Dangerous Aging Transition in a Network of Coupled Oscillators
aa r X i v : . [ n li n . AO ] J u l Dangerous Aging Transition in a Network of CoupledOscillators
Biswambhar Rakshit, a) Niveditha Rajendrakumar, b) and Bipin Balaram c) Department of Mathematics, Amrita School of Engineering, Coimbatore,Amrita Vishwa Vidyapeetham, , India Aerodynamics and Wind Energy Department, Faculty of Aerospace Engineering,Delft University of Technology Department of Mechanical Engineering, Amrita School of Engineering, Coimbatore,Amrita Vishwa Vidyapeetham, , India (Dated: 28 July 2020)
In this article, we investigate the dynamical robustness in a network of relaxationoscillators. In particular, we consider a network of diffusively coupled Van der Poloscillators to explore the aging transition phenomena. Our investigation reveals thatthe mechanism of aging transition in a network of Van der Pol oscillator is quitedifferent from that of typical sinusoidal oscillators such as Stuart-Landau oscillators.Unlike sinusoidal oscillators, the order parameter does not follow the second-orderphase transition. Rather we observe an abnormal phase transition of the order param-eter due to sudden unbounded trajectories at a critical point. We call it a dangerousaging transition. We provide details bifurcation analysis of such abnormal phasetransition. We show that the boundary crisis of a limit-cycle oscillator is at the helmof such a dangerous aging transition.PACS numbers: Valid PACS appear hereKeywords: Dynamical Robustness, Aging transition, Van der Pol oscillator
I. INTRODUCTION
Studying diffusively coupled oscillators have provided great insights to elucidate a plethora ofimportant self-organising activities in various complex systems ranging from physics, chemistry,biology, and engineering . Collective dynamics of coupled oscillators crucially depends not onlyon the intrinsic dynamics of individual oscillators but also on the nature of coupling topology. Inmany natural and man made systems robust oscillatory dynamics is an essential requirement fortheir proper functioning. Emergent rhythmic dynamics of such large-scale systems should be robustagainst various local degradation or deterioration.In recent past, researchers have spent considerable amount of time to explore the robustness ofrhythmic activities in a network of coupled oscillators when a fraction of the dynamical componentsare deteriorated or functionally degraded but not removed . If this degradation reaches a certaincritical point, the regular functioning of such systems may hamper and face severe disruption. Thisemergent behaviour is described as aging transition and is an active area of research . The ag-ing transitions might cause catastrophic effects in many natural and real-world systems such asmetapopulation dynamics in ecology, neuronal dynamics in brain, cardiac oscillations, and power-grid network . Therefore, it is of immense practical interest to understand the mechanisms ofaging transition in various complex systems which will eventually help us to propose some controlmechanisms to avoid such catastrophes.Till now studies have focussed on dynamical robustness by considering different coupling topolo-gies of the network or using different coupling functions . Some possible remedial measureshave also been proposed by some researchers to enhance the dynamical robustness . Most ofthese studies are based on Stuart-Landau limit cycle oscillator which describes dynamics near a a) Electronic mail: [email protected] b) Electronic mail: [email protected] c) Electronic mail: b [email protected] supercritical Hopf bifurcation. Nevertheless, Kundu et al studied the aging transition in ecologicalas well as neuronal model . In authors have studied the aging transition in a metapopulationwhere as in authors have shown the effects of chemical synapse to enhance the robustness in amultiplex network. To the best of our knowledge till now aging transition has not been well studiedfor relaxation oscillators. In our present study we explore how aging transition takes place in a net-work of Van der Pol oscillators. The Van der Pol equation which exhibits non-sinusoidal oscillationis a well known relaxation oscillator and it has been used widely to model phenomena in physical,engineering and biological sciences. Extensions of such oscillators are very useful to model theelectrical activity of the heart and action potentials of neurons .In this work, we have investigated the mechanism of aging transition in a network of a diffusivelycoupled van der Pol oscillators. Mathematically this situation can be modeled as a network of cou-pled oscillators where oscillatory nodes switch to equilibrium mode progressively . If the numberof nodes that perturbed to equilibrium state from oscillatory state reaches a critical level, the