Dynamic interaction induced explosive death
Shiva Dixit, Sayantan Nag Chowdhury, Dibakar Ghosh, Manish Dev Shrimali
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Dynamic interaction induced explosive death
Shiva Dixit , Sayantan Nag Chowdhury , Dibakar Ghosh , and Manish Dev Shrimali 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
PACS – Nonlinear dynamics and nonlinear dynamical systems
PACS – Interdisciplinary applications of physics
PACS – Structures and organization in complex systems
Abstract. - Most previous studies on coupled dynamical systems assume that all interactionsbetween oscillators take place uniformly in time, but in reality, this does not necessarily reflectthe usual scenario. The heterogeneity in the timings of such interactions strongly influencesthe dynamical processes. Here, we introduce a time-evolving state-space dependent couplingamong an ensemble of identical coupled oscillators, where individual units are interacting onlywhen the mean state of the system lies within a certain proximity of the phase space. Theyinteract globally with mean-field diffusive coupling in a certain vicinity and behave like uncoupledoscillators with self-feedback in the remaining complementary subspace. Interestingly due tothis occasional interaction, we find that the system shows an abrupt explosive transition fromoscillatory to death state. Further, in the explosive death transitions, the oscillatory state andthe death state coexist over a range of coupling strengths near the transition point. We exploreour claim using Van der pol, FitzHugh–Nagumo and Lorenz oscillators with dynamic mean fieldinteraction. The dynamic interaction mechanism can explain sudden suppression of oscillationsand concurrence of oscillatory and steady state in biological as well as technical systems.
Introduction. –
Explosive transition [1] in ensemblesof coupled dynamical systems grabs the attention of manyphysicists due to its relevance in various practical applica-tions. The transition from incoherence to coherence [2, 3]and the emergence of a giant connected component in thenetwork [4] in most of the cases are continuous and re-versible. However, abrupt emergence of a collective statedue to non-trivial interactions among coupled dynamicalsystems has also been widely reported [5–9]. The vast ma-jority of these studies are concerned with the discontinuoussynchronization transition (also known as explosive syn-chronization) [10–12], where the synchronization order pa-rameter exhibits an irreversible transition with respect tothe varying control parameter. Recently, explosive tran-sition from incoherent dynamics to frequency-locked stateis observed in heterogeneous Kuramoto models using at-tractive and repulsive interactions [13, 14].Besides, a new phenomenon of explosive oscillationquenching, termed as explosive death [15–19], is recentlyfound to provide a rich playground that can be exploredsuccessfully with the interdisciplinary approaches of com-plex systems. Despite its youth, this discontinuous and (a)
Email: [email protected] irreversibile transition during suppression of oscillation isenjoying widespread recognition. During this type of firstorder like transition, a mixed regime consisting of the os-cillatory state and the death state is found to coexist nearthe transition point. The coexistence of these two states isubiquitous in many physical systems [20,21], chemical sys-tems [22,23], biological systems [24] and in several numer-ical studies [25, 26] consisting of limit cycles and chaoticoscillators. Nevertheless, all these percieved studies onexplosive death [15–19] are done using static network for-malism, where the interactions among those oscillators areassumed to be invariant for the entire course of time.Here, a dynamic coupling configuration [27] is employedin this letter to inspect the explosive death phenomenon.Units in nature remain rarely isolated and the interactionamong those units are continuously updating dependingon various proximities. Information diffusion over commu-nication networks, data-packet transmission on the web,disease contagion on the social network of patients are per-haps few potential examples, which attests the fundamen-tal necessity of the temporal network approach [28, 29].Such time-varying interactions among coupled oscillatorsgive rise to fascinating collective phenomenon [30–38].p-1 a r X i v : . [ n li n . AO ] J a n . Dixit, S. Nag Chowdhury, D. Ghosh and M. D. ShrimaliEarlier, Majhi et al. [39] reported the emergence of deathstate in a temporal network of mobile oscillators. But, thefocus of our letter is completely different from that Ref.