Transitions among the diverse oscillation quenching states induced by the interplay of direct and indirect coupling
aa r X i v : . [ n li n . C D ] N ov Transitions among the diverse oscillation quenching states induced by the interplay ofdirect and indirect coupling
Debarati Ghosh and Tanmoy Banerjee ∗ Department of Physics, University of Burdwan, Burdwan 713 104, West Bengal, India. (Dated: July 2, 2018; Received :to be included by reviewer)We report the transitions among different oscillation quenching states induced by the interplayof diffusive (direct) coupling and environmental (indirect) coupling in coupled identical oscillators.This coupling scheme was introduced by Resmi et al . [Phys. Rev. E, 84, 046212 (2011)] as a generalscheme to induce amplitude death (AD) in nonlinear oscillators. Using a detailed bifurcation analysiswe show that in addition to AD, which actually occurs only in a small region of parameter space,this coupling scheme can induce other oscillation quenching states, namely oscillation death (OD)and a novel nontrvial AD (NAD) state, which is a nonzero bistable homogeneous steady state; moreimportantly, this coupling scheme mediates a transition from AD to OD state and a new transitionfrom AD to NAD state. We identify diverse routes to the NAD state and map all the transitionscenarios in the parameter space for periodic oscillators. Finally, we present the first experimentalevidence of oscillation quenching states and their transitions induced by the interplay of direct andindirect coupling.
PACS numbers: 05.45.Xt
I. INTRODUCTION
The suppression of oscillation is an important topicof research in the context of coupled oscillators and hasbeen studied in diverse fields such as physics, biology,and engineering [1]. Two distinct types of oscillationquenching processes are there, namely amplitude death(AD) and oscillation death (OD). The AD state is de-fined as a stable homogeneous steady state (HSS) thatarises in coupled oscillators under some parametric con-ditions [2],[3]; in the case of AD, oscillation is suppressedas all the coupled oscillators attain a common steadystate that was unstable in the uncoupled condition. Inthe case of OD, oscillators populate coupling dependentstable inhomogeneous steady states (IHSS) that are re-sulted from the symmetry breaking bifurcation; e.g., inthe case of two coupled oscillators, in the OD state oscil-lation is suppressed and they attain two different newlycreated coupling dependent steady states. In the phasespace OD may coexist with limit cycle oscillation. WhileAD is observed and characterized as a control mechanismto suppress oscillation in Laser application [4], neuronalsystems [5], electronic circuits [6], etc., the OD on theother hand is relevant in many biological and physicalsystems such as synthetic genetic oscillator [7], neuralnetwork [8], laser systems [9], etc (see [1] for an elabo-rate review on OD).In the earlier works on oscillation suppression, no cleardistinction between AD and OD was emphasized untilthe pioneering research by Koseska et al. [10], where itwas shown that the AD and OD are two dynamically dif-ferent phenomena, both from their origin and manifesta-tion. Ullner et al. [11] observed OD in genetic network ∗ The author to whom correspondence should be addressed:[email protected] under phase-repulsive coupling with realistic biologicalparameters and proved its importance in biological net-work. Later, it has been reported that most of the cou-pling schemes, which were believed to induce AD onlycan induce OD, also: Ref.