Influence of dissipation on extreme oscillations of a forced anharmonic oscillator
aa r X i v : . [ n li n . C D ] A ug International Journal of Non-Linear Mechanics 00 (2020) 1–15
Int JNonlinMech
Influence of dissipation on extreme oscillations of a forcedanharmonic oscillator
B. Kaviya a , R. Suresh a, ∗ , V. K. Chandrasekar a , B. Balachandran b a Centre for Nonlinear Science and Engineering, School of Electrical and Electronics Engineering, SASTRA Deemed University, Thanjavur 613401, India b Department of Mechanical Engineering, University of Maryland, College Park, MD 20742, USA
Abstract
Dynamics of a periodically forced anharmonic oscillator with cubic nonlinearity, linear damping, and nonlinear damping, is studied.To begin with, the authors examine the dynamics of an anharmonic oscillator with the preservation of parity symmetry. Due to thissymmetric nature, the system has two neutrally stable elliptic equilibrium points in positive and negative potential-wells. Hence,the unforced system can exhibit both single-well and double-well periodic oscillations depending on the initial conditions. Next,the authors include position-dependent damping in the form of nonlinear damping ( x ˙ x ) into the system. Then, the parity symmetryof the system is broken instantly and the stability of the two elliptic points is altered to result in stable focus and unstable focusin the positive and negative potential-wells, respectively. Consequently, the system is dual-natured and is either non-dissipativeor dissipative, depending on location in the phase space. The total energy of the system is used to explain this dual nature of thesystem. Furthermore, when one includes a periodic external forcing with suitable parameter values into the nonlinearly dampedanharmonic oscillator system and starts to increase the damping strength, the parity symmetry of the system is not broken rightaway, but it occurs after the damping reaches a threshold value. As a result, the system undergoes a transition from double-well chaotic oscillations to single-well chaos mediated through a type of mixed-mode oscillations called extreme events (EEs)in which the small-amplitude single-well chaotic oscillations are interrupted by rare and recurrent large-amplitude (double-well)chaotic bursts. Furthermore, it is found that the large-amplitude oscillations developed in the system are completely eliminated ifone incorporates linear damping into the system. Hence, it is believed that a novel means has been identified for controlling theEEs that occur in forced anharmonic oscillator system with nonlinear damping. The numerically calculated results are in goodagreement with the theoretically obtained results on the basis of Melnikov’s function. Further, it is demonstrated that when oneincludes linear damping into the system, this system has a dissipative nature throughout the entire phase space of the system. Thisis believed to be the key to the elimination of EEs. Keywords:
Anharmonic oscillator, Position dependent damping, Nonlinear damping, mixed-mode oscillations, Extreme events,
1. Introduction
Forced and damped nonlinear oscillators have been considered as paradigms for mimicking the dynamics of vari-ous physical and engineering systems such as Josephson junctions, electrical circuits, optical systems, macromechan-ical and microelectromechanical oscillators, and so on [1, 2, 3, 4, 5].
Damping , which is used to model loss of energy ∗ Corresponding author
Email addresses: [email protected] (R. Suresh ), [email protected] (V. K. Chandrasekar), [email protected] (B.Balachandran) . Kaviya et al. / International Journal of Non-Linear Mechanics 00 (2020) 1–15 due to friction and viscous forces, is ubiquitous in many mechanical systems and this characteristic influences the per-formance of the oscillators in di ff erent ways. Most oscillatory systems are subject to di ff erent damping combinationsand each one of them has di ff erent e ff ects on the considered dynamical systems. In general, it is common to use alinear damping model for describing damping or dissipation experienced by a system. However, in many oscillatorysystems such as microelectromechanial and nanoelectromechanical oscillators, nonlinear damping is found to play asignificant role. For example, in nanoelectromechanical systems made from carbon nanotubes and graphene, dampingis found to strongly depend on the amplitude of motion, and the damping force is nonlinear in nature [6]. These sys-tems are being used for mass and force sensing applications [7, 8]. Researchers have exploited the nonlinear nature ofdamping in these systems to improve the figures of merit for both nanotube and graphene resonators. In reference [9],the ion steady-state motion is well described by the Du ffi ng oscillator model with an additional nonlinear dampingterm. Both the linear damping and nonlinear damping can be tuned with the laser-cooling parameters helping oneto investigate the mechanical noise squeezing in laser cooling. Recently, the influence of nonlinear damping on themotion of a nanobeam resonator was studied and it was found that nonlinear damping can have a significant impacton the dynamics of micromechanical systems [10]. In fluid mechanics, linearly forced isotropic turbulence can bedescribed by an anharmonic oscillator model with nonlinear damping [11]. From a dynamics viewpoint, it has beenshown that nonlinear damping can be used to suppress chaos in oscillatory systems [12, 13, 14]. The stability ofresponses of nonlinearly damped, hard and soft Du ffi ng oscillators have also been analyzed [15, 16]. In addition, thee ff ect of nonlinear damping in forced Du ffi ng and other types of nonlinear oscillators has been extensively studied[17, 18, 19, 20, 21, 22].Nonlinear damping plays a significant role in the dynamics of systems driven by a direct external periodic forcingor a parametric excitation [2, 23, 24, 25]. Specifically, the development of mixed-mode oscillations and extremeevents (EEs) have been recently reported in systems influenced by nonlinear damping [26, 27, 28, 31, 32]. The rareand recurrent occurrence of large-amplitude events in system variables with heavy tails in the probability distributionis a signature of EEs. Examples of EEs that occur in natural and engineering systems include rogue waves in opticalsystems and oceans, epidemics, large-scale power black-outs in electrical power grids, harmful algal blooms in marineecosystems, jamming in computer and transportation networks, stock market