Aging transition in the absence of inactive oscillators
K. Sathiyadevi, I. Gowthaman, D. V. Senthilkumar, V. K. Chandrasekar
aa r X i v : . [ n li n . AO ] A ug Aging transition in the absence of inactive oscillators
Aging transition in the absence of inactive oscillators
K. Sathiyadevi, I. Gowthaman, D. V. Senthilkumar, a) and V. K. Chandrasekar b) Centre for Nonlinear Science & Engineering, School of Electrical & Electronics Engineering, SASTRA Deemed University,Thanjavur - 613 401, Tamil Nadu, India. School of Physics, Indian Institute of Science Education and Research, Thiruvananthapuram - 695551, Kerala,India. (Dated: 2 August 2019)
The role of counter-rotating oscillators in an ensemble of coexisting co- and counter-rotating oscillators is examined byincreasing the proportion of the latter. The phenomenon of aging transition was identified at a critical value of the ratioof the counter-rotating oscillators, which was otherwise realized only by increasing the number of inactive oscillatorsto a large extent. The effect of the mean-field feedback strength in the symmetry preserving coupling is also explored.The parameter space of aging transition was increased abruptly even for a feeble decrease in the feedback strengthand subsequently aging transition was observed at a critical value of the feedback strength surprisingly without anycounter-rotating oscillators. Further, the study was extended to symmetry breaking coupling using conjugate variablesand it was observed that the symmetry breaking coupling can facilitating the onset of aging transition even in theabsence of counter-rotating oscillators and for the unit value of the feedback strength. In general, the parameter spaceof aging transition was found to increase by increasing the frequency of oscillators and by increasing the proportionof the counter-rotating oscillators in both the symmetry preserving and symmetry breaking couplings. Further, thetransition from oscillatory to aging transition occurs via a Hopf bifurcation, while the transition from aging transitionto oscillation death state emerges via Pitchfork bifurcation. Analytical expressions for the critical ratio of the counter-rotating oscillators are deduced to find the stable boundaries of the aging transition.
Aging is a kind of deterioration which occurs in diversecomplex systems. It is evident from our daily life that liv-ing organisms (and its efficiency) degrade as it becomesolder. For instance, Alzheimer’s disease is an example of acause of failure of neurons due to the aging process. In thiscontext, the phenomenon of aging transition was reportedin an ensemble of oscillators by increasing the proportionof inactive oscillator . In the present work, we showthat even an appropriate proportion of counter-rotatingoscillators in an ensemble of coexisting co- and counter-rotating oscillator is capable of inducing the phenomenonof aging transition. Further, we find that either the mean-field feedback or the symmetry breaking coupling alonecan facilitate the onset of the aging transition.
I. INTRODUCTION
Complex systems are abundant, omnipresent in nature andare rarely isolated. Coupled nonlinear oscillators serve as anexcellent framework to unravel distinct collective patterns ob-served in such complex systems, which include chemical and biological systems , and so on. The network architec-ture and the strength of the interaction of the coupled sys-tems facilitate the onset of various collective behaviors suchas synchronization , chimera , clustering , and dis-tinct oscillation quenching states , having strong resem-blance with many natural processes. The phenomenon of ag-ing transition is one such a phenomenon of serious concern a) Electronic mail: [email protected] b) Electronic mail: [email protected] in the complex networks, degrading its dynamical activity. Inneural networks, de-actuation of a single node may cause se-ries concern in the entire network, which cascades continuousdegradation of neighboring nodes . A similar issue hasalso been reported in the power grids . Initially, the phe-nomenon of aging transition was reported by increasing thenumber of inactive oscillators from the active group of os-cillators in a globally coupled network. Originally, the ef-fect of aging behavior was analyzed through globally and dif-fusively coupled oscillators . Further, the emergence ofdesynchronization horn was identified for sufficiently strongnonisochronicity where the active oscillators desynchronizeresulting in local clustering . The phenomenon of aging tran-sition was also