[12] proved that the dynamicand conjugate coupling can induce OD; Refs. [13] and[14] reported the occurrence of OD induced by mean-fieldand repulsive coupling, respectively. More significantly,Refs. [10, 12–14] show that these coupling schemes caninduce an important transition phenomenon, namely thetransition from AD to OD that resembles the Turing-type bifurcation in spatially extended systems, which isbelieved to have connection with the phenomenon of cel-lular differentiation [15]. Further, in the recent studiesnew oscillation quenching state [13] and also new routesto oscillation quenching states [14] are reported. More re-cently, in Ref. [16] the important rigorous conditions forthe onset of AD and OD in a system of identical Stuart-Landau oscillators has been reported. Thus, search forthe transitions between different oscillation quenchingstates and identification of new oscillation quenching phe-nomena are an active area of recent research on coupledoscillators.In this paper we report the occurrence of the AD toOD transition and a new transition from AD to a novelnontrivial amplitude death (NAD) state induced by thesimultaneous presence of diffusive coupling (i.e. directcoupling) and environmental coupling (i.e. indirect cou-pling). To the best of our knowledge, the AD to NADtransition has not been observed earlier for any other cou-pling configuration. This direct-indirect coupling schemewas originally proposed by Resmi et al. [17] as a gen-eral scheme for inducing AD in coupled oscillators (laterextended for a network of oscillators [18]), and attractsimmediate attention due to its ease of implementationand generality to induce AD in any synchronizable units.Although diffusive coupling is widely studied in the con-text of synchronization but environmental coupling is aless explored topic; it is particularly important in biologi-cal systems, e.g., populations of cells in which oscillatoryreactions are taking place interact with each other viachemicals that diffuse in the surrounding medium [19].Since the authors of Ref.[17] rely mainly on time inte-gration of the dynamical equations and linear stabilityanalysis, thus the complete dynamical features inducedby this coupling scheme were not explored and only theAD state was identified and characterized there.In the present paper we employ a detailed bifurcationanalysis to show that apart from AD, which actually oc-curs in a small zone of parameter space, there exists di-verse oscillation suppression states, namely OD and anewly observed nontrivial AD (NAD) state. This NADstate has not only a nonzero homogeneous steady statebut, more significantly, in this state the system becomes bistable (will be elaborated later in this paper). We ex-plore the properties of the NAD state and identify threedistinct routes to NAD. More importantly, we recognizedifferent transition scenarios, e.g. AD to OD transition,and a novel transition from AD to NAD state. Theimportance of this AD-NAD transition lies in the factthat it gives the evidence of direct transition from themono-stability (AD) to bistability (NAD) in dynamicalsystem that may improve our understanding of the ori-gin of bistability arises in biological processes [20][21].In this study we choose the paradigmatic Stuart-Landauand Van der Pol oscillators and derive the range of cou-pling parameters where the transitions are occurred. Fi-nally, we report the first experimental evidence of theNAD state and the AD-NAD transition; also, we exper-imentally observe AD, OD and AD-OD transitions thatsupport our theoretical findings.
II. GENERIC OSCILLATORS WITHDIRECT-INDIRECT COUPLINGA. Direct-indirect coupled Stuart-Landauoscillators
