Environment induced Symmetry Breaking of the Oscillation-Death State
EEnvironment induced Symmetry Breaking of the Oscillation-Death State
Sudhanshu Shekhar Chaurasia , Manish Yadav , and Sudeshna Sinha Indian Institute of Science Education and Research (IISER) Mohali,Knowledge City, SAS Nagar, Sector 81, Manauli PO 140 306, Punjab, India
We investigate the impact of a common external system, which we call a common environment, onthe Oscillator Death (OD) states of a group of Stuart-Landau oscillators. The group of oscillatorsyield a completely symmetric OD state when uncoupled to the external system, i.e. the two ODstates occur with equal probability. However, remarkably, when coupled to a common external sys-tem this symmetry is significantly broken. For exponentially decaying external systems, the symme-try breaking is very pronounced for low environmental damping and strong oscillator-environmentcoupling. This is evident through the sharp transition from the symmetric to asymmetric stateoccurring at a critical oscillator-environment coupling strength and environmental damping rate.Further, we consider time-varying connections to the common external environment, with a frac-tion of oscillator-environment links switching on and off. Interestingly, we find that the asymmetryinduced by environmental coupling decreases as a power law with increase in fraction of such on-offconnections. The suggests that blinking oscillator-environment links can restore the symmetry ofthe OD state. Lastly, we demonstrate the generality of our results for a constant external drive,and find marked breaking of symmetry in the OD states there as well. When the constant environ-mental drive is large, the asymmetry in the OD states is very large, and the transition between thesymmetric and asymmetric state with increasing oscillator-environment coupling is very sharp. Soour results demonstrate an environmental coupling-induced mechanism for the prevalence of certainOD states in a system of oscillators, and suggests an underlying process for obtaining certain statespreferentially in ensembles of oscillators with environment-mediated coupling.
I. INTRODUCTION
Complex systems has been a very active area of re-search over the past few decades, initiated by the dis-covery that even systems with low degrees of freedomcan show a wide range of dynamical patterns. For in-stance, two or more oscillators, when coupled to eachother can show completely synchronized oscillations, in-phase or anti-phase synchronized oscillations, oscillationquenching to homogeneous steady states or inhomoge-neous steady states, with transitions between differentdynamical behaviours obtained by parameter tuning.In general, oscillation quenching is categorized into ho-mogeneous steady state (HSS) or amplitude death (AD)and inhomogeneous steady state (IHSS) or Oscillationdeath (OD) [1]. AD refers to the situation where thecoupled oscillator systems, under oscillation quenching,evolve to the same fixed point. This type of quenching isrelevant in laser systems [2–4], and is important in situa-tions involving stabilization to a particular fixed point. Alot of mechanisms leading to amplitude death have beenfound, such as time-delay in the coupling [5, 6], couplingvia conjugate variables [7], introduction of large varianceof frequencies [8] and coupling to a dissimilar external os-cillator [9]. On the other hand, oscillation quenching cangive rise to oscillation death, a phenomenon that is com-pletely different from AD. Here the oscillators split intotwo sub-groups, around an unstable fixed point via pitch-fork bifurcations, generating a set of stable fixed points.Oscillation death is very relevant to biological systems,as this oscillation quenching mechanism can lead to theemergence of inhomogeneity in homogeneous medium.So, for instance, OD has been interpreted as a mechanism for cellular differentiation [10, 11]. Thus a lot of researcheffort has centered around transitions from AD to OD[16, 17], and mechanisms that steer the dynamics to theOD state have been investigated. For example, OD canbe achieved via parametric modulation in coupled non-autonomous system [12], parameter mismatch (i.e. de-tuning of parameters) in coupled oscillators [13, 14] andthe introduction of local repulsive links in diffusively cou-pled oscillators [15]. In a complementary direction, somestudies have also shown how OD states are eliminatedwhen gradient coupling is introduced in delay inducedOD [18].Our work here focuses on oscillation quenching mech-anisms that give rise to inhomogeneous steady states.Our test-bed will be a group of oscillators, coupled to acommon external system, which is dynamically very dis-tinct from the oscillators. This common external systemprovides a common “environment” and allows a groupof oscillators to be indirectly coupled via an externalcommon medium. When uncoupled, the oscillators haveequal probability to go to either of the OD states. How-ever, we will show that this system displays symmetrybreaking when coupled. That is, a specific oscillatordeath state is preferentially achieved. This state selec-tion leads to asymmetric distribution of OD states inthe ensemble of oscillators, suggesting a natural mecha-nism that allows the emergence of a favoured set of fixedpoints. Further we will explore the effect of the oscillatorgroup connecting to the environment through links thatswitch on and off. We will demonstrate that blinkingoscillator-environment connections will remarkably workto towards partial restoration of the symmetry of the os-cillator death states, though the presence of some blink- a r X i v : . [ n li n . AO ] J un ing connections reduces the symmetry of the dynamicalequations. II. COUPLED OSCILLATORS
