Parametric excitation induced extreme events in MEMS and Lienard oscillator
aa r X i v : . [ n li n . AO ] A ug Parametric excitation induced extreme events in MEMS and Liénardoscillator
R. Suresh a) and V. K. Chandrasekar b) Centre for Nonlinear Science and Engineering, School of Electrical and Electronics Engineering, SASTRA Deemed University,Thanjavur 613 401, India (Dated: 27 August 2020)
The two paradigmatic nonlinear oscillatory models with parametric excitation are studied. The authors provide theo-retical evidence for the appearance of extreme events (EEs) in those systems. First, the authors consider a well knownLiénard type oscillator that shows the emergence of EEs via two bifurcation routes: Intermittency and period-doublingroutes for two different critical values of the excitation frequency. The authors also calculate the return time of twosuccessive EEs, defined as inter-event intervals, that follow Poisson-like distribution, confirm the rarity of the events.Further, the total energy of the Liénard oscillator is estimated to explain the mechanism for the development of EEs.Next, the authors confirmed the emergence of EEs in a parametrically excited microelectromechanical system. In thismodel, EEs occur due to the appearance of stick-slip bifurcation near the discontinuous boundary of the system. Sincethe parametric excitation is encountered in several real-world engineering models, like macro and micromechanicaloscillators, the implications of the results presented in this paper are perhaps beneficial to understand the developmentof EEs in such oscillatory systems.
The study of bursting oscillations (large-amplitude oscilla-tions alternated with small-amplitude oscillations) is stillan active topic of research due to its ubiquitous nature.Bursting oscillations are encountered and reported in ex-periments as well as in models of dynamical systems rang-ing from physics to biology. In contrast with the con-ventional bursting patterns that occurred in dynamicalsystems, the development of certain types of oscillatorystates, in which the large-amplitude oscillations occasion-ally appeared in time with an entirely unpredictable na-ture. These rare and recurrent large-amplitude oscilla-tions are distinguished as extreme oscillations or extremeevents (EEs) in the literature. This topic receives signifi-cant attention in recent years among the scientific commu-nity, and the development of EEs have already been stud-ied and reported in various dynamical systems. In partic-ular, very recently, the emergence of EEs in nonlinear dy-namical models influenced by external periodic forcing isreported. Although the emergence of EEs and their mech-anism are studied in detail in nonlinear systems with ex-ternal forcing, very few studies have been reported the ap-pearance of EEs in systems with parametric excitation, inparticular, in laser models. However, the influence of para-metric excitation to induce EEs in other classes of dynam-ical systems also requires immediate attention. Therefore,in this paper, the authors provide the theoretical evidencefor the appearance of EEs in two paradigmatic nonlin-ear oscillatory models with parametric excitation. First,the authors consider a well known Liénard type oscillatorthat shows the emergence of EEs via two different bifur-cation routes: Intermittency and period-doubling routesfor two different critical values of the excitation frequency.Further, the rarity of the EEs is confirmed by calculat-ing the return time of the two successive EEs defined as a) Electronic mail: [email protected]. b) Electronic mail: [email protected]. inter-event intervals that follow the Poisson-like distribu-tion. The total energy of the system is analytically esti-mated to explain the emerging mechanism of EEs in thisoscillator. Next, the authors demonstrated the emergenceof EEs in a parametrically excited microelectromechani-cal system, in which EEs occurred due to the presence ofstick-slip bifurcation near the discontinuous boundary ofthe system.
I. INTRODUCTION
In dynamical systems, the complex oscillatory patterns withdifferent amplitudes are interspersed. In particular, the co-existence of small and large-amplitude oscillations knownas bursting has been encountered and reported in a varietyof fields from physics to biology . In contrast with theconventional bursting patterns, there exist intermittent large-amplitude oscillations that occasionally appeared in time areknown as extreme oscillations or extreme events (EEs). Thistype of events occurred in many natural systems and en-gineering models including oceanography , ecosystems ,geophysics , transportation networks , power supplynetworks , mechanical oscillators , neural networks ,plasma , optical fiber and lasers , etc and receives no-table attention among the scientific community during the pastdecade. The experimental demonstration of EEs has also beenevidenced in many scientific laboratory experiments .Though there is no exact mathematical definition for EEs, ac-cording to statistical perspective, it was generally admittedfact that EEs show a long-tailed probability distribution andthe peaks which are greater than the threshold height are char-acterized as EEs. The threshold height is equal to the time-averaged mean value of all the peaks in a measured time se-ries plus 4–8 times the standard deviation derived for the longrun. Extreme events with similar statistical properties havealready been studied and evidenced in nonlinear dynamicalsystems modelled by ordinary as well as partial differentialequations . Specifically, in refs , the authorshave reported the influence of external periodic force to in-duce EEs in a Liénard type oscillatory model and microelec-tromechanical system (MEMS) model.In general, an external periodic force might affect the os-cillatory system, either additively or multiplicatively. In theformer case, the entire system is driven by an external force,and in the