At first, we consider two identical Stuart-Landau oscil-lators interacting directly through diffusive coupling andindirectly through a common environment s , which ismodeled as a damped dynamical system [17][22]. Math-ematical model of the coupled system is given by˙ x , = P , x , − ωy , + d ( x , − x , ) + ǫs, (1a)˙ y , = ωx , + P , y , , (1b)˙ s = − ks − ǫ ( x + x )2 . (1c)Here, P i = 1 − x i − y i ( i = 1 , ω . The diffusive coupling strength isgiven by d , and ǫ is the environmental coupling strengththat controls the mutual interaction between the sys- tems and environment. k represents the damping fac-tor of the environment ( k > , , , , F IHSS ≡ ( x † , y † , − x † , − y † , x † = − ωy † dy † + ω , y † = ± q ( d − ω )+ √ d − ω d , and F NHSS ≡ ( x ∗ , y ∗ , x ∗ , y ∗ , s ∗ ), where x ∗ = − kωy ∗ kω + ǫ y ∗ , y ∗ = ± q ( ǫ − kω )+ √ ǫ − k ω ǫ and s ∗ = − ǫx ∗ k . Thefixed points F IHSS give the inhomogeneous steady states(IHSS); note that it depends only upon d and indepen-dent of ǫ and k ; stabilization of IHSS results in OD.The fixed points F NHSS represent nontrivial homoge-neous steady states (NHSS), stabilization of which givesrise to a novel nontrivial amplitude death (NAD) state,which is a nonzero bistable state. It can be seen that thisNHSS depends upon ǫ and k , and independent of d .The characteristic equation of the system at the trivialHSS, (0 , , , , λ + P ′ T λ + P ′ T )( λ + P T λ + P T λ + P T ) = 0 , (2)where P ′ T = 2( d − P ′ T = (1 − d + ω ), P T = ( k − P T = (1 + ω + ǫ − k ) and P T = ( k + kω − ǫ ).Since Eq.(2) is a fifth-order polynomial, it is difficultto predict bifurcation points from the eigenvalues; thus,we derive the bifurcation points from the properties ofcoefficients of the characteristic equation itself using themethod stated in Ref.[23]. A close inspection of the non-trivial fixed points reveals that the IHSS, F IHSS , appearsthrough a pitchfork bifurcation at d P B = 1 + ω . (3)Also, the NHSS, F NHSS , appears through a pitchforkbifurcation at ǫ P B = p k (1 + ω ) . (4)Further, from Eq.(2) we derive the following values of d (by setting P ′ T = 0 [23]) and ǫ (by setting P T P T − P T = 0 [23]) where Hopf bifurcations (of trivial HSS)occur, respectively: d HB = 1 , (5) ǫ HB = r k − + 2 ω k − , (6)with an additional condition: ω >
1. We find that allthe eigenvalues at the trivial HSS have negative real partand thus gives rise to AD when d HB < d < d P B , k > ǫ HB < ǫ < ǫ P B . To corroborate our analysis wecompute the two parameter bifurcation diagram in ǫ − d space using the XPPAUT package [24]. Figure.1 showsthis for k = 4 (i.e. k >
2) and ω = 2: we observethat the parameter space is divided into different zonesseparated by two horizontal lines at d HB and d P B , andtwo vertical lines at ǫ HB and ǫ P B . Also, in the ǫ − d FIG. 1. (Color online) Bifurcation diagram in ǫ − d space forStuart-Landau oscillator using XPPAUT ( ω = 2 and k = 4).Horizontal arrow shows AD-NAD transition and vertical ar-row indicates AD-OD transition. AD occurs in a small regionwhere ǫ HB < ǫ < ǫ PB and d HB < d < d PB (for a detaileddescription see text). parameter space the area of the rectangular zone whereAD occurs is: ( d P B − d HB )( ǫ P B − ǫ HB ).At this point we identify the following two distincttransition scenarios (for k > ǫ HB < ǫ < ǫ P B ,variation of d gives rise to a transition from limit cycle(LC) to AD through Hopf bifurcation (HB1) at d HB , andfrom AD to OD through pitchfork bifurcation (PB1) at d P B (see Fig.1). This is also shown in one dimensional bi-furcation diagram [Fig.2(a)] at an exemplary value ǫ = 4and k = 4 where we see that AD occurs at d HB = 1 andOD emerges at d P B = 2 .
5; the OD state is representedby x = − x = x † . (ii) If d HB < d < d P B , varia-tion of ǫ induces a transition from LC to AD throughHopf bifurcation (HB2) at ǫ HB , and from AD to NADthrough pitchfork bifurcation (PB2) at ǫ P B (see Fig.1);Fig.2(b) shows this for d = 1 . k = 4: here ADoccurs at ǫ HB = 2 .