nor-mal activities of such systems may hamper and face severe disruption. We have observed that thecharacteristic of the aging transition in a globally coupled network of Van der Pol oscillators is verydifferent from that of Stuart-Landau oscillators. Our investigation reveals that as the network un-dergoes aging transition the dynamics of the whole network become unbounded at a critical point.This is a matter of serious concern in practical systems, and we call it a dangerous aging transition.We have provided details bifurcation mechanisms responsible for such a catastrophic aging transi-tion. We have established the fact that this dangerous aging transition takes place as a result of theboundary crisis of the limit cycle attractor. II. MODEL
For our present study consider the prototypical Van der Pol oscillator with nonlinear dampinggoverned by a second-order differential equation. The mathematical form of this relaxation oscilla-tor is given by ¨ x + µ ( x − ) ˙ x + x = x is the dynamical variable, and µ is the parameter. For µ < µ = µ >
1. Equating ˙ x = y gives˙ x = y ˙ y = − µ ( x − ) y − x (2)Here the period of oscillations is determined by some form of relaxation time and it is quite differentfrom sinusoidal or harmonic oscillations. The Van der Pol equation given above can be used tomodel various real life phenomena in physical, engineering, and biological sciences. III. NETWORK OF COUPLED OSCILLATORS
Next we consider a network of N diffusively coupled Van der Pol oscillators represented by thefollowing equation: ˙ X i = F ( X i ) + κ N ∑ j = A i j ( X j − X i ) (3)where X i = ( x i , y i ) T denotes the state vector and F ( X i ) = ( f ( x i , y i ) , g ( x i , y i )) T represents the inherentdynamics of the i -th node as in equation (1) for i = , , ..., N . In our present study we consider F ( X i ) = ( y i , − µ ( x i − ) y i − x i ) T . Which implies that the inherent dynamics of each individual nodeis given by the Van der Pol oscillator. Second term is the diffusive coupling that describes nature ofinteractions between dynamical elements i and j . A i j represents the adjacency matrix where A i j = i -th and j -th nodes are connected, otherwise A i j =
0. Here we consider that a fraction p of nodesare inactive. That means out of N nodes, pN number of nodes are in inactive mode(equilibriumstate), whereas ( − p ) N number of nodes are in active mode(oscillatory mode). Whenever a fraction p of the nodes become inactive due to local deterioration we observe phase synchronised oscillation.The individual inactive oscillator, which cannot oscillate by itself, able to oscillate in the networkdue to continuous inputs from neighbouring active oscillators. To measure the dynamical activity inthe network we define an order parameter R , where R = N ∑ Ni = ( h x i , max i t − h x i , min i t ) . Here nonzero R signifies persistence of oscillation in the network whereas R =
1. Global Coupling
For our present study we consider global (all-to-all) coupling and investigate the aging effects.Without loss of generality, we set the group of inactive elements for j = , , ..., N p and rest of thenodes as active for j = N p + , ..., N . For our numerical simulations, we set N =
200 as total numberof nodes in the network. For active oscillators we set µ = . µ = − .
1. Ourchoice of parameter value is based on the assumption that as the active node becomes inactive(dueto perturbation) its parameter value will be in close proximity of the Hopf bifurcation point. Withthese choice of parameter values, an isolated node manifests either a stable equilibrium state at theorigin (inactive oscillator) or a stable relaxation oscillations (active oscillator). p D p c2 =0.92p c1 =0.87 FIG. 1. The normalized order parameter D = R ( p ) R ( ) is plotted against the inactivation ratio p for the couplingstrength k = . For studying the macroscopic oscillation level of the entire network in Fig.1 we have plottedthe the normalised order parameter D = R ( p ) R ( ) against inactivation ratio p for the coupling strength k =
20. There are two critical points p c and p c in the figure where explosive phase transitions aretaking place. From p = p = p c the the order parameter remains constant. At the critical point p c the order parameter becomes unbounded. This basically implies that the dynamics of individualnodes diverges to infinity. This scenario continues till we reach the second critical point p = p c . At p c another phase transition takes place and the order parameter D converges to zero and remainsthe same till the end point p =
1. In the range ( p c , p c ) we have considered the finite time length tosimulate the network dynamics in order to show the diverging nature of the order parameter D . In FIG. 2. Time series for different inactivation ratios:(a) p = .
8, (b) p = .