analyzed by different groups probing variousaspects of the network .On the other hand, coexisting co- and counter-rotatingdynamical activity can be found in various natural systemssuch as fluid dynamics , physical as well as biolog-ical systems . Hence special attention is deemed towardsenriching our knowledge on counter-rotation induced collec-tive dynamics in various systems. Originally, the coexis-tence of clockwise and anticlockwise rotations was identi-fied by Tabor . Later, the universal occurrence of mixedsynchronization was reported in Stuart-Landau and Rössleroscillators . Further, mixed synchronization was demon-strated both theoretically as well as experimentally in the var-ious limit cycle and chaotic systems . Very recently, thecounter-rotating frequency induced dynamical effects were re-ported in the coupled Stuart-Landau oscillator with symmetrypreserving as well as symmetry breaking couplings. Theseauthors identified the suppression of oscillation death statewhen the symmetry breaks in the counter-rotating system .In addition, feedback also plays an essential role in the vari-ous natural and man-made systems. One of the recent reportsdemonstrated that by introducing a feedback strength in theging transition in the absence of inactive oscillators 2coupling can switch the stability of the stable steady statesfacilitating the revival of oscillation in coupled nonlinear os-cillators, which was also further corroborated experimentallyusing coupled electrochemical oscillators . The addition ofexternal feedback in retaining and enhancing the dynamicalrobustness in the presence and absence of time-delay was alsoreported . In particular, mean-field feedback has a predomi-nant application in neuroscience in deep brain simulation .Motivated by the above, in this work, we investigate the ef-fect of coexisting co- and counter-rotating oscillators on theemerging collective dynamical behavior of such an ensemble.Further, we also aim to study the impact of the mean-fieldfeedback on the observed collective dynamical behaviors. Inorder to elucidate the above, we consider an array of globallycoupled Stuart-Landau oscillators. Primarily, the effect of thecounter-rotating oscillators will be analyzed in symmetry pre-serving coupling. Surprisingly, we find that the proportionof the counter-rotating oscillators play a vital role in facilitat-ing the onset of aging transition through the Hopf bifurcation,which was otherwise identified only due to increasing in thenumber of inactive oscillators . The spread of aging tran-sition is found to increase upon increasing the frequency ofthe oscillators. We have also estimated the critical ratio of thecounter-rotating oscillators to show the stable boundaries ofthe aging transition. In addition, we also investigate the effectof the mean-field feedback on the spread of the aging transi-tion. Interestingly, we find that even a very small decrease inthe mean-field feedback strength enhances the aging transitionregion. In particular, we find that the mean-field induces thephenomenon of aging transition in an ensemble of oscillatorseven without any counter-rotating oscillators. Further, the ro-bustness of the observed results will be analyzed for symmetrybreaking conjugate coupling. We observe that the dynamicaltransition takes place from oscillatory to oscillation death statevia aging transition. We deduce the critical ratio of the counterrotating oscillator to find the stable boundaries of the spreadof the aging transition. Enriching the observed results, we findthat the conjugate coupling, which breaks the rotational sym-metry, indeed induces aging transition via a Hopf bifurcationeven without the counter-rotating oscillators and for the unitvalue of the mean-field feedback. In this case, the mean-fieldfeedback enhances the aging transition region. It is also tobe noted that the transition from aging transition to oscillationdeath takes place through pitchfork bifurcation. We find thatthe aging transition region enlarges upon decreasing the feed-back strength and increasing the frequency of the oscillators.The rest of the article is organized as follows: We willdemonstrate the emergence of the aging transition regionthrough symmetry preserving coupling in Sec. II, where theinfluence of the mean-field feedback will be discussed. InSec. III, the robustness of the obtained results will be in-spected in a symmetry breaking conjugate coupling as well.Finally, we summarize our results and draw conclusions inSec. IV. II. AGING TRANSITION: SYMMETRY PRESERVINGCOUPLING