Complex systems often undergo Hopf bifurcations andsufficiently close to such a bifurcation point, the variableswhich have slower time-scales can be eliminated. Thisleaves us with a couple of simple first order ordinarydifferential equations, popularly known as the Stuart-Landau system [19]. In this work we consider a groupof N globally coupled Stuart-Landau oscillators, with ω being the angular frequency of oscillator. Specifically,the oscillators are coupled via the mean field ¯ x of the x -variable, with ε intra reflecting the strength of intra-groupcoupling. Now, this oscillator group also couples to anexternal system, which we call the environment , denotedby u . The environment exponentially decays to zero,with decay constant k , when uncoupled from the oscil-lator group. However when coupled to the oscillators,the environment provides an input to the oscillators, aswell as receives a feedback proportional to the mean field¯ y of the y -variables of the oscillators. The strength ofthis feedback from the external system is given by thecoupling strength ε ext . So the complete dynamics of thegroup of oscillators, along with the external environment,is then given by the following evolution equations:˙ x i = (1 − x i − y i ) x i − ωy i + ε intra ( q ¯ x − x i )˙ y i = (1 − x i − y i ) y i + ωx i + ε ext u (1)˙ u = − ku + ¯ y where ¯ x = N (cid:80) Ni =1 x i and ¯ y = N (cid:80) Ni =1 ε ext y i .So the common external environment provides an in-direct coupling conjoining the different oscillators in thegroup. The idea is rooted in phenomena where a com-mon medium influences oscillators, such as in the popu-lation of yeast cells [20], where acetaldehyde is used as acommon medium, or in mechanical oscillators in a fluidenvironment [21]. Studies on the effect of an external en-vironment on coupled Stuart-Landau oscillators has re-vealed phenomena such as the revival of oscillations in agroup of oscillators at steady state by coupling to an os-cillating group via a common environment [22], phase-fliptransitions in a system of oscillators diffusively coupled tothe environment [23] and co-existence of in-phase oscil-lations and oscillation death in environmentally coupledoscillators [24].In this work we will first explore in Section III thesymmetry-breaking effect of the common external envi-ronment on the oscillatory patterns. Further, we will ex-plore the spatiotemporal effects of the time variation ofthe oscillator-environment links in Section IV. Lastly inSection V, we will demonstrate that a constant commonenvironment also leads to pronounced symmetry break-ing in the Oscillator Death states, suggesting the gener-ality of our central result. III. SYMMETRY BREAKING IN THEOSCILLATOR DEATH STATES
We first present the bifurcation sequence of the oscilla-tors as a function of the oscillator-environment couplingstrength ε ext . The values of ε intra and q are fixed at 6.0and 0.4 so that uncoupled oscillators are in the oscilla-tion death (OD) state in the absence of coupling to theenvironment. Here one of the oscillator death states haspositive x and negative y , and the other oscillator deathstate has and negative x and positive y (cf. Fig. 1). Wecall the steady state solution with x > x < . . . . . . . ε ext − . − . − . . . . . x FIG. 1: Bifurcation diagram of x of one of the Stuart-Landau oscillator in a group, with respect to the cou-pling strength ε ext of the group with the environment(cf. Eqn. 1). The diagram displays the superposition ofthe system evolving from a large range of random ini-tial states, with x i , y i ∈ [ − ,
1] and the environmentalvariable u ∈ [0 , ω =2 . q = 0 . ε intra = 6 . k = 0 . N = 20.In the absence of coupling to an external environ-ment, the states of the group of oscillators are sym-metrically distributed between the positive and negativestates. That is, starting from generic random initial con-ditions, the group of oscillators will have equal proba-bility to evolve to a positive state or a negative state.So one typically observes an equi-distribution of positiveand negative oscillators in a group of Stuart-Landau os-cillators in the oscillator death regime, when uncoupledto the environment. This is evident from the bifurcationdiagram, which shows equal probability to be in either ofthe two OD states at ε ext = 0 (as reflected by symbols ofthe same size in the positive and negative states in thefigure at ε ext = 0). This behaviour is also clear from thetime series of the oscillator group displayed in Fig. 2a.Interestingly however, when the oscillator group is cou-pled to the external environment we observe symmetrybreaking in the Oscillator Death states . Namely, thegroup of oscillators in the presence of the environment,preferentially go to one of the oscillator death state. So,typically we do not obtain an equal number of positiveand negative states. Rather there is now a pronouncedprevalence of one of the Oscillator Death states . t − . − . − . − . . . . . . x (a) t − . − . − . − . . . . . . x (b) FIG. 2: Time series of twenty oscillators in the group(shown in distinct colours), (a) in the absence of cou-pling to an external environment and (b) when the groupis connected to the external environment with couplingstrength ε ext = 0 .