latter situation, the periodic force influences on anyone of the system parameters, yielding quite different types ofsolutions. For example, the slow reduction of the catalytic ac-tivity in chemical reactions due to chemical erosion decreasesthe reaction performance . Periodic modulation in one of thesystem parameter in the microelectromechanical device in-duces the parametric resonance, which is used as vibration en-ergy harvesters . Electrostatically driven microelectrome-chanical systems are used to design highly effective bandpassfilters . In all these examples, there are certain control pa-rameters of the system vary periodically as a function of timeor manually altered between a specific range. Further, slowlyvarying parameters can lead to unusual and counter-intuitiveeffects such as stabilizing the unstable fixed points in the in-verted pendulum , periodic delay bifurcation in nonlineardynamical systems , etc. Therefore, the study of parametricexcitation or slowly varying control parameters in nonlineardynamical systems have been and continued to be an activetopic of research in many fields (See the refs and the ref-erences therein).Even though the study of EEs and their emerging mech-anism in the systems with external forcing are studied andacknowledged by the scientific community, very few stud-ies have been reported the emergence of EEs in parametri-cally excited nonlinear dynamical systems, especially in lasermodels . However, the influence of parametric exci-tation to induce EEs in other classes of dynamical systems,in particular, macro and micromechanical oscillator modelsalso require immediate attention. Therefore, in the presentstudy, the authors aim to investigate the dynamical changesthat occurred in the nonlinear systems in response to the para-metric excitation. In particular, they report the evidence ofthe occurrence of EEs in two paradigmatic examples of non-linear oscillators, a Liénard type model and in a cantilever-based MEMS model with parametric excitation. When onefirst considers the Liénard oscillator, for the suitable param-eter values, the large-amplitude oscillations appeared via twobifurcation routes, namely intermittency and period-doublingroutes at two critical values of the excitation frequency. Thethreshold height is estimated to classify the EEs from the fre-quent large-amplitude oscillations. In addition to that, the re-turn time of the two consecutive EEs is calculated, knownas inter-event intervals, which follows a Poisson-like distri-bution confirming the rare occurrence of the events. Further,the authors explained the mechanism responsible for the de-velopment of EEs in the Liénard oscillator using the totalenergy of the system. The occurrence of parametric excita-tion induced EEs is robust against system dynamics. To ver-ify this, next, the authors consider a MEMS model with dis-continuous boundaries to demonstrate the parametric exci- tation induced EEs. The models with discontinuous bound-aries are frequently encountered in mechanical oscillatorslike cantilever-based microelectromechanical oscillators ,mass-spring-damper oscillator , systems with friction , etc.Furthermore, the authors show that the existence of stick-slipbifurcation near the discontinuous boundary region is the un-derlying mechanism for the appearance of EEs in the MEMSmodel.One can also note here that, although when one implementsthe parametric excitation in the internal frequency of the Lié-nard and MEMS models, the underlying mechanism for theemergence of EEs is different for both the systems. Further-more, usually, the parametric resonance occurs when the ex-ternal excitation frequency equals to any integer multiples ofthe natural frequency of the oscillator. In both the systems,the emergence of EEs appeared for a specific region of the ex-citation frequency. However, the authors could not find anyrelation between them in the present study.The remainder of this paper has been organized as fol-lows: In Sec. II, the authors introduce the Liénard oscilla-tor with parametric excitation and demonstrate the emergenceand mechanism of EEs. Section III is devoted to the studyof EEs in a parametrically excited MEMS model. Finally, inSec. IV, the authors summarize their results with conclusions. II. EXTREME EVENTS IN THE LIÉNARD OSCILLATORA. Dynamical model
To start with, first, the authors consider a specific class ofLiénard-type nonlinear oscillator model with parametric exci-tation; that is,¨ x + α x ˙ x − γ [ + F cos ( ω t )] x + β x = , (1)where α is the magnitude of the position-dependent dampingor nonlinear damping, γ is the internal frequency parameterof the oscillator, and β represents the strength of cubic non-linearity. The parametric excitation amplitude and frequencyare represented by F and ω , respectively. In this model, theparametric excitation is introduced in the internal frequencyof the system. Thus, the intrinsic frequency parameter γ isperiodically time-varying as a function of F and ω .Equation (1) is equivalent to a well-known Mathieu’sequation when the damping is linear and β =
0. Mathieu’sequation exhibits unstable and stable behavior under para-metric excitation, which manifests into EEs in the stochasticgeneralization . Further, the mechanism causing EEs is alsoevident in the stochastic case . Nevertheless, in the presentstudy, the authors have considered a deterministic nonlin-ear equation (1) with parametric excitation and demonstratedthe emergence of EEs as a function of excitation parameters.When F = ,or as a conservative nonlinear oscillator perturbed by a nonlin-ear damping term ( α x ˙ x ). This type of oscillatory model withnonlinear damping arises in a broad class of physical, me- F ω LP1 LP2 PD1PD2 PD3SAC
FIG. 1. The two-parameter bifurcation diagram of the Liénard os-cillator (1) in the ( F , ω ) plane depicts the emergence of different bi-furcations. The system parameters are fixed as follows: α = . γ = .
5, and β = .