To find the impact of the counter-rotating oscillators in in-ducing the aging transition, we consider a general paradig-matic model of Stuart-Landau limit cycle oscillators withsymmetry preserving coupling. Various natural phenomenaincluding the breathing cycle, the circadian clock shows thelimit cycle behavior. Many nonlinear dynamical systems nearthe Hopf bifurcation can be approximated through Stuart-Landau oscillator . An array of globally coupled Stuart-Landau oscillators with symmetry preserving coupling can bewritten as˙ z j = ( λ + i ω j − | z j | ) z j + KN N ∑ k = ( α z k − z j ) , (1)where z j = re i θ j = x j + iy j ∈ C , j = , , ..., N . N be the num-ber of elements in the network which is chosen as N = x j and y j are the state variables of the j th system. λ is theHopf bifurcation parameter and α is the mean-field feedbackstrength. ω j is the frequency of the j th system. If the systemfrequency is + ω , the system rotates in a counter-clockwisedirection, while − ω indicates the clockwise direction. In or-der to study the role of the counter-rotating oscillators, wesplit the oscillators as one group of oscillators with the sys-tem frequency ω j = ω for j ∈ , ... N ( − p ) while the othergroup takes the value, ω j = − ω for j ∈ N ( − p ) + , ... N .The parameter p characterizes the fraction of the counter ro-tating oscillators. Now, we investigate the emergence of theaging transition as a function of the ratio of the counter ro-tating oscillators. The numerical analysis of the system (1) iscarried out using Runge-Kutta fourth order scheme with stepsize 0.01 time units.Initially, to unravel the onset of the aging transition as afunction of the ratio of the counter-rotating oscillators, we es-timate the order parameter | Z | , where Z = N ∑ Nj = z j , as inrefs. . Then the normalized order parameter can be ex-pressed as, Q ≡ | Z ( p ) | / | Z ( ) | . p is the ratio of the counter-rotating oscillators. p = p =
1, respectively, denotes thecompletely counter-clockwise and clockwise rotating oscilla-tors while p = . p ∈ ( , . ) to p ∈ ( . , ) or decreasing it from p ∈ ( , . ) to p ∈ ( . , ) . Hence, we examine the effect of the counter-rotating oscillator ratio from p = p = . p ) for two differentvalues of the mean-field feedback strength ( α ) in Figs. 1(a)and 1(b). In order to exemplify the above, we fix the sys-tem frequency ω = .
0. Finite non-zero value of the nor-malized order parameter is observed in the entire range of p for K = .
0, elucidating the oscillatory nature of the dy-namical state of all the oscillators. It is observed that theging transition in the absence of inactive oscillators 3 p Q p Q K = 10 . K = 7 . K = 4 . K = 1 . K p OS AG 12600.50.250 K p OS AG 12600.50.250 (a)(c) (b)(d)
FIG. 1. Normalized order parameter ( Q ) as a function of the ratio ofthe counter-rotating oscillators ( p ) for (a) α = . α = . ω = . λ = . N = K = . , . , . K = .
0. Thecorresponding two parameter diagram in ( K , p ) space as a functionof the coupling strength are depicted in Figs. (c) and (d). OS and AGrepresent the oscillatory and the aging transition region. normalized order parameter transit from finite value to nullvalue at the critical values p HB = . , . , and 0 .
19, forthe coupling strengths K = . , . , and 10 .
0, respectively.The null value of Q indicates the existence of the aging tran-sition region, which emerges through the Hopf bifurcation.From Fig. 1(a), it is also evident that increasing the couplingstrength decreases the critical value p HB , thereby illustratingthat the aging transition emerges even for smaller proportionof the counter-rotating oscillators, enhancing the aging tran-sition region, in strong contrast to increasing the number ofinactive oscillators to realize the aging transition as reportedso far . Further, to unravel the influence of the mean-fieldfeedback on the aging transition, we have also depicted thenormalized order parameter for α = .
95 in Fig. 1(b). Thecoupled systems exhibit oscillatory behavior for K =
1. Whenincreasing the coupling strength to K = . , . , and 10 .
0, theonset of the aging transition take place at p c = . , .
15 and0 .
1, respectively. Thus, it is evident from both Figs. 1(a) and1(b) that the critical value of p HB , for the onset the aging tran-sition, abruptly decreases even for a feeble decrease in themean-field feedback strength α , enhancing the aging transi-tion region. The corresponding global behavior of the coupledco- and counter-rotating system (1) is depicted in Figs. 1(c)and 1(d) in ( K , p ) space for α = . α = .