[39]. In this letter, we present the first comprehensiveanalysis of the effects of dynamic coupling configurationon the explosive death transition in an ensemble of iden-tical coupled self-excited oscillators [40–42]. This kind ofdiscontinuous transition from the oscillatory state to thedeath state using dynamic interaction is reported here forthe first time to the best of our knowledge. Moreover incase of mobile agents’ network [39, 43, 44], although themobility of mobile agents affects the collective dynamicsof the system, but the states of those oscillators situatedon the top of those agents usually do not influence theagents’ mobility. To get rid of this unidirectional affair, adynamic coupling scheme [27] is implemented here, wherethe mean state of all oscillators decides whether the inter-action among those oscillators will appear or not.The rest of the paper is organized as follows: In thenext section, we describe our model and details of theproposed dynamic coupling mechanism. After that, wesystematically investigate the transitions between the os-cillatory state and the death state in a system of coupledoscillators. To characterize the first-order like transition,we use the normalized average amplitude [45] as a mea-sure. To validate our claims, three different dynamical sys-tems (i) Van der Pol oscillator [46], (ii) FitzHugh-Nagumomodel [47], and (iii) Lorenz oscillator [48] are consideredas paradigmatic models. We also inspect numerically thathow the explosive death transition depends upon controlparameters of the systems. Using linear stability analysis,the backward transition point for this explosive transi-tion has been calculated. This backward critical couplingstrength does not depend on the size of the system andagrees completely with the numerics. Finally, we summa-rize our results and conclude. Mathematical framework. –
Here, N ( ≥
2) identi-cal nonlinear oscillators are considered and we couple themvia dynamic coupling. Each j -th oscillator ( j = 1 , , ..., N )is evolved by the set of differential equations [27],˙ X j = F ( X j ) + εβ ( H X − X j ) , (1)where ˙ X j = F ( X j ) reflects an isolated dynamics of j -th oscillator and ˙ X denotes differentiation of X with re-spect to time t . F ( X j ) : R d → R d is the vector fieldcorresponding to the d -dimensional vector X j of the dy-namical variables. ε ≥ β =diag( β , β , · · · , β d ) is a d × d diagonal matrix, wheredepending on β j ( j = 1 , , · · · , d ), the components of X j take part in the dynamic coupling. X = N (cid:80) Nj =1 X j de-notes the arithmetic mean of the state variables. The dy-namic nature of the coupling mechanism is contemplatedthrough the introduction of the step function H . Thisfunction H is chosen here as a function of the mean fieldterm X . Whenever the mean field term X lies within a pre-specified subset M ⊆ R d of the state space R d , themean field interaction is activated by setting H = 1. Onthe other hand, if X lies in the complementary subset R d \ M , then the mean field interaction is turned off bysetting H = 0. Thus, the range of H consists only twodistinct values 1 and 0 solely depending on the value of X . In absence of mean field interaction, the oscillatorsbecome uncoupled with negative self-feedback only. Inter-estingly, the oscillators are either completely independentof each other (when H = 0) or they are globally coupledwith each other (when H = 1). Hence, representing eachoscillator as a node and their interactions as edges [38],we have a time-varying network where the degree of eachnode at each time step is either 0 (when H = 0) or N − H = 1).The subset M can be defined in term of ∆ (cid:48) , whichis written in the normalized form as ∆ = ∆ (cid:48) ∆ a .Here, ∆ a is the width of the attractor alongthe clipping direction [37]. For numerical sim-ulation, ∆ ∈ [0 ,
1] assigns the closed interval[global minima of the attractor , global minima of theattractor + ( global maxima of the attractor − global minima of the attractor) × ∆] as the meanfield active region. Clearly, when ∆ tends to 0+, thenneither of those N oscillators get the suitable opportunityto interact among each other and as a consequence,only self-negative feedback is activated [49]. Besides, theglobal all-to-all interaction is established through meanfield diffusive coupling for ∆ → − . The value of ∆ = 1enables the entire phase-space as interaction active space.To demonstrate our findings, the d × d diagonal matrix β = diag(1 , , ...,
0) is considered by choosing β = 1 and β j = 0 for j = 2 , , ..., N . Since, time-independent andtime-varying diffusive interactions do not lead to stabilizethe unstable stationary point of the system [43, 44, 50–52]in general, our employed dynamic coupling is found to bebeneficial to stabilize the networked oscillators from oscil-latory behavior to death states. However, the nature ofthe emergent steady states may differ. The coupled oscil-lators may collapse into the existing stationary point ofthe uncoupled system. In the literature, this stationarypoint is known as amplitude death (AD) state [53]. Onthe other hand, it is also possible that coupled oscilla-tors may converge into a new coupling-dependent steadystate(s). This phenomenon is referred as oscillation death(OD) state [54, 55]. In the following section, we will ex-plore the dynamics of the oscillators under the proposeddynamic coupling formalism. More precisely, we investi-gate the interplay between the control parameter ∆ andthe coupling strength ε , for which the discontinuous and ir-reversible transition from the oscillatory state to the deathstate occurs in the ensemble of nonlinear oscillators. Theequation (1) is integrated using the Runge-Kutta fourth-order (RK4) method for a time of 10 units with a fixedintegration time step dt = 0 .