6, and k = 0 . ε ext . This is reflected by symbols of thedifferent sizes in the positive and negative states in thefigure at large ε ext . This behaviour is also clear from the time series of the oscillator group displayed in Fig. 2b,which shows the oscillators preferentially evolving to oneof the two OD solutions.In order to gauge the global stability of an OscillatorDeath state, say the positive state, we use the conceptof Basin Stability. We choose a large number of randominitial conditions, uniformly spread over phase space vol-ume. For each initial state, we calculate the fraction f + of oscillators that evolve to the positive OD state. Theaverage of f + over random initial conditions (cid:104) f + (cid:105) yieldsan estimate of the Basin Stability of the positive state,and indicates the probability of obtaining the positiveoscillator death state in a group of oscillators startingfrom random initial conditions in the prescribed volumeof phase space. The most symmetric distribution, namelyhalf the oscillators in the positive state and the other halfin the negative state, leads to a Basin Stability measureof 0 . Deviations from . indicate asymmetry in thedistribution of oscillator death states , with a prevalenceof the positive or negative state. So the quantity (cid:104) f + (cid:105) serves as an order parameter for symmetry-breaking ofthe Oscillator Death states.It is clearly evident from Fig. 3 that there is a sharptransition from a reasonably symmetric state (where (cid:104) f + (cid:105) is close to 0 .
5) to a completely asymmetric statecharacterized by (cid:104) f + (cid:105) ∼ ε ext increases. This suggeststhat the external environment plays a key role in breakingthe symmetry of the Oscillator Death state, as this phe-nomenon emerges only when the oscillator-environmentcoupling is sufficiently strong, with the sudden onset ofasymmetry in the group of oscillators occurring at a criti-cal coupling strength. Further, it is clear from Fig. 3 thatthe symmetry breaking of the Oscillator Death states isindependent of system size N , over a large range of sys-tem sizes. . . . . . . . ε ext . . . . . . h f + i FIG. 3: Basin Stability of the positive Oscillator Deathstate of coupled oscillators with ε ext , where groups ofoscillators of different sizes N =40 , , , ,
120 areshown in different colours. Here the damping constantof the external environment is k = 0 . IV. EFFECT OF BLINKING CONNECTIONS
In the section above we considered the effect of the ex-ternal environment on a group of oscillators, when the ex-ternal system was connected to all oscillators at all times,and we clearly demonstrated that this lead to markedasymmetry in Oscillator Death states. This is in con-tradistinction to the case where the group of oscillatorsare not connected to an external system, which leads tocomplete symmetry in the Oscillator Death states. Nowwe will consider the effect of oscillator-environment con-nections blinking on-off, and explore the effect of suchtime-varying links on the symmetry of the OscillatorDeath states.In order to model connections to the environmentblinking on and off, we consider a time-dependentoscillator-environment coupling strength ε i ext ( t ) = ε ext g i ( t ) in Eqn. 1, with the feedback from the os-cillator group to the external system given by ¯ y = N (cid:80) Ni =1 ε i ext ( t ) y i . If the connection of an oscillator tothe environment is constant, g i ( t ) = 1 for all t . Such alink is considered a time-invariant non-blinking connec-tion . If the connection of the i th oscillator in the groupand the external system periodically switches on and off,namely the link is a blinking connection , g i ( t ) is a squarewave. When oscillator i in the group is connected to envi-ronment g i ( t ) = 1, otherwise g i ( t ) = 0. So g i ( t ) switchesbetween 0 (off) and 1 (on), with time period T blink whichprovides a measure of the time-scale at which the linksvary. Here we will principally consider rapidly switchinglinks, i.e. low T blink .One of the most important parameters in this time-varying scenario is the fraction of blinking oscillator-environment connections in the group, which we denoteby f blink . If all oscillators are connected to the exter-nal, then f blink = 0 and if all connections are blinking,then f blink = 1. Here we will study the entire range0 ≤ f blink ≤