95, respec-tively. The co- and counter-rotating coupled Stuart-Landauoscillators exhibit oscillations for small values of the couplingstrength and for the smaller ratio of the counter-rotating oscil-lators, whereas the coupled Stuart-Landau oscillators showsthe aging transition for appreciable values of p and K (seeFigs. 1(c) and 1(d)). Further, to confirm the emergence of ag-ing transition analytically, we have performed the linear sta-bility analysis which will be detailed as follows.In the aging transition, z j is stabilized to trivial fixed point(i.e. z j = . Here, one group w = z j , ( j = , ... N ( − p )) rotates in clockwise direction whilethe other group w = z j , ( j = N ( − p ) + , ... N ) rotates incounter-clockwise direction.˙ w = ( + i ω − | w | ) w + K ( α ( − p ) w − ( − α p ) w ) , ˙ w = ( − i ω − | w | ) w + K ( α pw + ( α ( − p ) − ) w ) . (2)It is to be noted that the sign of ω , either positive or neg-ative depicts whether the oscillators are rotating in clock-wise or anti-clockwise direction, respectively. By perform-ing the linear stability analysis of Eq. (2) around the origin( w = w = p HB as p HB = α K ( + ( − + α ) K ) ω ± √ ∆ ( α K ( + ( − + α ) K ) ω ) . (3)where, ∆ = α K ( + ( − + α ) K ) ω ( − − ( − + α ) k +( − + α ) K − ω ) . Using the above expression, one can ob- K p AG ➝ p HBω = 5 . ω = 4 . ω = 3 . ω = 2 . K ω AG p = 0 . p = 0 . p = 0 . p = 0 . K p AG ➝ p HB K ω AG (a) (b) (c) (d) FIG. 2. Analytical boundaries of the aging transition region for α = .
0: (a) ( K , p ) space for different values of system frequency ω = . , . , . .
0, (b) ( K , ω ) space for different values of thecritical ratio of the counter rotating oscillators p for p = . , . , . p = . p HB is the critical curve corresponding to Hopf bi-furcation curve. The corresponding analytical aging boundaries for α = .
95 are shown in Fig. (c) and (d). tain the stable boundaries of the aging transition region as de-picted in Fig. 2. The critical curve p HB is the Hopf bifur-cation curve across which there is a change the stability ofthe limit cycle occurs. Figures 2(a) and 2(b) are plotted forthe mean-field feedback α = .
0. The distinct line-types inFig. 2(a) correspond to the different values of ω , namely as ω = . ω = . ω = . ω = . ( K , p ) plane. FromFig. 2(a), it is evident that increasing the system frequencyincreases the aging transition region as a function of the cou-pling strength and the ratio of the counter-rotating oscillators.Similarly, the aging transition regions are depicted in ( K , ω ) plane in Figure 2(b) for different ratio of the counter rotat-ing oscillators p = . , p = . , p = . p = .
5. Theging transition in the absence of inactive oscillators 4region enclosed by the curves correspond to aging transitionregion. Figure 2(b) clearly delineates the increase in the agingtransition region while increasing the ratio of counter-rotatingoscillators. Moreover, the aging transition region enhances toa large extent when both the co- and counter-rotating oscilla-tors balance each other. Further, to substantiate the observedresults in Fig. 1, we have also depicted the dynamical tran-sition observed for the mean-field feedback α = .
95. Thus,it is evident from Figs. 2(c) and 2(d) that the aging transitionregion is increased upon decreasing the mean-field feedbackstrength α . In particular, the aging transition region enhancesfor increasing the frequency of the oscillations, increasing theratio of the counter-rotating oscillators and while decreasingthe mean-field feedback strength. K p p HB ➝ AG α = 0 . α = 0 . α = 0 . α = 0 . α = 1 . α p K = 10 K = 8 K = 6 ➝ p HB AG (a) (b) FIG. 3. (a) The aging transition region, for symmetry preserving cou-pling, for different values of the mean-field feedback strength ( α = . , . , . , .
92 and 0 .
9) for the system frequency ω = .
0. (b)Critical ratio of counter rotating oscillators as a function of the feed-back strength for the system frequency ω = . K = . K = . K = . For further insight on the effect of the mean-field feed-back strength on the aging transition region and to show theenlarging of the aging transition region as a function of thefeedback strength, we have depicted the critical boundaries p HB of the aging transition for ω = . α = . , . , . , .
92 and 0 . p c as a function of α for three different values of the coupling strengths K = . K = . K = . K = . K = . K = .