01 after removing enoughtransients of 10 units throughout this letter.p-2ynamic interaction induced explosive death Results. –
In this section, we discuss the explosivedeath state using the above coupling configuration. To dothis, we consider three paradigmatic systems, namely Vander Pol oscillator (limit cycle), FitzHugh-Nagumo system(excitable system) and Lorenz system (chaotic system). -2020 0.5 1 1.5 2 2.5 3 x (a)00.510 0.5 1 1.5 2 2.5 3A ε (b)-2020 10 20 30 40 x i t (c) -2020 10 20 30 40 t (d) ε =1.57 ε =1.60 ε =1.01 ε =0.99 HBHB
Fig. 1: (a) Bifurcation diagram of the VDP oscillator is plottedwith respect to ε at ∆ = 0 using the software XPPAUT [56].Here, the red solid and black dotted lines show the stable andunstable steady states, respectively. Besides, green and bluecircles represent stable and unstable periodic solutions, respec-tively. Here, HB denotes subcritical Hopf bifurcation point. (b)Forward and backward continuation of the order parameter ¯ A is shown here for ∆ = 0 . ε for N = 100 coupled VDP oscillators. Time–seriesof the system (2) near the transition point for (c) forward and(d) backward continuations is contemplated here at ∆ = 0 . b = 3 . Van der Pol Oscillator.
We study a network of N Van der Pol (VDP) oscillators [46] with dynamic meanfield interaction as follows:˙ x j = y j + ε ( H ( x )¯ x − x j ) , ˙ y j = b (1 − x j ) y j − x j , j = 1 , , · · · , N. (2)Here, x = N (cid:80) Nj =1 x j and ( x j , y j ) is the state variable ofthe j -th VDP oscillator. For positive values of the damp-ing coefficient b , the VDP oscillator possesses a limit cycle. ε stands for the interaction strength among interacted os-cillators. All initial conditions are chosen randomly within[ − , × [ − , .
0, the mean field interaction is completelyinactive and thus, only negative self-feedback plays its role among those fully disconnected oscillators. To un-derstand the qualitative changes of the system (2) underself-feedback at ∆ = 0 . ε , we plotthe bifurcation diagram in Fig. 1(a). This bifurcation di-agram depicts that a symmetry breaking pitchfork bifur-cation gives birth to an ε -dependent nontrivial OD state( x ∗ , y ∗ , − x ∗ , − y ∗ ) at ε ≈ .
33. Here, x ∗ = ± (cid:114) − bε and y ∗ = εx ∗ . This OD state gains stability through sub-critical Hopf bifurcation at ε = 1 .
0, which is denoted byHB point in Fig. 1(a). This HB point also gives birth toan unstable limit cycle. In the bifurcation diagram, thered solid and black dotted lines represent the stable andunstable steady states, while the green and blue circlessignify stable and unstable periodic orbits, respectively.A shaded region is highlighted in this figure, where ODstate coexists with a stable periodic orbit and an unsta-ble limit cycle for ε ∈ [1 , . ε = 1 .
265 andlosses its stability. Thus, OD is the only remaining stablestate beyond ε = 1 .
265 and the bistable regime disappearsas the stable limit cycle (green circle) becomes unstable.To distinguish between the oscillatory state and steadystate, the difference between the global maximum andminimum values of the attractor at a particular value ofthe coupling strength ε is calculated and this is given by[45] a ( ε ) = N − N (cid:88) j =1 [ (cid:104) x maxj (cid:105) t − (cid:104) x minj (cid:105) t ] , (3)after averaging it over the N oscillators. Here, (cid:104)· · · (cid:105) t in-dicates the sufficiently long time average. The normalizedaverage amplitude A is now treated here as an order pa-rameter [16] and it is given by A = a ( ε ) a (0) . (4)Thus, this average amplitude parameter A lies withinthe closed interval [0 , A reflects the oscillatory state of the system (1). On theother hand, the death state is indicated through A = 0. Inorder to study the effect of dynamic interaction, ∆ = 0 . A at ∆ = 0 . ε is plotted for both forward and backwardcontinuations. The values of A are calculated followingthe steps provided in the Ref. [16]. The initial value of A at ε = 0 is calculated using any random initial condition( x j (0) , y j (0)) within [ − , × [ − , ε is graduallyincreased (in the case of forward continuation) adiabati-cally upto ε = 3 .