1, and gauge the effect of the fraction ofblinking connections on the symmetry of OD state. No-tice that the presence of connections switching on-off in asub-set of oscillators results in the dynamical equations ofthe oscillator groups being less symmetric, as the groupsplits into two sub-sets having distinct dynamics. So itis most relevant to investigate if this lack of symmetry inthe dynamical equations leads to more asymmetry in thesteady states. However, what we will demonstrate in thisSection is the following result: counter-intuitively, blink-ing links partially restore the symmetry of the emergentOscillator Death states.Fig. 4 shows the Basin Stability of the positive oscil-lator death state, as a function of the fraction f blink ofoscillators with blinking connections to the environment.We find that when there are no blinking links, namelythe connections of the oscillators to the external systemare always on, the emergent state is the most asymmet-ric . That is, the deviations of the Basin Stability from0 . f blink = 0. Increasing thenumber of blinking connections reduces the asymmetry and restores the symmetry of the oscillator death statesto a large extent, yielding states that are almost equi-distributed between positive and negative states. Thetransition from the asymmetric state (where (cid:104) f + (cid:105) ∼ (cid:104) f + (cid:105) is significantlydifferent from 0) occurs sharply at a critical fraction ofblinking links, which we denote by f c blink . Further, it isevident from Fig. 4 that the symmetry breaking dynam-ics of the system, and f c blink in particular, is independentof number N of oscillators in the group. . . . . . . f blink . . . . h f + i FIG. 4: Dependence of the Basin Stability of the positiveOscillator Death state (cid:104) f + (cid:105) , on the fraction of oscillators f blink with blinking oscillator-environment connections inthe group. Different system sizes N = 40 , , ,
100 areshown in different colours. Here the time period of theon-off blinking T blink = 0 .
02, oscillator-environment cou-pling strength ε ext = 0 . k = 0 . f blink , and the oscillator-environment coupling strength ε ext . It is evident that for weaker coupling strengths thegroup of oscillators evolve to the positive and negativeOD states with almost equal probability, i.e. (cid:104) f + (cid:105) isquite close to 0 .
5. On the other hand, for strong cou-pling strengths, there is a sharp transition from a veryasymmetric situation (where (cid:104) f + (cid:105) ∼
0) at low f blink , toa more balanced situation (where (cid:104) f + (cid:105) is closer to 0 . f blink . Further, one observes that a system witha large fraction of blinking connections does not becomemarkedly asymmetric even for large coupling strengths,i.e. (cid:104) f + (cid:105) is not close to 0 even for ε ext close to 1. How-ever, in a system with few blinking connections, thereis a sharp transition to asymmetry for sufficiently highcoupling strengths. . . . . . . f blink . . . . . . h f + i (a) . . . . . . . ε ext . . . . . . h f + i (b) FIG. 5: Basin Stability of the positive Oscillator Deathstate, (a) as a function of fraction of oscillators withblinking oscillator-environment connections f blink , forcoupling strength ε ext = 0 . ε ext = 0 . ε ext , for f blink = 0 .
25 (green) and f blink = 0 .
75 (blue). Here the time period of blinking T blink = 0 .
