Enhancement of dynamical robustness in a mean-field coupled network through self-feedback delay
aa r X i v : . [ n li n . AO ] J u l Enhancement of dynamical robustness in a mean-fieldcoupled network through self-feedback delay
Amit Sharma a) and Biswambhar Rakshit b) Department of Physics, Central University of Rajasthan, Ajmer 305 817,India Department of Mathematics, Amrita School of Engineering, Coimbatore,Amrita Vishwa Vidyapeetham, , India (Dated: 28 July 2020)
In this article, we propose a very efficient technique to enhance the dynamical robust-ness for a network of mean-field coupled oscillators experiencing aging transition. Inparticular, we present a control mechanism based on delayed negative self-feedback,which can effectively enhance dynamical activities in a mean-field coupled networkof active and inactive oscillators. Even for a small value of delay, robustness gets en-hanced to a significant level. In our proposed scheme, the enhancing effect is morepronounced for strong coupling. To our surprise even if all the oscillators perturbedto equilibrium mode delayed negative self-feedback able to restore oscillatory activ-ities in the network for strong coupling strength. We demonstrate that our proposedmechanism is independent of coupling topology. For a globally coupled network,we provide numerical and analytical treatment to verify our claim. Also, for globalcoupling to establish the generality of our scheme, we validate our results for bothStuart-Landau limit cycle oscillators and chaotic R¨ossler oscillators. To show thatour scheme is independent of network topology, we also provide numerical resultsfor the local mean-field coupled complex network.PACS numbers: Valid PACS appear hereKeywords: Dynamical Robustness, Aging transition, Self-feedback delay, CoupledOscillators
I. INTRODUCTION
Exploring emergent dynamics of a vast network of coupled oscillators got considerable attentionin recent years due to its applicability to understanding various self-organized complex systems .The emergent behaviors of such complex systems depend on the individual dynamics of local sub-units as well as the network topology. The normal functioning of many natural systems requiresstable and robust oscillatory dynamics. Therefore, the macroscopic dynamics of such a large-scalesystem should exhibit resilience against local perturbations.In recent times, much attention has been invested in understanding the dynamical robustness,which is defined as the ability of a network of oscillators to regulate its rhythmic activity when afraction of the dynamical units is malfunctioning . Physically this situation can be modeled as anetwork of coupled oscillators where oscillatory nodes switch to equilibrium mode progressively .If the number of nodes which perturbed to equilibrium mode reaches a critical level, the normalactivities of such systems may hamper and face severe disruption. This emergent phenomenon isdescribed as aging transition . The aging transition might have catastrophic effects in many natu-ral and real-world systems such as metapopulation dynamics in ecology, neuronal dynamics in brain,cardiac oscillations, and power-grid network . Therefore, it is of great practical significance topropose some remedial measures or control mechanisms to enhance the dynamical robustness ofthe coupled systems against the aging or deterioration of the individual unit. Till now, researchersmainly studied aging transition not only considering different network topologies but also usingdifferent coupling functions . Some recent efforts have been directed to explore the possible a) Electronic mail: [email protected] b) Electronic mail: [email protected] mechanisms to enhance the dynamical robustness to avoid catastrophic transition. Liu et al. pro-posed a mechanism for robustness enhancement that involves an additional parameter to controlthe diffusion rate. Kundu et al. have shown that the robustness can be enhanced by a posi-tive feedback mechanism as well as asymmetric couplings. Bera has employed low pass filteringmechanism for enhancing dynamical robustness. Despite such attempts, enhancement mechanismsare yet to be explored fully and deserve significant attention.All the previous studies on the aging transition considered nearest neighbour diffusive coupling,which effectively depicts many real-world complex systems. However, for many biological andphysical systems, the mean-field coupling is also relevant. It has been shown that the diffusionprocess for a network of genetic oscillators can be modelled as mean-field coupling through a reg-ulation known as quorum-sensing. . A network of mean-field coupled oscillators can model thedynamics of SCN neurons , which produces