0, re-spectively. It is evident that the critical value p c facilitatingthe onset of the aging transition region drops down to zeroeven for a small decrease in the mean-field feedback strengths.Substantiating that the aging transition can indeed induced bythe mean-field feedback strength even in the absence of thecounter-rotating oscillators. One can also note that increasingthe coupling strength decreases the value of the mean-fieldfeedback strength for the onset of the aging transition.From symmetry preserving coupling, we have unraveledthe emergence of the aging transition via Hopf bifurcation as aresult of increasing the ratio of the counter-rotating oscillators.It is also established that the mean-field feedback can facilitatethe onset of the aging transition without any counter-rotatingoscillators. In addition, the spread of the aging transition re- gion is increased upon either increasing the system frequencyor decreasing the mean-field feedback strength. Further, to ex-amine the robustness of the observed transition, we have alsoanalyzed the dynamical transitions in a symmetry breakingcoupling in the following. III. AGING TRANSITION: SYMMETRY BREAKINGCOUPLING
It is established fact that the symmetry breaking couplingcan be responsible for the emergence of various collectivepatterns including frequency cluster, amplitude cluster, ampli-tude chimera, frequency chimera, various kinds of oscillationquenching states and so on . We introduce the symmetrybreaking in the coupling using the conjugate variables in thecoupling. In this section, we investigate the effect of conjugatecoupling on the aging transition as a function of the counter-rotating oscillators, the frequency of the oscillators, and themean-field feedback. In order to illustrate the above, we con-sider an array of globally coupled Stuart-Landau oscillatorswith conjugate coupling described as˙ z j = ( λ + i ω j − | z j | ) z j + KN N ∑ k = ( α Im ( z k ) − Re ( z j )) . (4)Here, the coupling among the oscillators are implementedthrough dissimilar variables. The proposed coupling breaksthe system rotational symmetry explicitly, where, the rota-tional invariance is not preserved under such transformation z j → z j e i θ . K p ODAGOS 8400.50.250 K p ODAGOS 8400.50.250 (a) (b)
FIG. 4. Two parameter diagram for an array of conjugately coupledStuart-Landau oscillators (symmetry breaking coupling) in ( K , p ) space for (a) α = . α = .
96. Other parameters: λ = . ω = . N = Initially, the dynamical transitions are analyzed numeri-cally as a function of the coupling strength in symmetry break-ing conjugately coupled system (4). Figures 4(a) and 4(b) areplotted for two different values of the mean-field feedbackstrength α = . α = .
96, respectively, for the systemfrequency ω = .
0. From both the figures, we found thatthe transition takes place from oscillatory region to oscilla-tion death state through the aging transition. The transitionfrom oscillatory to aging transition transition takes place viaHopf bifurcation while transit from aging to oscillation deathoccurs via pitchfork bifurcation. Further, it is to be noted thatthe symmetry breaking coupling using the conjugate variablesging transition in the absence of inactive oscillators 5induces the aging transition without any counter-rotating os-cillators even for the unit value of the mean-field feedbackstrength. Comparing Fig. 4(b) with Fig. 4(a), it is clear thatthe feedback strength α increases the aging transition regionin the ( K , p ) space but the oscillatory region remains unal-tered. Furthermore, the observed numerical boundaries of theaging transition region are confirmed analytically in the fol-lowing. K p p PF ➝ p HB ➝ ω = . ω = . ω = . ω = . K ω AG p = 0 . p = 0 . p = 0 . p = 0 . K p p PF ➝ p H B ➝ ω = . ω = . ω = . ω = . K ω AG (a) (b)(d) (c) FIG. 5. Analytical boundaries of the aging transition region (a) ( K , p )plane for distinct values of the system frequency ω = . , . , . . ( K , ω ) plane for p = . , . , . p = . p HB and p PF are the critical value of the counter rotating oscillators, whichseparate the boundary between the oscillatory and the aging transi-tion region, and the aging transition and the oscillation death, re-spectively. The corresponding analytical boundaries for α = .
96 isshown in Fig. (c) and (d).