0, i.e., the simulations are carried out forthe next increased value of ε using the final state of thestate variable as the initial condition. The same methodis also applied in the reverse direction (i.e., from ε = 3 . .
0) for the backward continuation. Fig. 1(b) revealsa sudden and discontinuous fall of the order parameter A = 0 in forward continuation at ε = 1 .
6. Similarly, thebackward continuation also shows a sharp transition from A = 0 to a finite value at ε = 1 .
0. These two transitionpoints occur at different values of ε , and thus a hysteresisarea is observed, which is the typical evocative for a first-order phase transition. The corresponding time-series ofthe system (2) near both the forward and backward tran-sitions is portrayed in Figs. 1(c) and 1(d), respectively for N = 100 oscillators. The oscillatory behavior is lost afterthe forward transition point ε = 1 . ε = 1) and the system stabilizes to thestable OD state. It indicates the explosive transition ofthe amplitude as the strength of the coupling is changed.This fact is also supported in Fig. 1(b), where the valuesof A exhibit hysteresis revealing the coexistence of ODand stable periodic attractor for a given coupling strengthwithin the interval [1 , .
6] of ε . b ε OS ODADHA
Fig. 2: Different dynamical domains of N coupled VDP os-cillators in the parameter plane ( ε − b ). The regimes markedOS, HA, OD, and AD represent the oscillatory state, hystere-sis area, oscillation death state, and amplitude death state,respectively. The other parameters are ∆ = 0 .
5, and N = 100.∆ = 0 . x ∈ [ − , The effect of the damping coefficient b on the explo-sive transition of the N = 100 coupled VDP system(2) is depicted in Fig. 2. The phase diagram ( ε − b ) isdrawn at ∆ = 0 . ε adiabat-ically in both forward and backward directions. In thisfigure OS , AD , and OD describe oscillatory, amplitudedeath, and oscillation death states respectively. One canobserve clearly that the system stabilized at AD state( x j = 0 , y j = 0, j = 1 , , ..., N ) with increasing couplingstrength via second-order transition for b < .
0. A dif-ferent scenario is observed for b > .
0, where the coupledsystem is stabilized to OD states via the first-order liketransition. During this explosive death transition, a hys-teresis region is found, where OS and OD solutions co-exist. The co-existence of OS and OD states is denotedas HA in the Fig. 2. Interestingly, we can see that incre-ment of parameter b helps to enhance the hysteresis areain parameter space.The backward transition point for this explosive transi-tion can be computed using linear stability analysis around the OD states of the system (2) at H ( x ) = 0. The ODstates for this system (2) at H ( x ) = 0 are x i = x ∗ , y i = y ∗ , ∀ i = 1 , , · · · , N , where x ∗ = ± (cid:113) − bε and y ∗ = εx ∗ .For this stationary state, Jacobian matrix can be writtenas the block diagonal matrix J ⊕ J ⊕ J · · · ⊕ J ( N times),where J is the Jacobian matrix of the isolated system withonly negative self-feedback at ( x ∗ , y ∗ ) given by J = (cid:34) − ε − bε ε (cid:35) and the corresponding eigenvalues are λ , = 1 − ε ± √ ε − bε + 6 ε + 12 ε . (5)It gives the Hopf bifurcation point through which theOD solutions are stabilized at ε HB = 1. A close inspec-tion of the OD states and the corresponding eigenvalueanalysis also help to detect the bifurcation point ε P B = 1 b ,where the pitchfork bifurcation occurs. These Hopf bifur-cation point ε HB = 1 and the pitchfork bifurcation point ε P B = 1 b match perfectly with our numerically found bi-furcation diagram given in Fig. 1(a). Note that the Hopfbifurcation point ε HB = 1 is not only independent of thenumber of oscillators N , but also it does not depend ex-plicitly on ∆. To validate our analytical finding and tounderstand the role of ∆, we numerically investigate thephase diagram in the parameter plane ( ε − ∆) for b = 3 . ε is portrayed in Fig. 3.The backward transition point fits exactly with our ana-lytically calculated ε HB = 1. ∆ ∼ .
71 creates two distin-guished zones in this parameter space. For ∆ ≥ .