943 and NAD at ǫ P B = 4 . bistable , i.e., depend-ing upon the initial conditions it may attain two differentsteady states, either x = x = x ∗ or x = x = − x ∗ .We also examine the change in environment in the abovetwo cases. It is interesting to note that in the OD statesince x = − x , the effect of the system on the environ-ment vanishes [see Eq.(1c)] and s remains in its trivialzero steady state [Fig.2(c), left panel]. But in the NADstate, since x , = ± x ∗ the effect of the system on theenvironment is very much present there, thus like theNAD state the environment s also becomes bistable: de-pending upon initial conditions it attains either s + or s − state beyond PB2 [Fig.2(c), right panel].Outside these horizontal and vertical rectangular re-gions [of width ( d P B − d HB ) and ( ǫ P B − ǫ HB ), respec-tively] we identify several other bifurcation curves that S d -0.2-0.100.10.2 x , x HB1
PB1 AD OD x x (cid:1) =4 -0.2-0.100.10.2 0 2 4 6 8 10 (cid:1) x , x x -x AD NAD
PB2HB2 d=1.5 (a)(b) (cid:1)
HB1
HB2 PB2 (cid:1) =4 d=1.5 (c) s+s- d -0.200.2 S FIG. 2. (Color online) (a) Transition from LC to AD (atHB1), and AD to OD (at PB1) with the variation of d ( ǫ = 4, k = 4). Black line: unstable steady state, deep gray (vi-olet) line: OD state, light gray (golden) line: NAD state,(green) solid circles: stable limit cycle. (b) Transition fromLC to AD (at HB2), and AD to NAD (at PB2) with thevariation of ǫ ( d = 1 . k = 4). Here, the NAD state isa bistable state: depending upon initial conditions one gets x , = ± x ∗ . (c) Variation of the environment, s : (left panel)in the OD state ( ǫ = 4) s attains the stable zero steady statebeyond HB1; (right panel) between HB2 and PB2, s attainsthe stable zero steady state but beyond PB2 the environment s becomes bistable: depending upon initial conditions s mayattain either s + or s − state. mark the distinct regions of occurrence of oscillation sup-pression states and their coexistence. For this purpose atfirst we consider the characteristic equation correspond-ing to the fixed point F IHSS ≡ ( x † , y † , − x † , − y † ,
0) whichis given by( λ + P ′† N λ + P ′† N )( λ + P † N λ + P † N λ + P † N ) = 0 , (7)where P ′† N = 2( d −
1) + 4 x † + 4 y † , P ′† N = (1 − x † − y † )(1 − x † − y † − d ) + ω − x † y † , P † N = 4( x † + y † ) − k , P † N = 1 + ω + ǫ − k + ( x † + y † ) { k − x † + y † ) } . P † N = k { − x † + y † ) + 3( x † + y † ) + ω } − ǫ (1 − x † − y † ). From Eq.(7) we find that ǫ and k appear only in the term ( λ + P † N λ + P † N λ + P † N ),i.e., they control only three eigenvalues.We find the characteristic equation corresponding tothe fixed point F NHSS ≡ ( x ∗ , y ∗ , x ∗ , y ∗ , s ∗ ) as:( λ + P ′∗ N λ + P ′∗ N )( λ + P ∗ N λ + P ∗ N λ + P ∗ N ) = 0 , (8)where P ′∗ N , P ′∗ N , P ∗ N , P ∗ N and P ∗ N are identical to the P ′ and P values of the above paragraph with the † signis now replaced by the ∗ sign. FIG. 3. (Color online) (a) Variation of ǫ ( d = 0 . x , = ± x ∗ ) appears through subcritical Hopf bifur-cation at HBS2. grey (red) line: stable steady state. Open(blue) circles: unstable limit cycle. (b) Variation of ǫ ( d = 4):The OD arises at HBS1 through subcritical Hopf bifurcation;a new NAD state emerges at PBS1 through subcritical pitch-fork bifurcation and coexists with OD [gray (green) filled re-gion]. (c) Variation of d ( ǫ = 2): The OD state appearsthrough subcritical Hopf bifurcation at HBS1. (d) Variationof d ( ǫ = 8): The OD state appears through subcritical pitch-fork bifurcation at PBS2 that coexists with the NAD state[gray (green) filled region]. For d < d HB , the NAD state appears through Hopf bi-furcation at HBS2 (Fig.1) which is obtained from Eq.(8)by putting P ′∗ N = 0 [23] and is given by ǫ HBS = s k ( d + 1) + 4 kω d + 1) . (9)This is shown in Fig.3(a) for d = 0 .