For finding the analytical boundaries of the aging transitionregion in the symmetry breaking conjugate coupling, we havededuced the critical ratio of the counter rotating oscillatorsfrom the linear stability analysis as, p HB = ( α ( − + K ) K w ± √ ∆ ) α ( − + K ) K w , (5)and p PF = α K ω ± √ ∆ ) α K w , (6)where ∆ = α ( − + k ) K ( − K − ( − + α ) K ) ω ( K +( − + α ) K − ( + ω )) , and ∆ = α K ω (( α − ) K − ( α − ) K + ( α − − ω ) + K ( + ω ) − ( + ω ) ) . p HB and p PF are the critical curves corresponding to the Hopfbifurcation and the pitchfork bifurcation curves. Across p HB there is a change in the stability of the limit cycle oscillators,which separate the boundary between the oscillatory state andthe aging transition region. p PF is the pitchfork bifurcationcurve on which there is a transition from homogeneous steadystate (AT) to inhomogeneous steady state (OD state). UsingEq. (5) and Eq. (6), we have plotted Fig. 5 in ( K , p ) spaceand ( K , ω ) space. Enhancing of the aging transition region is clearly evident (see Fig. 5(c) and 5(d)) for decrease in thevalue of the mean-field strength.Finally, to show the nature of the aging transition re-gion upon decreasing the mean-field feedback strength in thesymmetry breaking coupling, we have depicted the analyt-ical curves of different mean-field feedback strength α = . , . , . , .
94 for ω = . ω = . ( α = . , . , . , . ) increases the aging transitionregion. K p p P F ➝ p H B ➝ α = 0 . α = 0 . α = 0 . α = 1 . K p p P F ➝ p H B ➝ (a) (b) FIG. 6. The aging transition region for the different value of thefeedback strength in conjugately coupled Stuart-Landau oscillators.Analytical boundaries of the aging transition region for distinct val-ues of feedback strength ( α = . , . , . , . ) for (a) ω = . ω = . From the above discussed results from the symmetry pre-serving and symmetry breaking coupling, it is confirmed thateven the proportion of the counter-rotating oscillator in an en-semble of coexisting co- and counter-rotating oscillators, in-deed, facilitates the emergence of the aging transition in con-trast to the earlier reports, where a large proportion of inactiveoscillators are introduced to realize the same. In addition, it isshown that the mean-field feedback strength facilitates the on-set of the aging transition despite the absence of the counter-rotating oscillators. Region of the observed aging transition isenhanced while increasing the frequency of the system or de-creasing the mean-field feedback strength. Further, it is clearthat such transition is robust in both symmetry preserving aswell symmetry breaking coupled oscillators.
IV. CONCLUSION
Degradation of dynamical activity (is also called as aging)is inevitable in many complex networks including power gridsand neural networks. In the present work, we have exploredthe onset of the aging transition in a network of the globallycoupled limit cycle oscillators where the system frequenciesare distributed to a distinct ratio of co- and counter-rotatingoscillators. We found that even the counter-rotating oscilla-tors also can induce the aging transition in an ensemble ofcoexisting co- and counter-rotating oscillators unlike the ear-lier reports, which increases a large proportion of the inactiveoscillators. In order to show this, initially, we have consideredging transition in the absence of inactive oscillators 6an array of globally coupled Stuart-Landau limit cycleoscillators with symmetry preserving coupling. We haveanalyzed the influence of the mean-field feedback strengthon the aging transition. Interestingly, we have observed thataging transition region is enhanced abruptly even for a smalldecrease in the feedback strength and subsequently the ratioof the counter-rotating oscillators necessary to induce theaging transition drops down to zero corroborating that themean-field feedback can indeed induce the phenomenon ofthe aging transition. Further, we have also analyzed the effectof the system frequency on the aging transition region, whichincreases the aging transition region. The observed resultsare further confirmed analytically through the linear stabilityanalysis by deducing the critical ratio of the counter-rotatingoscillators for which change in the stability of the limit cycleoscillators occurs via the Hopf bifurcation. Additionally, therobustness of such aging transition is also analyzed in thesymmetry breaking conjugate coupling which also exhibitssimilar dynamical transitions. Surprisingly, we found thatthe symmetry breaking conjugate coupling itself is capableof facilitating the onset of the aging transition without anycounter-rotating oscillators even for the unit value of themean-field feedback strength. Furthermore, we have alsodeduced the analytical critical curves corresponding tothe Hopf bifurcation and the pitchfork bifurcation curves.Finally, from the obtained results, we have identified that theobserved aging transition behavior is robust in both symmetrypreserving as well as symmetry breaking coupling.The proposed work also leads to many open problems. Dueto complex interactions in real-world systems, the study ofcollective dynamics under distinct network geometries is in-triguing research for many years and is inevitable. There-fore, it is a valuable insight to extend our analysis to varioustopological structures including scale-free, small world and soon . Since, feedback is a general mechanism to revoke thedynamism from the deteriorated dynamical units in a com-plex network and is also essential to investigate the influenceof various feedbacks . From the earlier reports, it isidentified that the processing delay is used to destabilize thehomogeneous as well as an inhomogeneous steady state ei-ther in the presence or the absence of propagation delay .Thus, it will also be interesting to investigate the effect of vari-ous delays including propagation delay, processing delay, andlow pass filter . Furthermore, the role of nonlinearparameter is inevident in the observed dynamical transitionsand which plays a crucial role in inducing various collectivepatterns . In view of the above, it is also intriguingto investigate the dynamical transition with the addition ofnonisochronicity parameter, as a function of co- and counter-rotating frequencies of a dynamical system.