5. For other details, see the text. chanical, chemical systems and engineering models with ap-propriate transformations. For instance, various micro and na-noelectromechanical systems having nonlinear damping coef-ficient in the form of restoring force or stiffness coefficient ,likely the nonlinear elements in the electronic circuits. Be-sides, the autonomous Liénard oscillator shows the bistablenature of the coexistence of dissipative and conservative dy-namics depending on the initial conditions . Hence, thetrajectories either dissipate and approach to the stable fixedpoint or exhibit self-sustained periodic oscillations subject tothe initial conditions. Thus, the nonlinear damping term inthe Liénard oscillator act as damping, and also pumping term(which gives rise to self-sustained oscillations) depending onthe amplitude of the oscillations.Further, it is worth to emphasize that the autonomous Lié-nard oscillator is an example of the reversible system underthe transformation S : x → − x , t → − t + T , where T ( period ) = πω . Due to this reversible property, Liénard oscillator plays animportant role in Hamiltonian and non-Hamiltonian dynam-ics. One can also note here that the Liénard oscillator hasisochronous oscillation property and exhibiting parity-time( PT )-symmetric nature . In the next section, the dynamicsof Eq. (1) will be studied in detail in terms of the excitationparameters. The development and generation mechanism ofEEs will also be explained. B. Results and Discussion
The dynamics of an autonomous Liénard oscillator and itsdual characteristic nature of dissipative and conservative dy-namics are well studied and reported. Further, the effect of ex-ternal periodic forcing in the Liénard oscillator is examined,and the appearance of periodic mixed-mode oscillations andEEs in the model with external periodic forcing has also beenrecently evidenced with experimental confirmations . In
FIG. 2. The maxima of the dynamical variable ˙ x of the oscillator (1)are plotted against the parametric excitation frequency in the range of ω ∈ ( . , . ) with the fixed value of F = .
25 (where the horizontalline is marked in Fig. 1). The other system parameters are fixed as inFig.1. The vertical dashed lines indicate the ω values where differentbifurcations occur. The arrow marks and the corresponding labelsfrom (a) to (d) represents the ω values where the time evolution plotsare plotted which are depicted in Fig. 3. the present paper, the authors have considered a Liénard os-cillator model. Still, instead of the external force driving thesystem, it acts only on the internal frequency ( γ ), resulting inthe periodic change in γ .Equation (1) is numerically integrated using the fourth-order Runge-Kutta method with the time step of 0 .
01, and forthe present numerical study, the system parameters are fixedas α = . γ = .
5, and β = .
5. The initial conditions arechosen from the dissipative region. Therefore, the trajectoriesof the autonomous oscillator damp and approach to the stablefixed-point, or the forced oscillator exhibit periodic or non-periodic solutions based on the values of the excitation param-eters. The system dynamics is examined by varying the exci-tation amplitude in the range of F ∈ [ . , . ] and frequency ω ∈ [ . , . ] . The two-parameter bifurcation is depicted inFig. 1 to show the occurrence of various bifurcations. Fur-ther, in Fig. 2, the qualitative changes that occurred in the Lié-nard oscillator at different bifurcation points are portrayed asa one-parameter bifurcation diagram for ω ∈ [ . , . ] by fix-ing F = .
25 (at which the horizontal line is marked in Fig. 1).The curves marked as PD1, PD2 and PD3 in Fig. 1 representsthe parameter values at which the period-doubling cascadesoccur. These states are manifested in Fig. 2 by the verticaldotted lines for ω = . . . ω = . ω = . . Further, the amplitude of the large-sizedattractor is slowly decreasing in size with a reduced value of ω and when ω ≈ . ω from lower to higher val-ues. The authors emphasize here that the intermittency routeto the onset of EEs have already been reported in several otherdynamical models such as neuron models, in semiconductorand optically injected laser systems .In a nutshell, it is evident from Fig. 2 that the chaotic dy-namics have emerged through two distinct routes at two dif-ferent critical values of the excitation frequency. Also, it isclear from Fig. 2 that for ω ∈ ( . , . ] , the Liénardoscillator exhibit large-sized chaotic attractor in which thelarge-amplitude chaotic oscillations alternates with the small-amplitude chaotic oscillations in the time domain. Specifi-cally, for a certain range of ω , the large-amplitude oscillationsoccurred occasionally and at random time intervals, which arethen characterized as EEs. On the other hand, for other val-ues of ω , the system exhibits frequent large-amplitude chaoticoscillations. Therefore, to distinguish EEs, the authors haveused a threshold height, first proposed by Massel to defineextreme rouge waves that occurred in oceans, which has thenbeen widely used to characterize EEs in the literature of ex-treme value theory in recent times . The thresh-old height can be defined from the dynamical aspect as a de-viation of several standard deviations away from the averagevalue of the system observable, given as, H T = h P i + n σ , (2)where h P i is the time-averaged