5. We can see thatthe NAD state here coexists with an unstable limit cycle.For d > d
P B , three more bifurcation curves appear,namely HBS1, PBS1 and PBS2 (Fig.1). We derive thelocus of all the curves in the ǫ − d space. First, for ǫ < ǫ HB , the OD appears through subcritical Hopf bi-furcation at HBS1 whose locus is derived from Eq.(7) byputting P † N P † N − P † N = 0 as, ǫ HBS = s G [ ω + ( k − + G (3 G + 4( k − k − x † + G , (10)with G = ( x † + y † ) and G = (1 − G ). This isshown in Fig.3(b) for d = 4 ( d > d P B ): between HBS1and HB2 the OD state coexists with an unstable limitcycle, between HB2 and PBS1, OD is the only stablestate. Beyond PBS1, OD is accompanied by a NAD statethat is created by a subcritical pitchfork bifurcation (atPBS1).For ǫ > ǫ
P B (and d > d
P B ), we have two subcriticalpitchfork bifurcation curves, PBS1 and PBS2, locus ofwhich are derived by putting P ′∗ N = 0 and P † N = 0 in FIG. 4. (Color online) Phase diagram in ǫ − k space ( d = 1 . k ≤ Eq.(8) and Eq.(7), respectively: d P BS = 1 + G (3 G −
4) + ω − G − y ∗ ) , (11a) ǫ P BS = s K [1 + ω + G (3 G − − G − y † . (11b)Here G = x ∗ + y ∗ . Between the curves PBS1 andPBS2, OD and NAD states coexist (see Fig. 1). Notethat bifurcation curves given by Eq.(9)–Eq.(11) dependupon both d and ǫ (for a given ω and k ). Figure 3(c-d)show the variation of the system dynamics for variable d .For ǫ = 2, OD emerges along with an unstable limit cycleat HBS1; for ǫ = 8 ( > ǫ P B ), NAD and LC coexists upto d = d HB (HB1), and between HB1 and PBS2, NAD isthe only stable state, then beyond PBS2, NAD and ODcoexist. Here the OD state emerges through subcriticalpitchfork bifurcation (at PBS2).It is important to note that, unlike conventional ADthat has only two routes, namely Hopf and saddle-nodebifurcation [2], we identify three distinct routes to NADstate: (i) supercritical pitchfork bifurcation route whichoccurs at ǫ P B for d HB < d < d P B . (ii)
Subcritical pitch-fork bifurcation route (PBS1) that occur for ǫ > ǫ
P B and d > d
P B . (iii)
Subcritical Hopf bifurcation route thatoccurs for d < d HB along the HBS2 curve whose locusis given by Eq.(9). Although the subcritical pitchforkbifurcation route to a different nontrivial AD state wasobserved earlier in the mean-field coupled oscillators [13]the other two routes to NAD were not observed earlierfor any other coupling schemes.We also examine the effect of environment (i.e. k ) inthe oscillation quenching and transition scenarios. FromEq.(4) and Eq.(6) it is clear that ǫ HB and ǫ P B collide at k = 2, thus the AD region (and hence AD-OD and AD-NAD transitions) vanishes for k ≤
2. Fig.4 shows this inthe ǫ − k parameter space for a fixed d (we choose d = 1 . d HB < d < d P B ). We can see that for k > ǫ ; also the zoneof AD region surrounded by HB2 and PB2 curves getsnarrower for decreasing k , and vanishes at k = 2, whichis in accordance with our earlier result. Since d HB and d P B both are independent of k , thus variation of k doesnot affect the dynamics that is entirely controlled by d . B. Van der Pol oscillators with direct-indirectcoupling
Next, to verify the generality of the above oscillationquenching transitions in periodic oscillator, we considertwo identical Van der Pol oscillators under the same cou-pling scheme; mathematical model of the coupled systemis given by ˙ x , = y , + d ( x , − x , ) + ǫs, (12a)˙ y , = a (1 − x , ) y , − x , , (12b)˙ s = − ks − ǫ ( x + x )2 . (12c)From Eq.(12) we can see that the origin (0 , , , ,
0) isthe homogeneous steady state (HSS), and also we havetwo coupling dependent nontrivial fixed points, F IHSS ≡ ( x † , y † , − x † , − y † , x † = y † d , y † = ± q d − da and F NHSS ≡ ( x ∗ , y ∗ , x ∗ , y ∗ , s ∗ ), where x ∗ = ky ∗ ǫ , y ∗ = ± q ǫ ( aǫ − k ) ak and s ∗ = − ǫx ∗ k .The characteristic equation of the system at the fixedpoint (0 , , , ,
