, solid line corresponds to the stable behavior and dashedline represents unstability of the origin. HB and PB stand for Hopfbifurcation and pitchfork bifurcation, respectively. (c)-(d)
Tempo-ral evolution of x i for i = , , ··· , N : Time traces are shown in (c)for ε = . ε = .
1. Complete synchronization (blue)or partial synchronization (red) is observed in the subfigure (c) de-pending on the suitable initial conditions. Similarly, depending uponthe initial states, completely synchronized limit cycle (blue) or ADstate (red) is appeared in the subfigure (d). The other parameters are ρ = , ω =
2, and N = fit exactly with our numerically obtained bifurcation diagramgiven in Fig. 3 (b). The dashed line in Fig. 3 (a) correspondsto the choice of ∆ = .
45 at which the bifurcation diagram(Fig. 3 (b)) is scrutinized. An inverse Hopf bifurcation occursat ε HB = ε PB = ∆ < . ∆ < .
49, the coupling activated re-gion does not contain the unstable stationary point O , and thusit is highly probable that the only coupling remains active for ∆ < .
49 is negative self-feedback. However, a bi-stable re-gion BS x i is shown in Figs. 3 (c) and 3(d) at ε = . ε = . ∆ = .
45, respectively. Figure3 (c) depicts incoherent nature of the trajectories (red curves),where each SL oscillator exhibits stable periodic orbit, but of
FIG. 4. (a)
Map of dynamic regimes for N = coupled SLoscillators in the parameter plane τ − ε : The regimes marked asNS, CS, AD, OD and BS2 representing the desynchronized oscilla-tory state, complete synchronized oscillatory state, amplitude deathstate, oscillation death state, and bistable state (oscillatory state andoscillation death state), respectively. The solid black line is the ana-lytically derived relation (8). (b) Bifurcation diagram of N = coupled SL oscillators with respect to coupling strength ε for τ = .
45: HB and PB are Hopf bifurcation and pitchfork bifurca-tion point, respectively. The coupled system is first stabilized at theorigin through inverse Hopf bifurcation and subsequently, OD statesare born through a pitchfork bifurcation. For details, please see thetext. various radii (amplitudes). Depending on the suitable initialconditions, N =
100 trajectories may collapsed into a singletrajectory (blue curve) as shown in Fig. 3 (c). This region ofmixed state (MX) is highlighted in Fig. 3 (a). The resultingtime series for ε = . BS B. Temporal dynamic interaction (TDI):
The continuous interaction is not always existent and man-ageable in many real systems, such as the transmissions ofbiological signals between synapses and the communicationsof ant colonies in the processing of migration, as well as theseasonal interactions between predator–prey in the ecosys-tem, leading to the discontinuous and intermittent couplingrelationship . Therefore, it is of essential importance to in-vestigate the oscillation patterns in the coupled system con-taining temporal discontinuous coupling. Figures 1 (c) and 1(d) represent the effect of our considered time-varying interac-tion through TDI. Figure 1 (c) exhibits that the interaction re-mains constant in one part of the period where τ = .
4, and forthe remaining part ( − τ ) = .
6, the interaction disappears. InFig. 1 (d), the same process is repeated for τ = .
8. Here, wehave taken T = . H = H ( t ) is taken into consideration as definedin Eq. (3). Here, H = H ( t ) depends on the interaction activetime τ and time period T of the system on the network (1)of SL oscillators. Here, we consider smaller values of T ascompared to the oscillation time period ( T SL ∼ .