circadian oscillations. In this article, we explore apossible mechanism to augment the dynamical robustness of a mean-field coupled oscillators.The feedback mechanism is considered as one of the main themes of scientific understanding ofthe last century and has been widely used in control theory . The positive feedback which favors thesystem’s instability in the dynamical system has been used extensively in neural networks, geneticnetworks, etc . On the other hand, negative feedback promotes stability and has been widely usedto model biochemical systems . It is one of the fundamental mechanisms in cellular networks andis shown to be present in many biochemical systems including bacterial adaptation , mammaliancell cycle , etc. But the negative feedback with time delay favours oscillatory dynamics in thesystem , a scenario that we explore in this work as well.It is a well-established fact that time delays are an essential part of many natural systems. Inrecent years, more and more systems are being recognised to be influenced by or to be describablevia a delayed coupling. Time delay due to the finite propagation speed of the external signal, whichis described as propagation delay has been widely used to control the dynamics of coupled nonlinearsystems . The internal self-feedback delay appears because sometimes the system needs a finitetime to process the received signal and then act on it. It is demonstrated that such type of localself-feedback delay acts as negative feedback and plays a crucial role in reviving oscillations oramplitude death and oscillation death . In comparison to propagation delay, the effects of self-feedback delay in coupling are very less explored. There are only a few instances where systemswith self-feedback delay have been investigated.In the context of network of coupled oscillators feedback has been widely used for control ofnetwork dynamics and synchronization . However, the effects of feedback in the aging transitionhave not been well explored. Only recently, Kundu et al. bring out an in-depth study on the effectsof external positive feedback to increase the dynamical persistence of a network of oscillators. Inthis work, we study the ability of negative self-feedback with a time delay to enhance the dynamicalpersistence of a network that is experiencing an aging transition. We have shown that delayed self-feedback is an effective control mechanism to enhance the dynamical robustness in a mean-fieldcoupled network. We have shown that self-feedback delay very effectively enhances the robustnessfor global as well as local mean-field coupled network. Even when the amount of the local self-feedback delays is minimal, it effectively enhances the dynamical robustness of the network. Wepresent an in-depth study of the global network. For a global network to show that our enhancementmechanism is independent of the model, we provide results for the Stuart-Landau limit cycle systemas well as the chaotic R¨ossler system. We elucidate our results, both analytically and numerically.To show that our method is independent of coupling topology, we also present numerical results formean-field coupled oscillators interacting via complex network topology. II. GLOBAL MEAN-FIELD COUPLED NETWORK
In this section, we present an extensive study of the effect of negative self-feedback with a timedelay to elevate the dynamical robustness in a global mean-field coupled network. Here globalcoupling signifies that all the dynamical units have equal contribution to the mean-field, which actsequally on all the units. We consider N mean-field coupled Stuart-Landau oscillators with timeddelayed self-feedback. Mathematically one can write the governing equation of motion as˙ z j ( t ) = ( ρ j + i ω − | z j ( t ) | ) z j ( t ) + k [ z − z j ( t − τ )] , (1)for j = , , ..., N . Where z j = x j + iy j is the complex amplitude of the j th oscillator, and z isthe average mean-field. τ accounts for the delay in the local self-feedback term z j ( t − τ ) , whichbasically acts as negative feedback. Here ω (= ) is the internal frequency of each oscillators, and k for the coupling strength. ρ j is the bifurcation parameter of the j th oscillator and gives rise to asupercritical Hopf bifurcation at ρ j =
0. Each individual Stuart-Landau oscillator displays a stablesinusoidal type oscillation for ρ j = a ( > ) and converges to stable equilibrium point z j = ρ j = b ( < ) . (c)80 85 90 95 100 Time -101 R e ( z j ) (d)-1 0 1 x j -101 y j (a)80 85 90 95 100-101 R e ( z j ) (b)-1 0 1-101 y j FIG. 1. Time series of real part of z j and phase portrait of N mean-field coupled Stuart-Landau oscillatorsconsisting of active and inactive nodes are plotted with (a,b) k = k = p = .