0) is given by,( λ + Q ′ T λ + Q ′ T )( λ + Q T λ + Q T λ + Q T ) = 0 , (13)where Q ′ T = (2 d − a ), Q ′ T = (1 − ad ), Q T =( k − a ), Q T = (1 + ǫ − ka ), Q T = ( k − ǫ a ). Likethe Stuart-Landau oscillator case, F IHSS and F NHSS emerge through pitchfork bifurcation at d P B = a and ǫ P B = q ka , respectively. Also, using the similar argu-ments used in the previous subsection, we derive twoHopf bifurcation points from Eq.(13) as d HB = a , (14) ǫ HB = r a (1 + k ) − ka k . (15)Two-parameter bifurcation diagram using XPPAUT isshown in Fig.5(a) (for k = 1, a = 0 . ǫ P B and ǫ HB collide at k = a , thus AD region vanishes for k ≤ a .We identify the following two distinct transition scenarios(for k > a ): (i) If ǫ HB < ǫ < ǫ P B , variation of d givesrise to a transition from limit cycle (LC) to AD (at d HB ),and from AD to OD (at d P B ). (ii) If d HB < d < d P B ,variation of ǫ induces a transition from LC to AD (at ǫ HB ), and from AD to NAD (at ǫ P B ).Outside this region, we have identified other bifurca-tion curves which are qualitatively equivalent to that ofthe Stuart-Landau oscillator. The characteristic equa-tions for the nontrivial fixed points is given by( λ + Q ′ iN λ + Q ′ iN )( λ + Q iN λ + Q iN λ + Q iN ) = 0 , (16)where Q ′ iN = a ( x i −
1) + 2 d , Q ′ iN = 2 ax i y i + 1 − da (1 − x i ), Q iN = k + a ( x i − Q iN = 1 + ǫ + FIG. 5. (Color online)(a) Two-parameter bifurcation diagramin ǫ − d space for Van der Pol oscillator ( k = 1). AD-NADand AD-OD transitions are indicated with arrow. (b) Two-parameter bifurcation diagram in ǫ − k space ( d = 0 . k at ǫ = 1 . a =0 . ax i y i − ka (1 − x i ), Q iN = k (1 + 2 ax i y i ) − ǫ a (1 − x i ).Where i will be replaced by † and ∗ for the nontrivialfixed points F IHSS ≡ ( x † , y † , − x † , − y † ,
0) and F NHSS ≡ ( x ∗ , y ∗ , x ∗ , y ∗ , s ∗ ), respectively. The locus of different bi-furcation curves are derived from Eq.(16) using the samemethod discussed in the coupled Stuart-Landau oscilla-tors and they are given by ǫ HBS = r k d , (17a) d HB = (1 − k ) + p ( k − + k (8 a − kǫ )8 a − kǫ , (17b) ǫ P BS = s k + p k (1 + 16 ad )4 a , (17c) d P BS = k + √ k + 8 akǫ ak . (17d)All the oscillation quenching scenarios remain qualita-tively same as that of the Stuart-Landau (SL) oscillatorexcept now the HBS1 curve of SL case is replaced byHB3 curve. We also show the two-parameter bifurcationdiagram in ǫ − k space [Fig.5(b)], which depicts that for k < a no AD is possible (here a = 0 . d = 0 . k at ǫ = 1 . III. EXPERIMENT
We experimentally verify the occurrence of all the tran-sitions predicted in the above section. For this purposewe implement an electronic circuit that mimics the cou-pled Van der Pol oscillators [25] with the direct-indirectcoupling given by Eq. (12); Fig. 6 depicts the said cir-cuit. In this circuit diagram the sub unit associated withthe op-amp “Ad” acts as an differential amplifier andemulates the diffusive coupling part. The sub circuit ofthe op-amp “AS” mimics the environmental part whosedamping parameter ( k ) is controlled by the resistor R k .The voltage equation of the circuit of Fig. 6 can be writ-ten as: CR dV xi dt = V yi + RR d (cid:18) V xj − V xi (cid:19) + RR ǫ V s , (18a) CR dV yi dt = R R a (cid:18) V α − V xi (cid:19) V yi − V xi , (18b) CR dV s dt = − RR k V s − RR ǫ (cid:18) V xi + V xj (cid:19) . (18c)Here, i, j = 1 , i = j . Equation (18) is normal-ized with respect to CR , and thus now becomes equiva-lent to Eq. (12) for the following normalized parameters:˙ u = dudτ , τ = tCR , ǫ = RR ǫ , d = RR d , k = RR k , a = R R a ,10 V α = 1, x i = V xi V sat , y i = V yi V sat , and s = V s V sat . Thus, R d determines the diffusive coupling strength d and R ǫ determines the environmental coupling strength ǫ . V sat is the op-amp saturation voltage. To make our circuitequivalent to Eq. (12) we take V α = 0 .