26) of un-coupled oscillator . The results are shown here for T = . ε − τ for coupled SL oscillator in Fig. 4 (a). A transition is witnessed from inco-herent state (NS) to synchronized state (CS) with increasingcoupling strength ε . With further increment of ε , either ODstates or coexistence of oscillatory state and OD state is founddepending on the value of τ . The bistable region is marked asBS2 in Fig. 4 (a). To further understand the scenario, τ = . τ in Fig. 4 (b).Increment of ε reveals the suppression of stable limit cycleand AD appears at ε = τ =
0, through inverse Hopf bi-furcation as shown in Fig. 4 (a). While at ε = h ( t ) ) in Eq. (4) is time dependent, weconsider an average eigenvalue λ = [ τ (cid:48) λ on +( T − τ (cid:48) ) λ of f ] T , where λ on and λ o f f are the numerically largest eigenvalues of thestability matrix at the stationary point over the period τ (cid:48) and T − τ (cid:48) , respectively . Therefore, the linear stability analy-sis at a stationary point (zero in Eq. (4)) provides nontrivialcharacteristic equations, λ = ρτ + ( − τ ) (cid:32) ρ − ε (cid:33) . (6)Letting λ = α + i γ , where α and γ are real and imaginary partof the eigenvalues, Eq. (6) leads to α = ρ − ε + ετ . (7)The solid black line in Fig. 4, corresponds to the locus ( α = τ = − ρε , ε (cid:54) = . (8)Analytical condition (solid black line) of Eq. (8) matches per-fectly with the numerically calculated amplitude death (AD)region in Fig. 4 (a). Here the dynamics changes from peri-odic attractor to AD via Hopf bifurcation as a real part of theeigenvalue α becomes negative. C. Average interaction time in spatial dynamic interaction
In addition, we have explored the relation between the aver-age interaction time in SDI. It is obvious that whenever thereis a discontinuous interaction in space, the temporal discon-tinuity must accompany it. In SDI, the interaction term isswitched on or off depending on the mean state of the trajec-tories in the phase space, but in TDI, the on-off factor appearsin a completely periodic manner. Here, we try to provide acorrelation between the average interaction time in the spatialscheme ( τ avg ( ∆ ) ) and the active interaction time in temporalframework ( τ ).The order parameter A , which is the normalized averageamplitude is now defined as A = a ( ε ) a ( ) , (9) ε (a)(b) τ a v g ( Δ ) A ( ε ) A ( ε ) τ a v g ( Δ ) , , τ τ ,, FIG. 5.
Average interaction time and average amplitude due toSDI : The τ avg ( ∆ ) (orange), A (blue) and Eq. (8) (black solid line)are plotted as a function of coupling strength ε at (a) ∆ = .
35 and (b) ∆ = .
5, respectively. Gray shaded region is the bistable region.Surprisingly, SDI yields both first-order and second-order transitionsfrom oscillatory state to death state as observed in subfigure (a) de-pending on the initial conditions as well as the velocity fields near theinteraction switching on-off region for ∆ = .
35. Although, that dis-continuous and abrupt transition is completely vanished for ∆ ≥ . ∆ = .
5. For details, please see the text. where a ( ε ) = ∑ Ni = ( (cid:104) x i , max (cid:105) t − (cid:104) x i , min (cid:105) t ) N . Here, a ( ε ) denotesthe difference between the global maximum and minimumvalues of the attractor at a particular value of the couplingstrength ε and (cid:104)· · · (cid:105) t indicates the sufficiently long time aver-age. Thus, A measures the average amplitude of the oscillatorsin the coupled system and for an oscillatory state, the value of A will be greater than zero, while for a death state A = .We calculate the average on time τ avg ( ∆ ) in space for a chosenregion ( ∆ ) over large number of initial conditions ( ≈ ) .In Fig. 5, τ avg ( ∆ ) (orange), average amplitude A (blue) for agiven ∆ and the critical curve (8) (black) separating the steadystate and oscillatory regions in temporal interaction are plot-ted as a function of coupling strength ε . As depicted in Fig. 5(a), we find that at ε = ∆ = τ avg ( ∆ ) = .
35. While withincreasing ε , the value of τ avg ( ∆ ) eventually decreases andat a critical coupling strength ε c ( ∆ ) , the τ avg ( ∆ ) reaches to0. The τ avg ( ∆ ) crosses the analytical curve (Eq. (8)) of thetemporal interaction at ε (cid:117) .
23, and after that, system com-pletely ceases down to steady state. Beyond ε =
2, there isa gray shaded region of bistability in Fig. 5 (a), where boththe stationary point attractor and limit cycle coexist. In thisbistable region, where two behaviors exist side-by-side overa parameter region, a first order phase-transition to AD stateis also uncovered in Fig. 5 (a) through an abrupt transition of A . The justification behind this discontinuous jump is due tothe bistable behavior of the system as shown in Figs. 2 (a) and3 (a). The traditional continuous transition is feasible basedon the suitable choices of initial conditions as shown in Fig. 5(a). But, there still exists a suitable set of initial conditions asshown in Figs. 2 (a) and 3 (a), for which the system may stillexhibit oscillation with small amplitude beyond the couplingstrength ε HB =
2. However, with increasing coupling strength ε beyond the critical value ε ≈ .
23, the variation of A ( ε ) clearly indicates an abrupt transition from oscillatory state to FIG. 6.