positive peak value of the ˙ x component of the Liénard oscillator, σ is the standard devi-ation of the ˙ x variable, and n is an integer, which is system-dependent. For the Liénard oscillator, the value of n is chosenas 8 . In order to calculate the threshold height H T , Eq. (1)is numerically integrated for a long run with the iterationsof 2 × time units of the system variable ˙ x , after leavingsufficient transients. During the emergence of frequent large-amplitude oscillations, the average value of the large peaks isvery high, and so H T becomes larger than the largest peaks.On the other hand, the occasional occurrence of large peakswhose amplitude is larger than H T are identified as EEs.The temporal evolution of the ˙ x variable of the Liénrd sys-tem is plotted in Figs. 3 (a) - (d) for different values of excita-tion frequency ω . Here, the authors have plotted the maximaof ˙ x variable ( ˙ x max >
0) for better clarity. The red (dark gray)horizontal line in these figures represent the value of H T . Foran illustration, if one notices the bifurcation diagram in Fig. 2with increasing ω , the periodic orbit suddenly bifurcates into alarge-sized chaotic attractor at ω ≈ . -6 -4 -2
0 1 2 3 P D F x . max (e)10 -6 -4 -2
0 2 4 P D F x . max (f)10 -6 -4 -2
0 1 2 3 P D F x . max (g)10 -6 -4 -2
0 1 2 3 P D F x . max (h) 0 1 2 3 5 6 7 8 9 10 x . m a x t (a) × x . m a x t (b) × x . m a x t (c) × x . m a x t (d) × FIG. 3. (a) - (d) Temporal evolution of the maxima of ˙ x variable( ˙ x max ), and (e) - (h) their respective probability distribution functions(PDF) of the oscillator (1) depict various dynamical states observedin the Liénard oscillator for different values of ω . (a) and (e) showsEEs emerged from the intermittency route for ω = . ω = . ω = . ω = . H T and the long-tail behavior of the PDF in (e) and (g)corroborate the occurrence of EEs. route. At this value of ω , the system displays a combina-tion of small-amplitude oscillations along with occasional in-termittent large-amplitude chaotic bursts. The temporal evo-lution of the system for ω = . H T . If one closely look into Fig. 3(a), the bursting orspikes (arrows mark some of them) have occurred almost atperiodic intervals. Nevertheless, the time difference betweenthe two successive EEs occurred at random intervals of time.Further, the authors also estimated the probability distributionfunction (PDF) of the time series (Fig. 3(a)), which shows thelong-tail behavior (trademark property of EEs). The PDF isplotted in Fig. 3(e) and the vertical dashed line indicates thevalue of H T .Further, when one starts increasing the excitation frequency( ω ) for higher values, the occurrence of large-amplitude os-cillations is also increasing, and the switching from a small-amplitude oscillation to large-amplitude oscillation is very -5 -3 -1 P D F IEI (a) × -5 -3 P D F IEI (c) ×
048 0 4 8 I E I n + × IEI n × (b)036 0 3 6 I E I n + × IEI n × (d) FIG. 4. (a) and (c) Inter-event interval (IEI) histogram of the Liénardoscillator shows the Poisson-like distribution confirming the rare na-ture of the EEs for ω = . ω = . frequent. The temporal dynamics of such a dynamical stateis illustrated in Fig. 3(b) for ω = . H T has a large value than the largest peaks,and no events are qualified as EEs, which is also evidentfrom Fig. 3(b). The corresponding PDF of Fig. 3(b) is plot-ted in Fig. 3(f), which almost has an equal number of largeevents compare to the small events. Figure 3(c) is plottedfor ω = . H T denotes the EEs dynamics. Moreover, one can alsoobserve that unlike in Fig. 3(a), here (in Fig. 3(c)) the timedifference between successive bursts (events) appeared com-pletely at random intervals of time. The long-tail behavior ofPDF (Fig. 3(g)) once again confirming the occurrence of EEs.Beyond ω = . ω = . . The IEI se-quences (IEI n ) is defined as the history of time intervals be-tween consecutive events in the event train. The authors con-sidered those bursts as events that are higher than the thresholdheight H T . Let t n be the occurrence time of the n th event in aset of N events, then the IEI n is a variable: IEI n = t n + − t n , n = , , · · · , ( N − ) . (3)The JIEI histogram can be defined as the serial correlationof IEI following an event (IEI n + ) as a function of the pro-ceeding one (IEI n ). To estimate the IEI and JIEI, the authorshave numerically generated large data (2 × time units) by leaving sufficiently large transient. The PDFs of the IEI areplotted in Figs. 4(a) and 4(c) using a semi-log scale, corre-sponding to the time evolution plots presented in Figs. 3(a)and 3(c), for ω = . ω = . P ( x ) = λ e − λ x , where x is the inter-event interval and λ > λ = . λ = . ω . The dashed lines in Figs. 4(b) and4(d) indicates the mean N − ∑ n = IEI n / ( N − ) value of JIEI. TheJIEI analysis reveals that most of the event intervals are foundto fall close to the origin of x - and y -axis, indicating that mostof the time there is little change in consecutive event intervals.When one moves away from the origin, the points in the JIEIhistogram follows a negative exponential distribution.The global dynamics of the oscillator (1) as a function of α , F and ω is portrayed as a two-parameter diagram depictedFig. 5. In particular, in Fig. 5(a) the authors have varied the pa-rameters α ∈ [ . , . ] and ω ∈ [ . , . ] on a fine grid andintegrated the system for a long run for F = .