1; also, a = 0 . R a = 200 Ω) is taken to match with the parametervalue used in Fig.5; V α and a determine the amplitudeand shape of the limit cycle. Also, we choose C = 10 nFand R = 10 kΩ that determine the frequency of individ-ual oscillations, which is, in this case, 1 .
54 kHz (for theuncoupled case), and are shown in Fig. 6 (inset). Weexperimentally verify that the choice of V α , a and τ (i.e., C and R ) does not affect the qualitative features of the coupled dynamics .In this experiment at first we take R d = 30 kΩ( d = R/R d = 0 .
33, i.e., d HB < d < d P B ) and k = 1(i.e. R k = 10 kΩ): we observe a continuous transitionfrom limit cycle to AD, and AD to NAD for decreasing R ǫ (i.e. increasing ǫ ). We notice that in the limit cy-cle state two systems are in complete synchronized state.In Fig. 7 (a), using the experimental snapshots of thewaveforms [with a digital storage oscilloscope (TektronixTDS2002B, 60 MHz, 1 GS/s)], we demonstrate differentdynamical behaviors for the following parameter values:limit cycle at R ǫ = 11 . R ǫ = 9 kΩ, and NADfor R ǫ = 2 .
75 kΩ. Two bistable NAD states V x , and − V x , at R ǫ = 2 .
75 kΩ are found by random parame-ter sweeping around R ǫ = 2 .
75 kΩ. For the comparisonpurpose, we also show the numerical results (using thefourth-order Runge-Kutta method, 0 .
01 step size) tak-ing ǫ and d values that are equivalent to R ǫ and R d , FIG. 6. (Color online) Experimental circuit diagram of di-rectly and indirectly coupled VdP oscillators. AS, Ad, A1-A5 op-amps are realized with TL074 (JFET). All the unla-beled resistors have value R = 10 kΩ. C=10 nF, R a = 200Ω, R X = 10 kΩ, R k = 10 kΩ , V α = 0 . ±
12 v power suppliesare used; resistors (capacitors) have ±
5% ( ± ⊗ sign indi-cates squarer using AD633. Inset (in the middle part) showsthe oscillation from the uncoupled VdP oscillators: uppertrace (yellow) V x , lower trace (cyan) V x ( y -axis:10 v/div, x -axis:500 µ s/div). respectively: for d = 0 .
33, Fig.7 (b) shows the limit cy-cle for ǫ = 0 .
85, AD for ǫ = 1 .
11, and bistable
NAD for ǫ = 3 .
64 (using proper initial conditions). Next, we take R ǫ = 8 .
33 kΩ (i.e., ǫ = R/R ǫ = 1 . ǫ HB < ǫ < ǫ HB )and observe a continuous transition from limit cycle toAD, and AD to OD for decreasing R d . Fig. 7 (c), andFig. 7 (d), respectively demonstrate experimental andnumerical results: limit cycle (anti-phase synchronized)for R d = 43 kΩ ( d = 0 . R d = 30 .
50 kΩ( d = 0 . R d = 4 .
53 kΩ ( d = 2 . k ) on the coupled dynamics. For this wechoose the following circuit parameters: R d = 20kΩ (i.e., d = 0 .
5) and R ǫ = 6 . ǫ = 1 .