The impact of internal parameters ρ and ω on thesuppression of oscillations in an ensemble of SL oscillators : Thevalues of the parameters are taken as (a) ∆ = . ω = .
0, (b) ∆ = . ρ =
1, (c) τ = . ω = .
0, and (d) τ = . ρ =
1. Our proposed SDI and TDI are found to be robust over a largeinterval of internal parameters ρ and ω in order to obtain the generaltransition from incoherent oscillatory state to stable death state. Thedynamic coupling seems to break the inherent symmetry of the oscil-lator and thus gives rise to stable AD or OD states depending on theparameters. The regions NS and CS depict incoherent domain andcoherent regime of synchronized limit cycle respectively. death state. Such a sudden transition is also presented for twocoupled SL oscillators with ∆ = . ∆ = .
35. However, this interestingfeature of first-order transition from oscillatory state to deathstate is completely lost for ∆ ≥ .
5. In Fig. 5 (b), one can seethat ∆ = τ avg ( ∆ ) = . ε increases, then the τ avg ( ∆ ) gradually reduces andcrosses the boundary condition (Eq. (8)) of the temporal in-teraction at ε (cid:117) .
62. On the basis of the above analysis, inboth the cases it is clear that whenever the τ avg ( ∆ ) crosses theboundary (Eq. (8)) of the temporal interaction, there is alwaysa steady state arises in spatial interaction. This attests the wellagreement of the numerical simulation with our analyticallyderived result (Eq. (8)). D. Effect of ρ and ω Till now, the numerical results are presented with fixed in-ternal parameters ρ = ω =
2. These values are high-lighted through black dashed lines in the subfigures of Fig. 6.Figure 6 demonstrates the consequence of different choices of ρ and ω . The subfigures (a) and (b) of Fig. 6 are drawn withfixed ∆ = . τ = .
5. All these subfigures portray thefact that the transition from the oscillatory dynamics to steadystate is generic for all values of internal parameters. Althoughthat steady states portray AD or OD depending on the valuesof the parameters. ω = ρ = ω ≤
1, AD state isnot found in Fig. 6 (b), which agrees well with our eigenvalueanalysis given in Sec. III A.
BS1 OD OD FIG. 7. (a)
Coupling strength ε vs SDI parameter ∆ for identicalRössler oscillators with N = x as a function of coupling strength ε with fixed ∆ = . x isplotted for stationary point solutions and extremum values for timedependent solutions of the coupled system (10). BS1 (the shaded re-gion) shows the bistable regime, where complete synchronized stateand oscillation death state may co-exist. (c)-(d) The time-series ofvariables x i : Different trajectories are converging to different at-tractors for different initial states in the subfigure (c). The couplingstrength ε is 0 .
15 for the subfigure (c) and 0 . a = . b = . c = . N = E. Chaotic System: Coupled Rössler Oscillator
In order to further validate the generic nature of the transi-tion from oscillatory state to death state, we examine the dy-namic interaction on the coupled chaotic system. We consider N =
100 coupled Rössler oscillators , interacting throughspatial or temporal mean-field diffusive interaction. The dy-namical equations are given as,˙ x i = − y i − z i + ε ( Hx − x i ) , ˙ y i = x i + ay i , ˙ z i = b + z i ( x i − c ) . (10)The parameters a = . , b = . c = . H reflects the space and time dependent interaction as de-scribed earlier through the relations (2) and (3), respectively.To illustrate the effect of space dependent interaction, we drawthe phase-diagram ε − ∆ for N =
100 coupled Rössler oscil-lators in Fig. 7 (a). For ∆ > .
49, the system traverses fromthe desynchrony to the synchrony regime. But for ∆ < . ε . This transition from oscillatory state to OD statecan take place via first order or second order depending uponthe value of ∆ at lower coupling strength. The coexistence ofthe oscillatory and OD states are marked as BS OD OD FIG. 8. (a)
Full ε - τ phase-diagram for system (10): Three dif-ferent dynamical domains of N =
100 coupled Rössler oscillatorsare distinguished in the parameter plane ( τ − ε ) . The notation NS,CS and OD are same as given in Fig. 7. (b) Bifurcation diagramagainst the coupling strength with fixed τ = .
45: The horizontaldashed line in subfigure (a) corresponds to τ = .