02, the damping constant of the environment k = 1 . N = 64.We will now focus on the effect of the dynamical fea-tures of the common environment on symmetry breakingof the OD state. Note that the common environment,when uncoupled to the oscillator group, is an exponen-tially decaying system u = u e − kt , where u is the am-plitude at time t = 0 and k is the damping rate.Fig. 6 shows the probability of obtaining the positiveOscillator Death state, as the fraction of oscillators withblinking connections is varied, for different environmen-tal damping constants k . It is evident that at high en-vironmental damping rates, the effect of environment isless pronounced, and the Oscillator Death states are se-lected with almost equal probability. However, there ispronounced asymmetry in OD states when the damping rate of the environment is low, with critical f c blink tendingto 1 as k increases. . . . . . . f blink . . . . . . h f + i FIG. 6: Basin Stability of the positive Oscillator Deathstate on the fraction of oscillators with blinking connec-tions f blink , for environmental damping constant k = 0 . k = 0 . k = 1 . T blink = 0 .
02, oscillator-environmentcoupling strength ε ext = 0 . N = 64. To estimate the Basin Stability werandomly sampled u ∈ (0 , k .Further we estimate the probability of an oscillator tobe in the positive Oscillator Death state, for the case ofthe sub-group of oscillators with blinking links to the en-vironment (see Fig. 7a), and for the case of the sub-groupof oscillators with static links to the environment (seeFig. 7b). For low environmental damping rates, there isa sharp boost in the probability of oscillators to be inthe positive OD state in the sub-group of oscillators withblinking oscillator-environment connections, at a critical f c blink (e.g. f c blink ∼ . k = 0 . f c blink ∼ . k = 0 . This implies that the sub-group of oscil-lators with blinking connections to the environment is thegroup that is vital to the restoration of symmetry . . . . . . . f blink . . . . . . . p + b li n k (a) . . . . . . f blink . . . . . . . p + s t a t i c (b) FIG. 7: Probability of obtaining the positive Oscilla-tor Death state for the sub-group of oscillators with (a)blinking oscillator-environment connections, denoted by p + blink , and (b) static oscillator-environment connectionsdenoted by p + static , as a function of the fraction of oscilla-tors with blinking connections f blink . Here the time pe-riod of blinking T blink = 0 .
02, the oscillator-environmentcoupling strength ε ext = 0 .
5, number of oscillators in thegroup N = 64, and the environmental damping constant k =0 . k = 0 . k = 1 . k of the environment, for different fractions ofblinking connections. It is evident that at low dampingconstants, there are very few oscillators in the positiveOD state. On increasing the damping constant of theenvironment there is a sharp jump in the fraction of os-cillators in the positive OD state. So there is a suddentransition from a very asymmetric state, where the frac-tion of oscillators in the positive OD state is close to zero,to a more symmetric state, where this fraction is closeto half. The critical k where this jump occurs dependsupon the number of blinking oscillator-environment links. When there is a higher fraction f blink of blinking connec-tions in the system, the critical k is lower, i.e. the jumpto a more symmetric situation occurs at lower dampingconstants. . . . . . . k . . . . . . h f + i FIG. 8: Basin Stability of the positive Oscillator Deathstate, as a function of the damping constant k of the en-vironment. Here time period of blinking T blink = 0 . ε ext = 0 . N = 64 andthe fraction of blinking oscillator-environment connec-tions are: f blink = 0 .
25 (red), f blink = 0 .
50 (green) and f blink = 0 .
75 (blue). . . . . . . f blink . . . . . . k c FIG. 9: Critical value of damping constant k c vs. frac-tion of blinking oscillator-environment connections f blink .Here the time period of blinking T blink = 0 .
02, oscillator-environment coupling strength ε ext = 0 . N = 64. The data points fromnumerical simulations are in blue, and the curve given byequation: k c = k c − c f blink , for k c ≈ .
26 and c ≈ . k where (cid:104) f + (cid:105) crosses a threshold value of 0 . k c . This critical valueindicates the damping constant below which significantsymmetry-breaking of the Oscillator Death states occurs.Fig. 9 shows critical k c as a function of the fraction ofblinking connections f blink . The critical damping k c de-creases with increasing fraction of blinking connections f blink . Specifically, in a large range of f blink we findthat k c decreases linearly with f blink (see Fig. 9). Thisdemonstrates that low environmental damping favoursenhanced asymmetry, while more blinking connectionstends to restore the symmetry of the OD states.Fig. 10 shows the fraction of oscillators in the posi-tive Oscillator Death state, in the parameter space of k and ε ext , for different fraction of blinking oscillator-environment connections. The black regions in the fig-ures represent the asymmetric state. Clearly, low en-vironmental damping k and high oscillator-environmentcoupling ε ext yields the greatest asymmetry in the emer-gent Oscillator Death states. . . . . . . k (a) (b) . . . ε ext0 . . . . . k (c) . . . . ε ext (d) . . . . . . FIG. 10: Basin Stability of the positive Oscillator Deathin the parameter space of oscillator-environment cou-pling strength ε ext and environmental damping constant k . The fraction of blinking oscillator-environment con-nections are: (a) f blink = 0 . f blink = 0 .