2. The red color shows the inactive oscillators, and blue color indicates the active node of oscillators inthe network.
Here aging refers to the incidence when an active oscillator with ρ j = a > ρ j = b < j = , , ..... N ( − p ) as active ones and the remaining oscillators j = N ( − p ) + , ..... N as inactive. Here N is the total number of oscillators in the network, and p is the fraction of inactive oscillators. As the inactivation ratio p reaches a critical value, the globaloscillation of the network dies out. Following Daido and Nakanishi , we define an order parameter Z = | Z ( p ) || Z ( ) | to quantify the the dynamics of the system, that defines the average magnitude of globaloscillation in the network, where | Z ( p ) | = N − ∑ Nl = | z l | . We have taken the network size N = a = b = − k = p = . z j is shown in the Fig.1(a-b). As we incorporate the coupling inthe network, the inactive oscillators start oscillation under the influence of active oscillators. Bothgroups of oscillators show the synchronized motion with a different amplitude which is shown inFig.1(c-d). In Figure. 2(a) we have plotted Z against the inactivation ratio p for different values oflocal self-feedback delay τ for a fixed coupling strength k =
5. One can observe that in the absenceof local self-feedback delay τ , the order parameter Z vanishes at a lower value of p = p c . Thisimplies that the aging transition takes place much faster. As we increase the local self-feedbackdelay τ , the aging transition can be observed at a higher value of p = p c . So one can conclude thatlocal delayed self-feedback plays an important role in enhancing the dynamical robustness. For abetter understanding of the effect of local self delay feedback in the coupled oscillators, we haveshown in Fig.2(b) the phase transition diagram in the plane τ − p for fixed k =
5. In this figure,region OS and AT denote the oscillatory state and death state (where Z = τ is a minimal change in the p c value insignificant. But as weincrease the value of τ on higher side, the critical value of p c increases and reach to the p c = τ = .
12. It shows that the local self-feedback delay τ dominates the aging transition in the coupledoscillator and enhances the dynamical robustness of mean-field coupled oscillators.Next, we find the critical value p c analytically. The global oscillation collapses at p c during agingtransition and the trivial fixed point z j = z j = A for the active (a)0 0.5 1 p Z =0=0.02=0.04=0.06=0.08=0.1 (b) ATOS0 0.5 1 p FIG. 2. (a) The order parameter Z as a function of the inactivation ratio p for various values of τ in a mean-field coupled network of N =
500 Stuart-Landau oscillators for a = , b = − , ω = k =
5. (b) The phasediagram in of coupled oscillator in ( p − τ ) parameter plane at fixed k =
5, where white and light pink regionbelongs to oscillatory state (OS) and aging transition (AT) region respectively. Solid black line is a fitting ofthe critical value of aging transition p c obtained from Eq.6. group and z j = I for the inactive group of oscillators, the original equation 1 reduces to the followingcoupled systems , ˙ A ( t ) = ( a + i ω + kq − | A ( t ) | ) A ( t ) − kA ( t − τ ) + kpI ( t ) , ˙ I ( t ) = ( b + i ω + kp − | I ( t ) | ) I ( t ) − kI ( t − τ ) + kqA ( t ) , (2)where q = − p . Now, we cary out linear stability analysis to reduce Eq.2 around the trivial fixedpoint A = I =
0. In order to find the critical value of the p c . The origin ( A = I = ) is stabilized ifthe real parts of the eigenvalues become negative. Therefore, the critical inactivation ratio p c can bederived when two complex conjugate eigenvalues of above equations intersect the imaginary axis.A linear stability analysis around the origin now gives a characteristic equation of the form, ( a + i ω + qk − ke − λτ − λ )( b + i ω + pk − ke − λτ − λ ) − pqk = , (3)Here, λ = λ R + i λ I . By taking the real part of the eigenvalue equal to zero ( λ R =
0) and separate thereal and imaginary part of Eqs. 3, we obtain the following equations, [ a + qk − kcos ( λ I τ )][ b + pk − kcos ( λ I τ )]= ( ω − λ I + ksin ( λ I τ )) + pqk , (4) [ a + b + k ( p + q ) − kcos ( λ I τ )][ ω − λ I + ksin ( λ I τ )] = , (5)where p + q =
1. Solving the above equations, we get critical value of inactivation ratio p c as, p c = − ab + k ( a + b ) β + k β − kb − k β k ( a − b ) , (6)where β = cos ( ατ ) and α = ω + k q − (cid:0) a + b + k k (cid:1) .The occurrence of the aging transition is featured by the existence of a critical parameter p c . Thecritical value of the p c (black solid line) also has a good agreement with the numerical result (shadedregion) for the aging transition shown in Fig. 2(b).When the local self-feedback delay τ =
0, the aging transition takes place for all k > p c decreases as coupling strength k increases. To explore the impact of k on the parameter p c , we plotthe p c as a function of coupling strength k for different τ values in Fig. 3. Surprisingly, we observethat for nonzero τ , the aging transition exists only for a finite interval of coupling strength k . Within = = . = . = . = . = . k p c FIG. 3. The critical value of aging transition p c as a function of coupling strength k for different values of localself-delay τ in the mean-field coupled oscillators, where p c monotonically decreases for increasing of k . (a)0 0.5 1 p R =0=0.1=0.2=0.3=0.4=0.5 (b) ATOS0 0.5 1 p FIG. 4. (a) The order parameter R is plotted as a function of the inactivation ratio p at the various values of τ in mean-field coupled network of N =