45, for which thisbifurcation diagram is explored. The other parameters are same aschosen in Fig. 7. For the smaller values of coupling strength ε , cou-pled system only exhibits the oscillatory behavior. For comparativelyhigher values of ε > . For H =
0, the stationary point solutions of the Eq. (10) aregiven by x ∗ = − (cid:32) P ± (cid:112) P − abQ Q (cid:33) , y ∗ = − x ∗ / a , and z ∗ = − b / ( x ∗ − c ) with P = c ( a ε − ) and Q = − P / c . The eigen-values of the system at approximate value of stationary pointfor a given set of parameter values are, λ = − c , λ , = (cid:16) a − ε ± (cid:112) ( a + ε − )( a + ε + ) (cid:17) (11)Thus equating the real parts of the complex eigen val-ues of (11), we obtain the condition of Hopf bifurcation as ε HB = a = .
1. This bifurcation point agrees quite well inFig. 7 (a). In fact, for ∆ = .
45 (the dashed line in Fig. 7(a)), we draw numerically the bifurcation diagram of coupledRössler oscillators in Fig. 7 (b). This bifurcation diagram of x displays quenching of oscillation and gives birth to stableOD through inverse Hopf bifurcation. Note that, there existsa region of ε ≈ [ . , . ] , where the system exhibits bistablebehavior (shaded region in Fig. 7 (b)). Figure 7 (c) demon-strates the temporal bistable phenomena for ε = .
15, whereoscillatory state and OD state may coexist. However, withenhancement of coupling parameter ε > . ε = . τ − ε for Rössler oscillator in Fig. 8 (a) for the temporal inter-action defined in Eq. (3). Proceeding to the way as proposed inSec. III B, a time-dependent linear stability analysis is carriedout. Linear stability analysis at the stationary point providesnontrivial characteristic equations, λ = (cid:16) a (cid:17) τ + ( − τ ) (cid:32) a − ε (cid:33) . (12) FIG. 9.
The phase diagram of Rosenzweig-MacArthur modelin (a) ( ∆ , ε ) and (b) ( τ , ε ) at T = .
6: Four different regimes areobserved, including no synchronization (NS) (gray), complete syn-chronization (CS) (sky-blue), bi-stable state (BS) (white) and steadystate (SS) (turquoise). The dynamics of coupled system changesfrom desynchronized state to the death state or the synchronized statedepending on the control parameters ∆ and τ as coupling strength ε isvaried. Coexistence of two attractors (BS) are noticed over a narrowregion of ∆ − ε in the subfigure (a). Setting λ = α + i γ , where α and γ are real and imaginaryparts of the eigenvalues, Eq. (12) leads to α = τε + a − ε . (13)The solid black line in Fig. (8) (a), corresponds to the locus ( α = τ = − a ε , ε (cid:54) = . (14)Analytical condition (solid black line) of Eq. (14) matcheswell with the numerically calculated OD region in Fig. 8 (a).Here the dynamics changes from periodic attractor to a sta-tionary point via Hopf bifurcation as a real part of the eigen-value α becomes negative. Figure 8 (a) reveals a transitionfrom incoherent (desynchronized) to coherent behavior hap-pens depending on the value of τ and ε . For comparativelylower values of τ , the same transition occurs, but the sys-tem settles down from desynchronized oscillatory state to ODstate. Our detailed eigenvalue analysis is also supported by bi-furcation analyses in Fig. 8 (b) for τ = .
45, where it is shownthat the dynamic coupling can induce a transition between pe-riodic attractor and OD even in identical Rössler oscillators.
F. Ecological System: Rosenzweig-MacArthur (RA) model
In order to further exemplify the dynamic interaction, weconsider the Rosenzweig-MacArthur (RA) model , inwhich dynamics belonging to each of the patch of a meta-population is described by the following equations˙ x i = rx i (cid:32) − x i K (cid:33) − α i x i x i + E y i + ε ( Hx − x i ) , ˙ y i = y i (cid:32) α i ξ x i x i + E − m (cid:19) , (15) where x i and y i are, respectively, vegetation and herbivoredensity, r is intrinsic growth rate, K is carrying capacity, α i = α is the maximum predation rate of the predator, E isthe half-saturation constant, ξ represents predator efficiency,and m is the mortality rate of the predator. The Rosenzweig-MacArthur model perhaps the simplest model that can actu-ally be applied in real ecosystems. As a result, this model be-comes a standard prey-predator model in consumer-resourcedynamics in theoretical ecology. Here, we choose the param-eter values as r = . K = . α = E = . ξ = .
5, and m = .