25. Differentdynamical states are separated as follows: The peaks whichare above the sub-threshold value of the system observable˙ x = H T is used. Here, the value of H T is calculated at each set of parameters. More specifically, forthe set of parameters to which the peak values of the oscillatorare larger than H T are marked as EEs region. Smaller than H T is identified as the region with frequent large-amplitude os-cillations. Furthermore, the chaotic region is separated fromthe periodic state by estimating the largest Lyapunov expo-nent of the Liénard oscillator for each set of parameters. Thepurple (dark gray) region in Fig. 5(a) indicates the EEs, lightgray represents the region of very frequent large-amplitudeoscillations. The black region stands for the small-amplitudechaos, and finally, the white area indicates the periodic os-cillations. The enlarged area of Fig. 5(a) (marked as a rect-angle) is depicted in Fig. 5(b) in which EEs region is mani-fested. Similarly, the same dynamical states as a function ofexcitation amplitude ( F ) and frequency ( ω ) is also identifiedand portrayed with the same color codes in Fig. 5(c) by fix-ing α = .
44. From Fig. 5, one can notice the two differentroutes (period-doubling and intermittency) to the emergenceof EEs for a wide parameter range. If one fixes α in Fig. 5(a)or F in Fig. 5(c) for a constant value and vary the excitationfrequency ω , the emergence of EEs is perceived via intermit-tency and period-doubling routes.In the next section, the authors will provide analytical rea-soning and mechanism for the development of EEs in the Lié-nard oscillator. FIG. 5. (a) The two-parameter diagram in the ( α , ω ) plane of the parametrically excited Liénard oscillator (1) shows the occurrence of differenttypes of oscillatory states for F = .
25. Other parameters are fixed as given in Fig. 1. The light-gray represents the region where very frequentoscillations occurred, purple (dark-gray) indicates the EE region, the black domain illustrates the region of small-amplitude chaotic oscillationsand the white area depict the periodic oscillations. (b) Magnified area of Fig. 5(a) clearly shows the occurrence of EEs (purple points). (c)Two-parameter diagram of the Liénard oscillator in ( F , ω ) plane with α = .
44 shows the occurrence of different oscillatory states.
C. Possible Mechanism for EEs
For F = X = ( , ) , X = (cid:16) + q γβ , (cid:17) and X = (cid:16) − q γβ , (cid:17) . For the chosen parameter values (as in Sec. II A),the system has a saddle point at X = (0, 0), a stable focusat X = (1, 0) and an unstable focus at X = (-1, 0). All thesethree fixed points are illustrated in Fig. 6 as filled triangle ( X ),filled circle ( X ), and an open circle ( X ), respectively. Inter-estingly, due to the presence of the nonlinear damping term( α x ˙ x ) in Eq. (1) the stable focus at (1, 0) is linearly stable andnonlinearly unstable. Hence, the trajectories dissipate and ap-proach to the stable focus if one chooses the initial conditionswithin some region of the phase-space. Otherwise, the sys-tem exhibit nonisochronous periodic oscillations. Therefore,based on the choice of initial conditions the system has eitherdissipative or conservative nature. One can examine the dualnature of the Liénard oscillator in terms of the total energyof the system. The total energy of the oscillator (1) withoutparametric excitation ( F =
0) can be written as E = (cid:20) ˙ x + α ˙ x (cid:16) x − γβ (cid:17) + β (cid:16) x − γβ (cid:17) (cid:21) × e αΩ tan − α ˙ x + β (cid:18) x − γβ (cid:19) Ω ˙ x − (cid:16) γ β (cid:17) e απ Ω , (4)where Ω = (cid:0) (cid:1) p β − α . For γ ≥ β > α , the sys-tem displays the dual nature of conservative and dissipativedynamics even if the system admits Hamiltonian. If one sub-stitutes the initial conditions for ( x , ˙ x ) in Eq. (4), for some ini-tial conditions E has negative values and for those initial con-ditions the system exhibit dissipative dynamics. On the con-trary, the system has conservative dynamics, when the totalenergy of the system E ≥
0. The boundary which separatesthe dissipative and conservative region is called homoclinicorbit. The black region in Fig. 6 displays the total energy of the oscillator when E < F = F sin ( ω t ) . To be more spe-cific, the stable and unstable equilibrium points move sym-metrically back and forth as a function of time (but in op-posite directions), and the saddle point X remains the same.For example, the stable equilibrium point X oscillates from X + = (cid:16) + q ( γ + F ) β , (cid:17) to X − = (cid:16) + q ( γ − F ) β , (cid:17) and the un-stable fixed point X oscillates from X + = (cid:16) − q ( γ + F ) β , (cid:17) to X − = (cid:16) − q ( γ − F ) β , (cid:17) . These points are marked in Fig. 6as a filled square ( X + ), open square ( X + ), filled diamond( X − ) and open diamond ( X − ) points for F = .