5) and vary R k .With decreasing R k (i.e., increasing k ) we observe a tran-sition from LC to NAD to AD, which is in accordancewith the numerical result of Fig.5 (c). Figure 7 (e) and(f) show this scenario experimentally and numerically,respectively (see caption for details). Thus, we see thatin a real system, despite the presence of inherent noise,parameter fluctuation and mismatch, the experimentalobservations are qualitatively similar to our theoreticalresults. -0.90.9400 420 440 460 LC AD NAD (cid:1)(cid:1)(cid:1) =0.85 =1.11 =3.64
LC AD ODAD ODLC
Rd=43 =30.5 =4.53 (cid:2)(cid:3) (cid:2)(cid:3) (cid:2)(cid:3) d d d =0.23 =0.33 =2.21 X V X1,2 t XX (a)(b)(c)(d) V X1,2 X V X1,2 (e) t Rd RdV X1 V X2 (cid:2)(cid:3) R (cid:1) =9 (cid:2)(cid:3) R (cid:1) =11.7 = (cid:1) (cid:2)(cid:3) (cid:2)(cid:3) =2.75R (cid:1) NAD NAD V X1,2 -V X1,2 -11400 420 440 460
NAD (cid:1) =3.64 -1.81.8400 420 440 460
LC NAD AD k=0.11 k=0.7 k=2 X t LC NAD AD
Rk=90.9 (cid:2)(cid:3)
Rk Rk=14.28 =5 (cid:2)(cid:3) (cid:2)(cid:3) (f)
FIG. 7. (Color online) Experimental real time traces [(a),(c)and (e)] of V x and V x along with the numerical time seriesplots [(b),(d) and (f)] of x and x . (a, b) Variation ofenvironmental coupling ( R d = 30 kΩ, d = 0 . R ǫ = 11 . ǫ = 0 . R ǫ = 9 kΩ ( ǫ = 1 . R ǫ = 2 .
75 kΩ ( ǫ = 3 . V x , and − V x , at R ǫ = 2 .
75 kΩ ( ǫ = 3 .
64) are shown. (c, d) Variation of diffusive coupling ( R ǫ = 8 .
33 kΩ, ǫ = 1 . R d = 43 kΩ ( d = 0 . R d = 30 .
50 kΩ ( d = 0 .
33) and OD state R d = 4 .
53 kΩ ( d =2 . (e, f) Variation of environment ( k ) [ R d = 20kΩ( d = 0 . R ǫ = 6 . ǫ = 1 . R k = 90 . k = 0 . R k = 14 .
28 kΩ ( k = 0 .
7) and AD at R k = 5 kΩ ( k = 2); for clarity only the V x , NAD state isshown. [(a),(c) and (e): y -axis: 5 v/div, x -axis: 500 µ s/div,for (e) x -axis: 250 µ s/div]. IV. CONCLUSION
We conclude that although the simultaneous applica-tion of direct and indirect coupling was proposed as ageneral scheme for inducing amplitude death state [17],in this paper we have shown that this coupling schemecan induce several other oscillation quenching states suchas OD and a new nontrivial AD (NAD) state that is a bistable state; this NAD state has different manifesta-tion and genesis compared to the conventional AD state.Further, we have shown that this coupling scheme caninduce transitions from AD to OD (with the variationin diffusive coupling strength), and AD to NAD (withthe variation in environmental coupling strength). Thelatter transition is reported for the first time. We havealso reported the first experimental evidence of oscilla-tion suppression states and their transitions induced bythis direct-indirect coupling.In the present paper we have investigated two genericperiodic oscillators that have certain type of symmetry.In biological systems (where negative concentrations areforbidden) and chaotic systems the bifurcation scenarioswill be much more complex. Since in biological oscil-lators non-negative asymmetric OD states have alreadybeen reported (see e.g. Ref.[1]), we hope that the theo-retical findings of this paper will be valid for biologicaloscillators, also. But, the exact bifurcation scenario maybe different and that requires further investigations. Fur-ther, we test the system dynamics for a regular network oflarge number of oscillators (not reported in the presentmanuscript) and find that the said AD, OD and NADstates are preserved for the larger system, also.Since in biological systems both diffusive and environ-mental coupling are omnipresent [18], thus we believethat this detailed study will improve our understandingof several biological processes. More specifically, as wehave shown that in the NAD state the system becomesbistable, thus, further research is required to explore anypossible connection of NAD to the bistability arises inmany biological processes such as the bistability in thebrain activity [20] (e.g. sleep-wake cycle of mammals andbirds), lac operon in the bacteria
E. coli [21], cell cycle[26], etc. Further, one of the central topics of recent re-search is to identify the coupling schemes that can imple-ment bistability in biological systems and many schemeshave already been identified and experimentally verifiedin this context [27]. Future research can be undertakenin order to explore the role of the direct-indirect couplingstudied here in inducing bistability in biological systems.
ACKNOWLEDGMENTS
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