2. Results for spatial and temporal dynamic interactionin parameter space ( ∆ , ε ) and ( τ , ε ) are shown in Figs. 9 (a)and 9 (b), respectively. In Goldwyn and Hastings , both thespecies can disperse between patches, but here for the sakeof simplicity we consider dispersal of only one of the species(i.e., vegetation) between patches. The numerical findings inFigs. 9 (a) and 9 (b) attest the generic signature of our pro-posed dynamic interaction schemes. Just like the earlier nu-merical simulations, here we also able to portray the transi-tion from desynchronized region (NS) to either synchronizedregime (CS) or to death states (SS) depending on the interplayof ε , ∆ and τ . Furthermore, a bistable region (BS) is found inFig. 9 (a), where the oscillatory dynamics coexist with deathstates. IV. CONCLUSION AND SUMMARY
The overarching motivation of this work is to explore theeffect of spatial as well as temporal dynamic interaction incoupled nonlinear oscillators. We have studied the transi-tion to steady state in interacting nonlinear oscillators witheffective dynamic mean-field interactions. Note that the focusis given on the interaction depending on the mean-field con-trol parameter in the phase space instead of considering mo-bile agents configuration, where the nodes are moving in thephase space neglecting the oscillator’s internal dynamics .We have found that the spatial and temporal dynamic con-trol parameters ∆ and τ play a vital role in the transitions tovarious synchronization regions as well as in the suppressionof oscillations. There is an optimal window in the parameterspace of coupling strength and spatial dynamic control param-eter, where the system undergoes to the steady state via firstor second order transitions. Recently, the attention has beenshifting away from continuous phase transition to first-orderdynamical transition due to its relevance in biologicaland chemical processes. However, our proposed dynamic in-teraction is a rare example of coupled system, which can offerfresh new insights due to its time-varying interaction strategy.And this important consideration may appeal at least few in-terested young researchers due to its applicability in variousnatural as well as social systems. We would like to explorethe first-order dynamical transitions in detail under the lime-light of phase transitions in near future. In temporal dynamicinteraction, using an approximate linear stability analysis, weobtained the threshold values of the coupling strength for thetransition to death state, and it is in agreement with the nu-merical results. The spatial interaction also brings discontin-uous occasional temporal interaction, which is analyzed bycalculating τ avg ( ∆ ) for a given ∆ . Analytic estimates are sup-plemented by numerics for several systems. Particularly, thethreshold of coupling strength, which can facilitate death stateof the whole system, is invariable with the change of networksize. Spatial dynamic interaction in general display multipleasymptotic dynamical states in relation to different initial con-ditions due to the interplay of ε and ∆ in a region of bistability.The coexistence of such two phases (the oscillatory state andthe steady state) at the same time is ubiquitous in many real-istic scenarios including biological and chemical systems .Our account of presented results may yield new insights andfoster the understanding of the temporal dynamics of coupledoscillators. ACKNOWLEDGMENT
MDS and AP acknowledges financial support (Grant No.EMR/2016/005561 and INT/RUS/RSF/P-18) from Depart-ment of Science and Technology (DST), Government of In-dia, New Delhi. S.N.C. would like to acknowledge the CSIR(Project No. 09/093(0194)/2020-EMR-I) for financial assis-tance.