9, and (c) p = . support of our claim, in Fig.2 we have plotted the time series of the network for different inactivationratio p . In Fig.2(a) we have considered p = . < p c and the network dynamics converges to astable limit cycle behaviour. For p = . p c , and p c weobserve diverging time series (Fig.2(b)) and for p = . > p c network dynamics converges to astable steady state(Fig.2(c)). Simulation of the network of oscillators while it is experiencing agingtransition reveals two surprising facts. We observe that even though the network is undergoingaging transition the average amplitude of the network remains almost same till the inactivation ratio p reaches its critical value p c . Our second observation is abnormal phase transition at p = p c which is explosive one and it has a far reaching consequence as far as the practical systems areconcerned. This explosive nature of the phase transition might cause great damage to the system.We call it a dangerous aging transition. At the same time since the average amplitude of the systemdoes not give any indication of the impending aging transition and the consequent abnormal phasetransition of the order parameter its is also very difficult to develop any early warning signal.For better understanding of this fascinatingly surprising phenomena and some analytical treat-ment of our results we use the system reduction technique developed by Daido et al . Synchronisedactivity among the oscillators permits us to reformulate the coupled system. By setting x i = A x , y i = A y for active oscillators and x i = I x , y i = I y for inactive oscillators we reduce the coupledsystem (3) as ˙ A x = A y + K p ( I x − A x ) , ˙ A y = − µ ( A x − ) A y − A x + K p ( I y − A y ) , ˙ I x = I y + Kq ( A x − I x ) , ˙ I y = − µ ( I x − ) I y − I x + Kq ( A y − I y ) (4)where q = − p .To obtain the value of p c analytically, a linear stability analysis of the reduced system (4) aroundthe origin ( A x = , A y = , I x = , I y =
0) can be carried out. The point p = p c is basically the Hopfbifurcation point of the reduced system. In Fig.3 we have plotted the bifurcation diagram (usingXPPAUT ) with respect to the inactivation parameter p and it reveals some interesting facts aboutthe aging transition. There is a stable limit cycle along with an unstable equilibrium point till thebifurcation parameter reaches the the value p = .
88. At this point, the stable limit cycle ceases p -3-2-10123 A x p -2-1012 A x (a)(b) FIG. 3. (a) Bifurcation diagram of the reduced system(using XPPAUT) with respect to the inactivation pa-rameter p . Here the black line corresponds to an unstable equilibrium while the red line represents a stableequilibrium. Green and blue line represents a stable and unstable limit cycle respectively. (b) Zoomed portionof figure (a). to exist and the trajectories diverge to infinity. This dynamical behaviour of the reduced systemperfectly matches with the actual network dynamics. The parameter value p = .
88 corresponds to p c in the actual network. At the p = . p = . p = . p c in the actual network.The existence of the small limit cycle, which is shown in Fig.3 for the reduced model in the range(0.9128, 0.9136), is not apparent in Fig.1 for the actual network dynamics. This is be because ofthe small size of the network we have considered due to lack of computational facility. However, inFig.4 we have plotted the normalised order parameter D for a small range of parameter value p byconsidering N = ( . , . ) .Next we investigate divergence of trajectories after the critical point p c . In Fig.5 by usingPoincare section representation we have plotted 2-dimensional cross sections( I x − I y plane) of 4-dimensional basin of attractions of the stable limit-cycle attractor for different values of the inacti-vation parameter p . We have calculated the 4-dimensional basin of attraction by varying I x , y in therange ( − , ) and equating A x , y = I x , y . White region represents the basin of attraction of the limitcycle. We have also plotted the limit-cycle attractor(blue circle). From this figure we can observethat as the parameter p tends to the critical value p c the basin of attraction gradually decreases andthe limit-cycle attractor approaches the basin boundary. So it clearly demonstrates that at the thepoint p = p c a boundary crisis for the limit cycle attractor takes place and this global bifurcationeventually leads to the unbounded behaviour of the trajectories. IV. CONCLUSION
In this paper, we have investigated the aging transition in a network of diffusively coupled Vander Pol oscillators. Our studies claim that the aging transition in coupled Van der Pol oscillatorsis qualitatively very different from that of sinusoidal or harmonic oscillations (Stuart-Landau os- p D FIG. 4. The normalized order parameter D = R ( p ) R ( ) is plotted against the inactivation ratio p in the range ( . , . ) . Here we consider N = k = . p . White region is the basin of attraction of the limit cycle. Onecan clearly see that the basin of attraction gradually decreases as the parameter p approaches the p c . cillators). In the case of Stuart-Landau oscillators, the order parameter decreases gradually and itfollows a typical second-order phase transition. But in the case of Van der Pol oscillators order pa-rameter does remain a constant over a large parameter range and then undergoes an abnormal phasetransition. In particular, it blow-up suddenly at a critical parameter value. As we vary the parameterfurther the order parameter becomes finite again. We have it a dangerous aging transition due tothe sudden blow-up of trajectories. By applying the system reduction technique we have exploredthe bifurcation mechanism responsible for this surprisingly interesting aging transition. Our studyshows that the boundary crisis of the limit cycle attractor, also known as the blue-sky catastrophe,is accountable for this dangerous aging transition.Our investigation provides significant new insight on the dynamical robustness of complex sys-tems which can be modelled as a network of coupled Van der Pol oscillators. Our study will havea significant impact to invoke broad interests in the community of nonlinear systems as well as invarious applications in the fields of science and technology. In the present study, we have consid-ered regular homogeneous (all to all coupling) network. However, many natural systems followcomplex network topology and it will be interesting to study the aging transition of coupled Van derPol oscillators on the top of a complex network. It will be part of our future study. V. DATA AVAILABILITY
The data that support the findings of this study are available from the corresponding author uponreasonable request.
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