500 R¨ossler oscillators for e = . k = .
2. (b) The phase diagramin of coupled oscillators in ( p − τ ) parameter plane at fixed k = .
2, where white and light pink region belongsto oscillatory state (OS) and aging transition (AT) region. this range, p c gradually decreases till it reaches its minimum value and then again monotonicallyincreases to unity. The above observation implies that strong coupling favors dynamical resilienceof the network against aging when feedback delay τ is large enough.To demonstrate that our enhancement method is independent of the dynamical model, we choosea chaotic dynamical system coupled through mean-field with local self-delay feedback. The gov-erning equation of N mean-field coupled R¨ossler oscillators with negative self-feedback delay canbe written as: ˙ x j ( t ) = − y j ( t ) − z j ( t ) + k [ x − x j ( t − τ )] , ˙ y j ( t ) = x j ( t ) + r j y j ( t ) + k [ y − y j ( t − τ )] , ˙ z j ( t ) = r j + z j ( t )( x j ( t ) − e ) + k [ z − z j ( t − τ )] , (7)where j = , , ... N . r j , e (= . ) are the intrinsic parameters of R¨ossler oscillator. x , y , and z are themeanfield term of x , y , z variables. We set the r j = − . j = , ..., pN for the inactive oscillatorsfalls into a fixed point and r j = . j = pN + , ..., N for active oscillators which shows the thechaotic oscillations. To study the aging transition in chaotic oscillators, we measure the amplitudeof oscillation R = M ( p ) M ( ) , where M is define as , M = q h ( X c − h X c i ) i , (8) X c = N − ∑ Nj = ( x j , y j , z j ) is the centroid, and h . i means it calculated for long time average.We have plotted the order parameter R in Figure 4(a) against the inactivation ratio p for the fixedvalue k = . τ . In the absence of local self-delay τ =
0, the brown line of R demarcated the critical aging transition point at p c = .
72, and when τ is increased, the transitionthreshold point p c increase to the higher value. It shows that local delayed self-feedback effectivelyenhances the robustness. In Fig. 4(b), we have shown the phase diagram in τ − p parameter plane for (c)OSAT0 0.5 1 k p (d)OSAT0 0.5 1 k p (b)OS AT0 0.5 100.51 FIG. 5. The phase diagram in ( p , k ) parameter plane for the network of mean-field coupled R¨ossler oscillators.The aging transition (AT) and oscillatory state (OS) regions are characterized by the light pink and white colorfor different value of local self-delay (a) τ =
0, (b) τ = .
2, (c) τ = .
4, and (d) τ = . a fixed coupling strength k = .