25. Due tothe movement of the equilibrium points, the dissipative andconservative regions of the oscillator oscillates in the timedomain as a function of the excitation amplitude and fre-quency. Consequently, the dissipative region shrinks and ex-pand in the phase-space like a balloon depending on the valueof F sin ( ω t ) . To confirm this, the total energy of the oscilla-tor (1) is again computed for maximum ( + F ) and minimum( − F ) values of F sin ( ω t ) which is given by E ± = (cid:20) ˙ x + α ˙ x (cid:16) x − ( γ ± F ) β (cid:17) + β (cid:16) x − ( γ ± F ) β (cid:17) (cid:21) × e αΩ tan − α ˙ x + β (cid:18) x − ( γ ± F ) β (cid:19) Ω y − (cid:16) ( γ ± F ) β (cid:17) e απ Ω . (5)The dissipative dynamics of E + and E − regions are depictedin Fig. 6 as purple (dark-gray) and aqua (light-gray), respec-tively. The system shows chaotic dynamics when it satisfiesthe condition F ' αγ / q γβ sinh (cid:16) πω √ γ (cid:17) √ ω , (6)which was analytically derived from Melnikov’s method byassuming the nonlinear damping ( α ) and the excitation ( F )terms in Eq. (1) as perturbations. For the suitable values ofexcitation parameters, the oscillator exhibits small-amplitudechaotic oscillations, which is confined in the small area ofthe phase-space for a long time. During the expansion andcontraction of the dissipative region, the system trajectoriesalso crossing the dissipative region near the saddle point, at ir-regular time intervals, is then strongly repelled towards theunstable direction into the conservative region. Therefore,the trajectories gain energy and make large excursions in thephase-space resulting in large-amplitude oscillations. The tra-jectories which are making large excursions are then return tothe dissipative region after a while and confined in the small-amplitude attractor. The next large excursion is possible onlywhen the above stated condition occurred again.It is important to emphasize that when one includes an ex-ternal force as an additive term into the Liénard oscillator, thenthe movement of the equilibrium points in the phase-space isentirely different than the present case. For instance, if theforcing amplitude ( F ) has a positive (negative) value, then thefixed points X ( X ) and X move towards each other. When F reaches the maximum (minimum) threshold, the X ( X ) and X equilibrium points collide each other and disappeared viasaddle-node bifurcation and only X ( X ) fixed point is feasi-ble. Hence, the three-fixed-point system is transformed intoa single-fixed-point system and vice versa as a function oftime. Nevertheless, as discussed earlier, when one includesthe forcing as a multiplicative term in the Liénard oscillator,then X and X equilibrium points move symmetrically backand forth as a function of time (but in opposite directions)and X remains in the same position. Thus, the movement ofequilibrium points and the emerging mechanism of EEs in theLiénard oscillator with the addictive forcing term is differentfrom the mechanism discussed in the present paper.Considering the current problem, one can also note that theexpansion and contraction of the dissipative region occurredperiodically regarding the excitation frequency. Whereas, thechaotic dynamics of the system depends on both excitationand internal frequency of the oscillator (in this study, the in-ternal and external excitation frequencies are dissimilar). Thelarge excursions only occurred when these two incidents co-exist at the same time near the saddle point. If these twoevents coexisted at infrequent intervals, then the oscillatorexhibit EEs. Otherwise, the large-amplitude oscillations oc-curred more frequently.As mentioned earlier, the oscillator has bistable naturebased on the choice of initial conditions. If a trajectory isoriginated from the initial condition that is very near to thedissipative region with energy E (which is near, but greaterthan E + ) in the phase-space, then the trajectory emerged fromthat initial condition is travelled into the E + region when theexcitation F sin ( ω t ) takes negative values. Since the internalfrequency of the oscillator and the excitation frequency aredissimilar, then the trajectories which are crossing the E + do-main are trapped into the dissipative region for a long time,and occasionally comes out of it to make large excursions inthe conservative region. This phenomenon is possible only -1 0 1 2-2.5 0 2.5 x . x E E - E + X X X X X X X FIG. 6. The phase-space diagram of the Liénard oscillator (1) de-picts the dissipative regions for three different situations. The blackregion ( E ) represents the dissipative region for F = E + ) indicates the dissipative region of theoscillator when the excitation amplitude is maximum ( + F ), and theaqua (light gray) region ( E − ) represents the dissipative region whenthe excitation amplitude is minimum ( − F ). The arrows represent theposition of the equilibrium points of the Liénard oscillator for theabove three cases that are depicted in different symbols and labels.For more details, see the text. when the initial conditions are originated or near the E + en-ergy region. Nevertheless, the trajectories that are begun awayfrom the dissipative region ( E + ) have another basin of attrac-tion with quasiperiodic dynamics. This is clearly illustrated inFig. 7 in which the dissipative energy regions ( E − , E and E + )for different values of F (as discussed in Fig. 6) are plottedfor comparison. An initial condition (marked as a filled tri-angle) with energy E ( E > E > E + ) originated near to theenergy region E + is considered. The trajectory evolved fromthis initial condition is trapped inside the dissipative region( E + ) for a long time and occasionally come out of it, exhibit-ing EEs. Nevertheless, another initial condition (indicated asa filled circle) with energy E , started away from the energyregion E + has quasiperiodic dynamics.Next, the authors have considered another dynamical sys-tem, a microelectromechanical cantilever system model, to il-lustrate the parametric excitation induced EEs phenomenon. III. EXTREME EVENTS INMICROELECTROMECHANICAL SYSTEM