DATA AVAILABILITY
The data that support the findings of this study are availablewithin the article. P. Holme and J. Saramäki, Physics Reports , 97 (2012). S. Nag Chowdhury, S. Kundu, M. Duh, M. Perc, and D. Ghosh, Entropy , 485 (2020). A. Pikovsky, J. Kurths, M. Rosenblum, and J. Kurths,
Synchronization:a universal concept in nonlinear sciences , Vol. 12 (Cambridge universitypress, 2003). A. Arenas, A. Díaz-Guilera, J. Kurths, Y. Moreno, and C. Zhou, PhysicsReports , 93 (2008). A. E. Hramov and A. A. Koronovskii, Chaos: An Interdisciplinary Journalof Nonlinear Science , 603 (2004). S. N. Chowdhury, S. Majhi, M. Ozer, D. Ghosh, and M. Perc, New Journalof Physics , 073048 (2019). S. N. Chowdhury, S. Majhi, and D. Ghosh, IEEE Transactions on NetworkScience and Engineering , 3159 (2020). A. Zakharova, M. Kapeller, and E. Schöll, Physical Review Letters ,154101 (2014). 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, PhysicalReview E , 052205 (2016). L. Khaleghi, S. Panahi, S. N. Chowdhury, S. Bogomolov, D. Ghosh, andS. Jafari, Physica A: Statistical Mechanics and its Applications , 122596(2019). A. V. Andreev, M. V. Ivanchenko, A. N. Pisarchik, and A. E. Hramov,Chaos, Solitons & Fractals , 110061 (2020). S. Majhi, B. K. Bera, D. Ghosh, and M. Perc, Physics of Life Reviews ,100 (2019). G. Saxena, A. Prasad, and R. Ramaswamy, Physics Reports , 205(2012). S. Nag Chowdhury, D. Ghosh, and C. Hens, Physical Review E ,022310 (2020). A. Koseska, E. Volkov, and J. Kurths, Physical Review Letters , 024103(2013). W. Zou, D. Senthilkumar, M. Zhan, and J. Kurths, Physical Review Letters , 014101 (2013). W. Zou, D. Senthilkumar, R. Nagao, I. Z. Kiss, Y. Tang, A. Koseska,J. Duan, and J. Kurths, Nature communications , 1 (2015). A. Prasad, Pramana , 407 (2013). K. Yadav, A. Sharma, and M. Shrimali, in
Proceedings of the Conferenceon Perspectives in Non-linear Dynamics (2017) p. 157. S. S. Chaurasia, A. Choudhary, M. D. Shrimali, and S. Sinha, Chaos, Soli-tons & Fractals , 249 (2019). S. Dixit and M. D. Shrimali, Chaos: An Interdisciplinary Journal of Non-linear Science , 033114 (2020). S. N. Chowdhury, S. Majhi, D. Ghosh, and A. Prasad, Physics Letters A , 125997 (2019). M. Schröder, M. Mannattil, D. Dutta, S. Chakraborty, and M. Timme,Physical Review Letters , 054101 (2015). S. N. Chowdhury and D. Ghosh, EPL (Europhysics Letters) , 10011(2019). A. T. Winfree, Journal of Theoretical Biology , 15 (1967). R. E. Mirollo and S. H. Strogatz, Journal of Statistical Physics , 245(1990). A. Buscarino, L. Fortuna, M. Frasca, and A. Rizzo, Chaos: An Interdisci-plinary Journal of Nonlinear Science , 015116 (2006). A. Tandon, M. Schröder, M. Mannattil, M. Timme, and S. Chakraborty,Chaos: An Interdisciplinary Journal of Nonlinear Science , 094817(2016). Y. Kuramoto,
Chemical oscillations, waves, and turbulence (Courier Cor-poration, 2003). O. E. Rössler, Physics Letters A , 397 (1976). E. E. Goldwyn and A. Hastings, Theoretical Population Biology , 395(2008). M. L. Rosenzweig and R. H. MacArthur, The American Naturalist , 209(1963). A. Sharma and M. D. Shrimali, Physical Review E , 057204 (2012). S. Dixit, A. Sharma, A. Prasad, and M. D. Shrimali, International Journalof Dynamics and Control , 1015 (2019). A. Koseska, E. Volkov, and J. Kurths, Physics Reports , 173 (2013). R. Karnatak, R. Ramaswamy, and A. Prasad, Physical Review E ,035201 (2007). Z. Sun, N. Zhao, X. Yang, and W. Xu, Nonlinear Dynamics , 1185(2018). U. K. Verma, A. Sharma, N. K. Kamal, J. Kurths, and M. D. Shrimali,Scientific reports , 1 (2017). E. E. Goldwyn and A. Hastings, Journal of Theoretical Biology , 237(2011). U. K. Verma, A. Sharma, N. K. Kamal, and M. D. Shrimali, Chaos: AnInterdisciplinary Journal of Nonlinear Science , 063127 (2019). H. Bi, X. Hu, X. Zhang, Y. Zou, Z. Liu, and S. Guan, EPL (EurophysicsLetters) , 50003 (2014). H. Chen, G. He, F. Huang, C. Shen, and Z. Hou, Chaos: An Interdisci-plinary Journal of Nonlinear Science , 033124 (2013). U. K. Verma, S. S. Chaurasia, and S. Sinha, Physical Review E , 032203(2019). A. Beuter, L. Glass, M. C. Mackey, and M. S. Titcombe,
Nonlinear dynam-ics in physiology and medicine (2003). B. D. Aguda, L. L. Hofmann Frisch, and L. Folke Olsen, Journal of theAmerican Chemical Society112