ACKNOWLEDGMENTS
KS sincerely thanks the CSIR for a fellowship underSRF Scheme (09/1095(0037)/18-EMR-I). The work ofVKC supported by a research project CSIR under Grant No.03(14444)/18/EMR-II. DVS is supported by the CSIREMR Grant No. 03(1400)/17/EMR-II. We also thank theSASTRA Deemed university for providing good infrastruc-ture lab facilities. Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence, Springer-Verlag Berlin Heidelberg (1984). R. Toth, A. F. Taylor, and M. R. Tinsley, J. Phys. Chem. B, , 10170-10176, (2006). J. Sawicki, I. Omelchenko, A. Zakharova and E. Schöll, Euro. Phys. J. B , 54 (2019). S. Majhi, B. K. Bera, D. Ghosh, and M. Perc, Phys. Life Rev. , 100-121(2019). A. Pikovsky, M. G. Rosenblum, and J. Kurths, Synchronization: A Uni-versal Concept in Nonlinear Sciences (Cambridge Univer- sity Press, Cam-bridge, 2001). S. Gupta, A. Campa, and S. Ruffo, Statistical Physics of Synchronization,Springer Briefs in Complexity, Springer International Publishing, (2018). M. Panaggio and D. M. Abrams, Nonlinearity , R67–R87 (2015). A. Zakharova, M. Kapeller, and E. Schöll, Phys. Rev. Lett. , 154101(2014). I. Omelchenko, A. Provata, J. Hizanidis, E. Schöll, and P. Hövel, Phys. Rev.E 91, 022917 (2015). E. Schöll, Eur. Phys. J. Special Topics , 891–919 (2016). K. Sathiyadevi, V. K. Chandrasekar, and D. V. Senthilkumar, Phys. Rev. E , 032301 (2018). S.Ghosha, A. Zakharova, and S. Jalan, Chaos, Solitons & Fractals , 56-60 (2018). S. Gil, Y. Kuramoto, and A. S. Mikhailov Europhys. Lett. , 60005 (2009). K. Sathiyadevi, V. K. Chandrasekar, D. V. Senthilkumar, and M. Laksh-manan, Phys. Rev. E , 032207 (2018). G. Saxena, A. Prasada, and R. Ramaswamy, Phys. Rep. , 205-228(2012). A. Koseska, E. Volkov, and J. Kurths, Phys. Rep. , 173-199 (2013). T. Banerjee and D. Ghosh, Phys. Rev. E (5), 052912 (2014). I. Schneider, M. Kapeller, S. Loos, A. Zakharova, B. Fiedler, and E. Schöll,Phys. Rev. E , 052915 (2015). W. Zou, M. Zhan, and J. Kurths, Phys. Rev. E , 062209 (2018). V. Resmi, G. Ambika, and R. E. Amritkar, Phys. Rev. E , 046212 (2011). A. Sharma, P. R. Sharma and M. D. Shrimali, Phys. Lett. A , 1562-1566(2012). L. Ambrogioni, M. A. J. vanGerven, E. Maris, PLoS Comput. Biol. (5),e1005540 (2017). S. Atasoy, L. Roseman, M. Kaelen, M. L. Kringelbach, G. Deco, R. L.Carhart-Harris, Sci. Rep. , 17661 (2017). C. L. DeMarco, IEEE Control Syst. Mag. , 40–51 (2001). H. Daido and K. Nakanishi, Phys. Rev. Lett. , 10 (2004). H. Daido and K. Nakanishi, Phys. Rev. E , 056206 (2007). H. Daido, Europhysics. Lett. , 40001 (2009). G. Tanaka, K. Morino, H. Daido, and K. Aihara, Phys. Rev. E , 052906(2014) B. Thakur, D. Sharma, and A. Sen, Phys. Rev. E , 042904 (2014). S. Kundu, S. Majhi, P. Karmakar, D. Ghosh and B. Rakshit, Euro. Phys.Lett. , 30001 (2018). R. Mukherjee and A. Sen, Chaos , 053109 (2018); Y. Murakami and H. Fukuta, Fluid Dyn. Res. , 449-59 (1998). M. Darvishyadegari and R. Hassanzadeh, Acta Mech. 229, 1783–1802(2018). K. Czolczynski, P. Perlikowski, A. Stefanski and T. Kapitaniak, Commun.Nonlinear Sci. Numer. Simulat. , 3658–72 (2012). J. Strzalko, J. Grabski, J. Wojewoda, M. Wiercigroch, and T. Kapitaniak,Chaos , 047503 (2012). S. Takagi and T. Ueda Physica D , 420-7 (2008). M. Tabor, Chaos and Integrability in Nonlinear Dynamics: An Introduction.John Wiley & Sons, New York; 1988. A. Prasad, Chaos, Solitons and Fractals , 42–46 (2010). S. K. Bhowmick, D. Ghosh, and S. K. Dana, Chaos A. Sharma and M. D. Shrimali, Nonlinear Dyn. , 371–377 (2012). ging transition in the absence of inactive oscillators 7 B. K. Bera, S. K. Bhowmick and D. Ghose, Int. J. Dynam. Control, ,269–273 (2017). N. Punetha, V. Varshney, S. Sahoo, G. Saxena, A. Prasad and R. Ra-maswamy, Phys. Rev. E , 022212 (2018). W. Zou, D. V. Senthilkumar, R. Nagao, I. Z. Kiss, Y. Tang, A. Koseska, J.Duan and J. Kurths, Nat. Commun. , 7709 (2015) S. Kundu, S. Majhi, and D. Ghosh, Phys. Rev. E , 052313 (2018). A. Franci, A. Chaillet, E. Panteley and F. L. Lagarrigue, Math. ControlSignals Syst. , 169–217 (2012). M. Luo, Y. Wu and J. Peng, Biol. Cybern. , 241–246 (2009). M. Frasca, A. Bergner, J. Kurths and L. Fortuna, Int. J. Bifurcat. Chaos, ,1250173 (2012). M. C. Thompson and P. L. Gal, Eur. J. Mech. B , 219 (2004) I. Schneider, M. Kapeller, S. Loos, A. Zakharova, B. Fiedler, and E. Schöll, Phys. Rev. E , 052915 (2015). Nannan Zhao, Zhongkui Sun and Wei Xu, Sci. Rep. , 8721 (2018). K. Premalatha, V. K. Chandrasekar, M. Senthilvelan and M. Lakshmanan,Phys. Rev. E , 052915 (2015). V. K. Chandrasekar, S. Karthiga, and M. Lakshmanan, Phys. Rev. E ,012903 (2015). Y. Liu, W. Zou, M. Zhan, J. Duan and J. Kurths, Europhysics. Lett, ,40004 (2016). K. Ponrasu, K. Sathiyadevi, V. K. Chandrasekar and M. Lakshmanan Euro-phys. Lett. , 20007 (2018). W. Zou, M. Zhan, and J. Kurths, Chaos , 114303 (2017). W. Zou, J. L. O. Espindola, D. V. Senthilkumar, I. Z. Kiss, M. Zhan and J.Kurths, Phys. Rev. E , 032214 (2019). X. Lei, W. Liu, W. Zou, and J. Kurths, Chaos29