25, (c) f blink = 0 .
50, (d) f blink = 1 .
0. Thetime period of blinking T blink = 0 .
02 and the number ofoscillators in the group N = 64.Now we focus on the line of transition from high (cid:104) f + (cid:105) to low (cid:104) f + (cid:105) , shown in Fig. 11. We find that k is pro-portional to ε along the lines of transition, with theproportionality constant depending on the fraction ofblinking oscillator-environment connections (see inset).Interestingly, comparing Fig. 9 and the inset of Fig. 11,reveals that both have the same dependence on f blink ,and the proportionality constant is equal to 4 k c . Thisimplies that the line of transition to asymmetry in thespace of k - ε ext is given by: k = 4 k c ε (2) where k c is inversely proportional to the fraction of blink-ing connections f blink . . . . . . . ε ext . . . . . k .
00 0 .
25 0 .
50 0 .
75 1 . f blink . . . . . a FIG. 11: Transition line in Fig. 10, fitted to the curve(solid lines): k = aε , where the different curves cor-respond to f blink = 0 . , . , . , . , . a with f blink .Lastly, we explore the effect of this symmetry-breakingon the external environment. The external system is adamped system influenced by the mean field of the groupof oscillators, and it settles down to a fixed point u (cid:63) whenthe OD state becomes stable. This is easily seen as fol-lows: when the OD state is stable, ¯ y is a constant. Sothe steady state solution of u is given by ¯ y/k . Denotingthe x variable of the positive OD state by x + and the y -variable as y + , and denoting the x variable of the neg-ative OD state by x − and the y -variable as y − , we have¯ y = f + y + +(1 − f + ) y − , yielding u (cid:63) = y + (2 f + − k . Clearlyif the probabilities of obtaining the positive and negativeOD states are equal, i.e. f + = 0 .
5, then u (cid:63) = 0. If theasymmetry is extreme and f + ∼ u (cid:63) = − y + /k .Since the positive OD state has x + > y + < u (cid:63) is positive. Also notice that the value of u (cid:63) is inverselyproportional to damping constant k (cf. Fig. 12a). So astrongly damped environment evolves to u (cid:63) close to zero,as is intuitive. Further, from linear stability analysis ofthe dynamics of the external system one can see that u (cid:63) is a stable steady state, as the derivative if the vectorfield governing ˙ u is − k which is always negative. Fur-ther, since increasing the fraction of blinking connectionsfavours symmetry, thereby pushing f + closer to half, u (cid:63) also decreases with increasing f blink (cf. Fig. 12b).So we can conclude that the state of the environment u (cid:63) is strongly correlated to the asymmetry . In fact, sim-ply observing u (cid:63) tells us if the symmetry-breaking is pro-nounced or not. . . . . . . k u ? (a) . . . . . . f blink u ? (b) FIG. 12: Environmental steady state u (cid:63) with respectto (a) damping constant k , for f blink = 0 . . . k = 0 .
08 (red), 0 .
12 (green), 0 .
16 (blue).Here the number of oscillators in the group N = 64. Effect of the frequency of blinking on Oscillation Death:
In the results discussed above the time-period of theblinking connections was small, i.e. the links switched on-off rapidly. Now we will investigate the influence of thetime-period T blink of the blinking oscillator-environmentconnections on the dynamics. Fig. 13 displays the ef-fect of increasing blinking time-period on the state of theoscillators. It is evident from the time-series of the os-cillators (cf. Fig. 13a) that after a critical blinking time-period the system starts to oscillate and the OD steadystate is destroyed. That is, slow blinking of links leadsto oscillation revival. This is also quantitatively demon-strated in Fig. 13b, which shows the amplitude of theoscillators. Clearly up to T blink ∼ . T blink increases further, the amplitude grows fromzero to a finite value, indicating the emergence of oscil-lations whose amplitude increases with T blink . After alarge value T blink ( ∼
10) the amplitude of the oscillationssaturate to a maximum value (cf. Fig. 13b). We findthat this maximum amplitude is the difference betweenthe steady state solution of the oscillator for the case of ε ext = 0 (i.e. when uncoupled from the external system)and the steady state arising for ε ext > t − . − . − . . . . x (a) − − T blink . . . . . . . . A m p li t ud e (b) FIG. 13: (a) Time series of one of the oscillators in groupfor T blink = 0 .