2, where two regions are observed, oscillatory state (OS) and agingtransition (AT). The critical value p c of aging transition from oscillatory to steady-state (where R =
0) increases with τ gradually. For R¨ossler model also we investigate the effects of couplingstrength on robustness. We have shown the phase diagram in ( p − k ) parameter space for differentvalue of τ in Figure 5. The light pink region in the parameter plane shows the aging transition wherethe order parameter R falls to zero. It clearly shows that the AT region decreases in size for the highervalues of τ , which is quite similar to the results we have obtained for Stuart-Landau oscillators. Itconfirms that self-feedback delay τ is also very useful in inflating the dynamical robustness of acoupled chaotic oscillator network. III. COMPLEX NETWORK TOPOLOGY
Next, we demonstrate the effectiveness of our enhancement technique when the oscillators areinteracting through a complex network topology . We Consider N mean-field coupled Stuart-Landau oscillators interacting via complex network topology. The mathematical model of the cou-pled system is given by˙ z j ( t ) = ( ρ j + i ω − | z j ( t ) | ) z j ( t ) + k (cid:18) ∑ Nl = A jl z l ( t ) r j − z j ( t − τ ) (cid:19) , (9)for j = , , ..., N and N = A jl is the adjacency matrix of the connection in the complexnetwork, i.e., A jl = j -th and l -th nodes are connected and zero otherwise. Here, we assume thatall coupled oscillators are interacting with each other through specific network topology and all theconnection between them have the same coupling strength. r j is the degree of node j, and k is thecoupling strength of interaction.Depending on the probability distribution of the degrees of nodes, we can categorize complexnetworks into two broad groups, namely homogeneous networks and heterogeneous networks .Degree distribution of homogeneous networks such as random graphs follows a binomial or Pois-son distribution. In contrast, heterogeneous networks such as scale-free networks obey a heavy-tailed degree distribution that can be approximated by a power law ( P ( k ) ≈ k − η , where P ( k ) is theprobability of having a node of degree m and η is the power-law exponent).In Fig. 6(a), we have plotted the order parameter Z as a function of inactivation ratio p for dif-ferent value of self-feedback delay term τ for a random network of N =
500 and the probability (a)0 0.5 1 p Z =0=0.04=0.08 (b)0 0.5 1 p Z =0=0.04=0.08 FIG. 6. The order parameter Z as a function of the inactivation ratio p for various values of τ in complexnetwork of N =
500 Stuart-Landau oscillators for a = , b = − , ω = k = γ = .
02, and (b) scale-free network of average degree h m i =8, and the the power law exponent η = . of connecting two distinct pair of nodes γ = .
02. It demonstrates the fact the p c value increasesas we increase τ . Even for a small amount of delay τ , dynamical robustness gets enhanced to asignificant amount. We have similar results for the scale-free network. In Fig. 6(b), we have shownhow p c value increases as we increase τ . These outcomes show that local self-feedback with timedelay is very efficient in enhancing the dynamical robustness when the oscillators are interactingthrough a complex network topology. It also establishes the fact that our enhancement technique isindependent of coupling topology. IV. CONCLUSION
In this paper, we have demonstrated that the introduction of a negative self-feedback with a delayinto the mean-field coupled oscillators can effectively increase the dynamical robustness, which isexhibited by enhancing the endurance of the oscillatory dynamics of the network against the agingof the individual nodes. We have demonstrated that our enhancement technique is applicable forboth global as well as local mean-field coupling. We have shown that the critical vale p c at whichthe aging transition takes place is positively correlated with the delay term τ . It means the networkbecomes more resilient to the effect of the local deterioration of oscillatory nodes with increases of τ . We have done a comprehensive, detailed study for the global network. We have seen that theenhancing effect of negative self-feedback with delay is more prominent for a strong coupling term k . To our surprise, we have found that for a non zero τ the aging transition exists for a finite lengthof the coupling strength k. Within this range, p c first diminishes from maximum to a minimumvalue and then again increases to the maximum. The global oscillations can be restored by localself-feedback with delays in coupled networks of purely non-oscillatory units. The introductionof delayed self-feedback in the coupling provides a straightforward but highly valuable techniquefor recovering dynamical activities in the network, whose oscillatory behavior has been weakeneddue to the aging of some elements. We have delineated the results using numerical simulations aswell as analytical findings. To state that our scheme is independent of local subsystems, we havesuccessfully employed our control mechanism to a network of coupled chaotic R¨ossler oscillators.We have also studied the role of negative self-feedback with a time delay to enhance dynamicalrobustness when the oscillators interact through a complex network topology. The results demon-strate the fact that our enhancement technique is independent of coupling topology.Our study might have applicability in enhancing the resilience of several natural systems thatcan experience aging in its local dynamical units. Lastly, our proposed framework widens theunderstanding of the roles of negative self-feedback delay in regulating oscillatory dynamics todesign network oscillators which can resist the aging transition. V. DATA AVAILABILITY
The data that support the findings of this study are available from the corresponding author uponreasonable request.
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