The normalized dimensionless equation of motion of theMEMS model is given as ¨ x + α ˙ x + γ [ + F cos ( ω t )] x − β ( − x ) = , (7)where x is the displacement of the movable cantilever struc-ture concerning the x -axis. α is the strength of the damp-ing force, γ belongs to the stiffness constant of the structure, -2 0 2 4 -3 0 3 x . x E E - E + E E FIG. 7. Evolution of the trajectories of the Liénard oscillator whenone chooses the initial conditions in different energy regions. Thechosen initial conditions are represented as a filled triangle and thefilled circle. The dissipative regions depicted in Fig. 6 for differentvalues of F are plotted here for comparison. The chaotic dynamicsof the oscillator is plotted as pink (light gray) lines when it is startedfrom very near to the region E + . The oscillator exhibits quasiperiodicdynamics when one chooses the initial conditions in the E region(away from E region). β represents the strength of the nonlinear electrostatic actu-ation force, F and ω are the amplitude and frequency of theparametric excitation, respectively. Equation (7) was numer-ically integrated using the Runge-Kutta integration methodwith adaptive step size and to obtain the numerical solutions,the system parameters are fixed as follows: α = 0.71, γ = 0.5,and β = 0.32. The system (7) has a discontinuous boundarydue to the presence of singularity at x = x > x < . Systems with discontinuousboundaries are frequently encountered in mechanical andlaser models . The nonlinear behavior of the system (7)has already been studied in the literature (refer to the reviewarticle and the references therein). In particular, theoreti-cal analysis and experimental results on the dynamical behav-ior of a bistable MEMS oscillator have been investigated .Also, the influence of super-harmonic excitation in the MEMSmodel is studied, and the emergence of chaotic oscillations viaperiod-doubling bifurcation is observed . In addition to that,the resonance property of the parametrically excited MEMSmodel without the nonlinear term ( β = .Further, in a recently reported study , the authors have in-corporated the external force as an additive term in the au-tonomous MEMS model and demonstrated the developmentof EEs. However, the influence of parametric excitation to in-duce EEs in the MEMS model and its developing mechanismare still unknown. Therefore, in the present paper, the au-thors have considered the MEMS model with parametric ex-citation (forcing as a multiplicative term as given in Eq. (7))and demonstrated the emergence of EEs as a function of theexcitation parameters F and ω .Without the parametric excitation ( F = FIG. 8. The one-parameter bifurcation diagram of the MEMS model(7) as a function of parametric excitation amplitude ω ∈ ( . , . ) .The other system parameters are fixed as α = . γ = . β = .
32, and F = . H T is plotted as a blue (dark gray) line. Theinset depicts the magnified view of the rectangular part to show thecritical value of ω in which the system exhibit EEs. system has three equilibrium points X = ( x , ) and X , = (cid:18) − x ± √ ( − x ) x , (cid:19) when γ > β /
4. Else, the system hasonly one fixed point of X . Here x is the solution of the equa-tion x − x + x − β / γ =
0. For the above-chosen parametervalues, the system has only one fixed point at X = (1.38248,0), which is a stable focus. If one incorporates the paramet-ric excitation and slowly decreases its frequency from higherto lower values by fixing the amplitude of the excitation, thenthe system exhibits various dynamical states. These dynam-ical transitions can be realized by plotting the bifurcation di-agram of the system (7), which is depicted in Fig. 8, showsthe qualitative changes that occurred in the dynamical vari-able x of the MEMS model as a function of ω ∈ [ . , . ] for F = .
48. The authors define EEs in the system (7) whenthe system variable x exceeds four times ( n =
4) the standarddeviation over mean peak value h P i in Eq. (2). The blue (darkgray) line in Fig. 8 indicates the threshold height H T calcu-lated using Eq. (2) in which h P i is the mean peak value ofthe x component of the MEMS model (7), and σ is the stan-dard deviation of the x variable. For ω ≥ . ω = . ω . For ω = . H T and do not qualify as EEs. Theamplitude of the large peaks is continuously increasing as ofdecreasing the frequency, and for ω = . H T , corroborating the development of EEs inthe system. The inset of Fig. 8 shows the magnified view ofthe small portion of the bifurcation diagram confirming theonset of EEs in the MEMS model.The time evolution, phase portraits and the PDFs of theMEMS model are plotted in Fig. 9 for two different valuesof ω . Figure 9(a) depicts the time traces of the maxima of the x m a x t × x m a x t × -10 0 10 0 5 10 x . x (c) 10 -6 -4 -2
0 10 20 30(f) P D F x max -20 0 20 40 60 0 15 30(d) x . x -6 -4 -2
0 10 20 30(e) P D F x max FIG. 9. Time evolution of the system (7) is plotted in the left verticalpanel for different values of forcing frequency (a) ω = .
856 and(b) ω = .
845 for F = .
48. The corresponding attractor and theprobability density function are depicted in middle [(c) and (d)] andright panels [(e) and (f)], respectively. The horizontal red (dark gray)line in the figures (a) and (b) and vertical dashed line in figures (e)and (f) represents the value of H T with n = system variable x for ω = . H T . The corresponding phase-space plot is drawnin Fig. 9(c) and the localized structure of the PDF plotted inFig. 9(e) also validating the small-amplitude oscillations ofthe system. The double-hump like the structure of the PDF inFig. 9(e) shows that the maxima of the system variable x areconfined in two different places of the phase-space, which isvisible in the attractor (Fig. 9(c)). If ω is decreased further,then the system exhibits intermittent large-amplitude oscilla-tions along with the small-amplitude chaos as illustrated inFig. 9(b) for ω = .