71, thesystem always settles down to stable oscillation for b = 3 . ε . For∆ ≥ .
71, the active interaction subspace M always con-tain these OD states and thus, the mean field couplingalways plays its role. Hence, the OD states of the system(2) at H ( x ) = 0 do not get the opportunity of being sta-bilized. For ∆ < .
71, an interval of ε is found, whereOD and oscillatory state coexist. This bistable region ishighlighted as HA in Fig. 3. In a previous study of meanfield interaction among identical VDP oscillators [16], anintensity of mean field Q ∈ [0 ,
1] is found to play an deci-sive role in the explosive death. Here, we have considereddynamic mean field interaction with control parameter ∆instead of density parameter Q , where interaction is onin a certain pre-defined subset of the state-space, other-wise remains off. This type of occasional interaction isrelevant in various circumstances including transmissionsof biological signals between synapses and the communi-cations of ant colonies in the processing of migration, aswell as the seasonal interactions between predator–preyin the ecosystem [57]. Robotic communication [58] andwireless communication systems are prominent specimensp-4ynamic interaction induced explosive deathof such applications, where continuous interaction amongthe units is not always feasible. Fig. 3: Phase diagram of N = 100 coupled VDP oscillatorsis represented here in the parameter plane ( ε − ∆) for b =3 .
0. The bistable region is highlighted as HA separating theoscillatory state (OS) and the stationary state (OD) regime.The coupling strength ε is varied adiabatically in both forwardand backward directions to obtain this figure. The forwardand backward transition points differ resulting in the explosivedeath phenomena. FitzHugh-Nagumo excitable system.
To concur theuniversality of the explosive death in the coupled sys-tem (1) under proposed dynamic interactions, we adopt amore realistic neuronal model, namely, FitzHugh-Nagumo(FHN) excitable system [47] for our study. The dynamicalequation for the network consisting of FHN neurons underconsidered dynamic framework can be written as,˙ x j = x j ( a − x j )( x j − − y j + ε ( H ( x )¯ x − x j ) , ˙ y j = bx j − cy j . (6)Here, x j represents the trans-membrane voltage and thevariable y j should model the time dependence of severalphysical quantities related to electrical conductances ofthe relevant ion currents across the membrane. In theFHN model, x j behaves as an excitable variable and y j acts as the slow refractory variable. We fix the parame-ters a = − . , b = 0 . c = 0 .
02. The initialconditions are chosen from [ − . , . × [ − . , . ε − ∆) as shown in Fig. 4(a). Interestingly,we report an explosive transition from oscillatory state toamplitude death state in certain parameter region. Tothe best of our knowledge, this novel explosive amplitudedeath phenomenon is reported for the first time in thisletter. The earlier all perceived results on explosive death[15–19] is concerned only with the explosive transitionfrom oscillatory states to OD states. Figure 4 is drawnmaintaining the same adiabatic process as discussed ear-lier. Just like our earlier observation with VDP oscillators,here we also find the hysteresis phenomenon, which is in-dicated in Fig. 4(a) as HA. The only difference is in thecase of VDP oscillators, the HA regime contains oscillatorystates and OD state, while in the case of FHN model, theAD state coexists with oscillatory states. The backwardtransition point is again found to be independent of ∆. Tocalculate this backward transition point, the eigen values of the N × N Jacobian matrix diag ( J, J, ..., J ) around theAD state ( x j = 0 , y j = 0 , j = 1 , , ..., N ) of the system(6) is calculated at H = 0. Here, J is the Jacobian of theisolated FHN model at (0 , J = (cid:20) − a − ε − b − c (cid:21) . ∆ OSHA AD (a)00 0.005 0.01 0.015 0.02 x ε (b) HB HA (a) Fig. 4: (a) Phase diagram of N = 100 coupled FHN sys-tem (6) in the ( ε − ∆) plane. The other parameters are a = − . , b = 0 . c = 0 .
02. Regions OS, HA, andAD indicate the oscillatory state, hysteresis area, and ampli-tude death state, respectively. In both forward and backwardtransitions, the system is integrated in an adiabatic fashion asdiscussed throughout the letter. The coexistence of AD and os-cillatory states is responsible for the explosive transition. (b)Bifurcation diagram of the FHN oscillator at H = 0. AD be-comes stable at ε = 0 .