02 (red), 200 . x -variable for different T blink . Here N = 20, f blink =0 . ε ext = 0 . k = 0 . T blink ∼ .
1, with fixed states transitioning to non-zero amplitudeoscillations, saturating around amplitude ∼ .
2. The sec-ond transition commences around T blink ∼ π/ω , wherethe amplitude starts to grow rapidly again, from ampli-tudes around 0 .
2, to the maximum amplitude.
V. CONSTANT COMMON ENVIRONMENT
We have shown the effect of exponentially decayingexternal environment on the OD state and the effect ofblinking connections. This mimics a situation where theexternal environment is a small bath, and so the dynam-ics of the oscillator group affects the dynamics of thecommon environment. In this section, we will considera common external environmental system mimicking alarge bath, where the external environment does not getaffected by the oscillator group. Rather it acts as a con-stant drive , which we denote by u c . The strength of thisoscillator-external drive connection is given by the cou-pling strength ε ext . So the complete dynamics of thegroup of oscillators is now given by the following evolu-tion equations:˙ x i = (1 − x i − y i ) x i − ωy i + ε intra ( q ¯ x − x i )˙ y i = (1 − x i − y i ) y i + ωx i + ε ext u c (3)where ¯ x = N (cid:80) Ni =1 x i . u c (a) (b) . . . ε ext02468 u c (c) . . . . ε ext (d) . . . . . . FIG. 14: Basin Stability of the positive Oscillator Deathstate in the parameter space of coupling strength ε ext and constant environment ( u c ), with fraction of blinkingoscillators (a) f blink = 0 .
0, (b) f blink = 0 .
25, (c) f blink =0 .
50, (d) f blink = 1 .
0. Here the time period of blinking T =0 .
02 and the number of oscillators in the group N =64.Fig. 14 shows the fraction of oscillators in positive Os-cillator Death state, in the parameter space of u c and ε ext . Different panels correspond to different fraction ofblinking connections, and the black regions in the fig-ures represent the asymmetric state. It is evident that even for the case of constant drive, the symmetry of theOscillator Death state is broken, and the system movespreferentially to the negative state. This further indi-cates the generality of our observations, and emergenceof symmetry-breaking in a group of oscillators due tocoupling to a common external system. VI. CONCLUSION
We investigated the impact of a common external sys-tem, which we call a common environment , on the Os-cillator Death states of a group of Stuart-Landau oscil-lators. First we consider external systems that exponen-tially decay to zero when uncoupled from the oscillatorgroup. Note that a group of oscillators yield a completelysymmetric Oscillator Death state when uncoupled to theexternal system, i.e. the positive and negative OD statesoccur with equal probability, and so in a large ensem-ble of oscillators the fraction of oscillators attracted tothe positive/negative state is very close to half. How-ever, remarkably, when coupled to a common externalsystem this symmetry is significantly broken. This sym-metry breaking is very pronounced for low environmen-tal damping and strong oscillator-environment coupling,as evident from the sharp transition from the symmet-ric to asymmetric state occurring at a critical oscillator-environment coupling strength and environmental damp-ing rate.Further, we consider a group of oscillators with time-varying connections to the common external environ-ment. In particular, we study the system with a frac-tion of oscillator-environment links that switch on-off.Interestingly, we noticed that the asymmetry induced byenvironmental coupling decreases as a power law withincrease in fraction of such on-off connections. This sug-gests that blinking oscillator-environment links can re-store the symmetry of the Oscillator Death state.Lastly, we demonstrated the generality of our resultsfor a constant external drive, i.e. a constant environ-ment, and found marked breaking of symmetry Oscilla-tor Death states there as well. 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