845 as a time series plot. From this fig-ure, one can notice the emergence of frequent large peaks thatare coexisted with the small-amplitude oscillations. Amongthem, only very few peaks are larger than the threshold height H T , qualified as EEs. The large excursions of the orbit inFig. 9(d) corroborate the appearance of EEs. Moreover, thelong-tail nature of the PDF in Fig. 9(f) is also confirming theoccurrence of EEs.The occasional occurrence of the EEs is again confirmedby estimating IEI and JIEI histograms of the MEMS modelfor ω = .
845 (equivalent to Fig. 9(b)) which are depicted inFigs. 10(a) and 10(b), respectively. The PDF of the IEI is thenfitted by P ( x ) = λ e − λ x with scaling parameter λ = . F , ω ) plane is plot-ted in Fig. 11 to identify and distinguish the EEs region fromthe non-extreme event region. The dark green (gray) areain Fig. 11 represent the non-extreme event region where themaximum peak value of the system observable x ( x max ) issmaller than H T . Other colored regions denote the large peaks,which are higher than H T , corroborating the extreme eventregion. Different color gradients indicate different peak val-ues of the MEMS model. One can also note that when theamplitude of the system is increased, then the corresponding -5 -3 -1 P D F IEI (a) ×
036 0 3 6 I E I n + × IEI n × (b) FIG. 10. (a) IEI histogram and (b) JIEI plot of the MEMS model (7)shows the Poisson-like distribution corroborating the emergence ofEEs for ω = .
845 and F = . ω F x max FIG. 11. The two-parameter bifurcation diagram of the system (7) asa function of the parametric excitation amplitude F ∈ [ . , . ] andexcitation frequency ω ∈ [ . , . ] . The maximum peak value, x max is indicated in different colors. The dark green (gray) regionshows the non-extreme event region where the peak values are below H T . Other colored regions are indicating extreme event dynamicswith different x max values. The white and black filled circles indicatethe set of parameter values ( F and ω ) at which the time series of theFigs. 9(a) and 9(b) are plotted. standard deviation value ( σ in Eq. (2)) is also increases result-ing in the increment of the threshold height H T accordingly.Therefore, as stated earlier, H T is calculated at each set ofparameters. The white and black filled circles in Fig. 11 indi-cates the parameter values ( F and ω ) at which the time evolu-tion plots in Figs. 9(a) and Fig. 9(b), respectively, are plotted.It is important to emphasize here that, although the systemis parametrically excited, the emerging mechanism of EEs issimilar to the mechanism discussed in ref. . Nevertheless,the parameter space in which EEs occurred is entirely differ-ent. As discussed earlier, the MEMS model has a discon-tinuous boundary at x =
1. This discontinuity induces thesliding motion of the trajectory in the phase-space. Hence,the trajectories which are entering into this sliding region un-dergo a stick-slip bifurcation. Therefore the trajectories aresliding over a long distance in the phase-space causes large-amplitude oscillations. The phase-space plot of the system (7)0 -20 0 20 40 60 0 15 30 45 x . x -6-3 0 3 1 1.05 1.1 FIG. 12. The phase-space diagram depicts the trajectories of theMEMS model (7) for three different values of ω . The black line rep-resents the attractor for ω = . ω = . ω = . x = is plotted in Fig. 12 for three different values of the excita-tion frequency ( ω ) to elucidate this phenomenon. First, for ω = . ω = . ω = . IV. CONCLUSION
To consolidate the results, the authors have investigatedthe dynamics of the parametrically excited two nonlinearoscillators, namely Liénard type oscillator and MEMSmodel to demonstrate the emergence of EEs in terms of theparametric excitation parameters. In the Liénrd oscillator,the authors confirmed that the appearance of EEs occurredvia two distinct dynamical routes: period-doubling andintermittency when varying the excitation frequency. Thethreshold height H T is calculated to distinguish EEs from theother dynamical states. Further, the long-tail behavior of the PDF additionally confirms the development of EEs in theLiénard oscillator. Furthermore, the authors also estimatedthe return time of the two successive EEs defined as IEI, forthe long run to quantify the rare occurrence of EEs. Theyshowed that the PDF of the IEI follows the Poisson-likedistribution, which reconfirms the emergence of rare events.The possible mechanism responsible for the development ofEEs in the Liénard oscillator is discussed based on the totalenergy of the system. Next, the authors have demonstratedthe emergence of EEs in a parametrically excited MEMSmodel with discontinuous boundary and found that theoccurrence of stick-slip bifurcation near the discontinuousboundary is the crucial mechanism for the appearance ofEEs. The results presented here may be advantageous tounderstand the emergence of EEs occurred in macro, andmicromechanical oscillators due to parametric excitation andalso these bursting oscillations are found to be beneficial forenergy harvesting applications . AUTHOR’S CONTRIBUTIONS
All authors contributed equally to this work.
ACKNOWLEDGMENTS
The work of R. S. is supported by SERB-DST FastTrack scheme for young scientists under Grant No.YSS/2015/001645. The work of V. K. C. is sponsored bythe SERB-DST-MATRICS Grant No. MTR/2018/000676 andCSIR EMR Grant No. 03(1444)/18/EMR-II.
DATA AVAILABILITY
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