005 and this is shown through red line.However, a stable periodic orbit (green circle) and an unstableperiodic orbit (blue circle) coexist with AD for ε ∈ [0 . , . ε = 0 . The Hopf bifurcation occurs at ε HB = − ( a + c ), wherethe real part of the eigen values of J become negative.For our choice of parameter values, the backward transi-tion point is given by ε HB = 0 . ≤ ∆ ≤ .
3. For ∆ > .
3, the system (6) exhibits theoscillatory behavior (OS) only. These trends are similarto those observed for coupled VDP oscillators suggestinggenerality of the explosive death phenomena.In Fig. 4(b), the bifurcation of the FHN oscillators isshown with respect to ε . Here, Hopf bifurcation point isdenoted by HB, which matches perfectly with our analyt-ically calculated value. The red and black lines representstable and unstable steady state, while the green and bluecircles represents stable and unstable limit cycle, respec-tively. The unstable origin stabilizes at ε = 0 . ε ∈ [0 . , . ε = 0 .
01, the sta-ble periodic orbit collides with the unstable periodic orbitand loses their stability and eventually disappears. Theorigin is the only stable state for ε > .
01 and 0 ≤ ∆ ≤ . Lorenz system.
To speculate this irreversible mech-anism, we now consider the case of N coupled chaoticLorenz oscillators [48] interacting via dynamic mean fieldinteraction. The mathematical equations representing thenetwork dynamics are,˙ x j = σ ( y j − x j ) + ε ( H ( x )¯ x − x j ) , ˙ y j = ( ρ − z j ) x j − y j , ˙ z j = x j y j − βz j , (7)where j = 1 , , ..., N is the index of oscillators. The systemparameters are chosen as σ = 10, ρ = 28, and β = 83 forwhich an individual system is in chaotic state. Randominitial conditions are chosen within [ − , × [ − , × [ − , Fig. 5: Interplay of ε and ∆ generates different dynamical do-mains including the oscillatory state (OS), hysteresis area (HA)and oscillation death states (OD). The simulation is carried outfor N = 100 coupled Lorenz oscillators with adiabatic increa-ment of ε both in forward and backward directions. Clearly,the backward transition point ( ε = 1 . By adiabatically changing ε in both directions withstep size δε = 0 .
01, we are able to capture the differentemerging behaviors of coupled Lorenz oscillators in the( ε − ∆) parameter space (Fig. 5). The Jacobian matrix at H = 0 corresponding to the OD states x i = x ∗ , y i = y ∗ , z i = z ∗ ( i = 1 , , ..., N ), where x ∗ = ± (cid:114) βσ ( ρ − − βεσ + ε , y ∗ = ( σ + ε ) x ∗ σ and z ∗ = x ∗ y ∗ β can be represented by a N × N block circulant matrix circ( J, , · · · , ) with J = − σ − ε σ ρ − z ∗ − − x ∗ y ∗ x ∗ − β . Negativity of the real part of all eigen values of thisblock circulant matrix determines the backward criticalpoint at ε = 1 . N , i.e., the number of oscillators present in the system. As we vary ε in the both directions with step size δε = 0 .
01 adiabatically, a first order like phase transitionwith hysteresis is found within the interval ε ∈ [1 . , . ∈ [0 , . ε − ∆) parameter space.However, one should notice that the backward transitionpoint remains unchanged with variation of ∆. Conclusion. –
The present work has demonstratedthat apart from the continuous interaction, dynamic in-teraction leads to an explosive death transition in cou-pled nonlinear oscillators. We have studied explosive firstorder like dynamical transition in nonlinear oscillators in-teracting through the state-space dependent dynamic cou-pling. Using limit-cycle and chaotic oscillators interactingthrough dynamic interaction, we have shown that the sys-tem possesses two distinct states, viz. the oscillatory stateand the steady state and the transition between these twostates is irreversible, abrupt, and associated with the pres-ence of a hysteresis region where such states co-exist. Thehysteresis region crucially depends on the interaction re-gion in state-space as well as the coupling strength. Theanalysis performed here may help to understand the effectof dynamic interaction leading to the interesting collectivedynamical behavior of coupled systems and its relevancewith the occurrence of such states in many natural sys-tems. ∗ ∗ ∗
MDS acknowledges financial support (Grant No.EMR/2016/005561 and INT/RUS/RSF/P-18) from De-partment of Science and Technology (DST), Governmentof India, New Delhi. S.N.C. would like to acknowledge theCSIR (Project No. 09/093(0194)/2020-EMR-I) for finan-cial assistance.
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