crashes, and epileptic seizures [33, 34,35, 36, 37, 38, 39, 40, 41]. Similar statistical behaviors of the appearance of sudden changes in the system variableshave been noticed in many dynamical systems governed by nonlinear equations with nonlinear damping [27, 31, 32].However, an understanding of the occurrence of EEs in such systems is still being developed and the significance ofnonlinear damping for the development of EEs has not received careful attention. Furthermore, an understanding of themechanism that triggers EEs in dynamical systems is crucial for developing strategies to control such events. Althoughthis is out of reach in natural systems, it may certainly be possible in several engineering systems, such as power gridnetworks, mechanical systems, optical systems, and so on. In these systems, one can design control techniques toavoid the emergence of EEs. In line with this, control of EEs in dynamical systems has been recently investigated[31, 36, 42, 43, 44]. However, the studies carried out in this direction are quite limited and a systematic study oncontrol of EEs is still in the early stages of research. Motivated by the above, in this paper, the authors investigate thedynamics of an anharmonic oscillator with cubic nonlinearity in the presence of linear damping, nonlinear damping,and periodic external forcing. A primary objective of this paper is to establish an understanding of the role played bythe nonlinear damping in the development of EEs and strategies to control such events.First, the authors study the dynamics of the undamped, anharmonic oscillator, in which the parity ( P )-symmetryis preserved. Due to this symmetric nature, the system has two neutrally stable elliptic equilibrium points in bothpositive and negative potential-wells. Therefore, the system has a conservative nature in the entire phase space andcan exhibit single-well periodic oscillations if the trajectories are started near one of the equilibrium points or double-well periodic oscillations when the initial conditions are chosen away from these equilibrium points. It is shown thatthe system motions are single-well periodic oscillations when the initial conditions are chosen from the region wherethe total energy is negatively valued. On the contrary, the system motions are in the form of double-well periodicoscillations if the initial conditions are chosen in the region where the total energy is positively valued.Next, the authors add a position-dependent damping or nonlinear damping term of the form α x ˙ x into the anhar-monic oscillator equation and investigate the system dynamics with respect to the nonlinear damping parameter. Dueto the inclusion of nonlinear damping term, the symmetry of the system is broken instantly and the system has a parityand time-reversal ( PT ) - symmetry; this alters the stability of the equilibrium points. For the positive values of α ,the equilibrium point in the negative potential-well becomes a source and repels nearby trajectories. The repelled2 . Kaviya et al. / International Journal of Non-Linear Mechanics 00 (2020) 1–15 trajectories are attracted by the fixed point in the right potential-well that acts as a sink. Therefore, the system has adissipative nature in some regions of the phase space in which the trajectories are damped and attracted to the rightpotential-well. At the same time, the system has non-dissipative dynamics in other areas of phase space where thetrajectories are in the form of periodic oscillations. The system has either a dissipative or a non-dissipative nature,depending on the location in the phase space. The underlying mechanism is explained in terms of the total energy ofthe system.Furthermore, when one considers an external periodic forcing of the anharmonic oscillator, in the absence ofnonlinear damping, the system preserves symmetry and the system motions are manifested as double-well chaoticoscillations for certain values of amplitude and frequency of the external forcing. As earlier mentioned, the inclusionof nonlinear damping makes the system asymmetric and the unstable focus in the left potential-well does not attractsystem trajectories. Therefore, the number of trajectories travelling into the left potential-well is gradually reducedas a function of the nonlinear damping strength. The system exhibits large-amplitude oscillations that are alternatedwith small-amplitude oscillations. These oscillations are named as bursting-like oscillations (BOs). Specifically, fora certain range of the nonlinear damping parameter, the large-amplitude (double-well) oscillations occur sporadicallyand recurrently with a highly unpredictable nature. These rarely occurring large-amplitude oscillations are character-ized as EEs. To di ff erentiate EEs from other dynamical states, the threshold H s = h P n i + σ has been numericallyestimated [27]. Here, h P n i is the time-averaged peak value of one of the system variables and σ stands for the meanstandard deviation. In other words, the threshold height is equal to the time-averaged mean value of the peak pluseight times the standard deviation derived for a long run with the iterations of 2 × time units (after leaving outtransients). During the occurrence of EEs the large-amplitude oscillations occur occasionally. Therefore, the peaksare larger than the threshold H s . By contrast, for the other dynamical states, the average peak value ( h P n i ) is quitehigh. Hence, H s becomes higher than the large peaks. Finally, the system exhibits single-well bounded chaotic oscil-lations when one increases the damping strength above the threshold value. In a nutshell, the authors have found thatby including the nonlinear damping term into the forced anharmonic oscillator system, the P –symmetry is not brokeninstantaneously. But this happens only when the damping parameter is taken beyond a threshold value. As a result,the system undergoes a transition from double-well chaotic oscillations to single-well chaos intervened by EEs, withrespect to variation in the nonlinear damping parameter.In accordance with the goal of suppressing large-amplitude oscillations and to identify means to control EEs, alinear damping term is included in the forced anharmonic oscillator along with nonlinear damping and the authorsexamine the responses of the resulting dynamical system. Interestingly, it is found that the large-amplitude oscillationsare completely eradicated from the system dynamics and only single-well small-amplitude oscillations are feasible.The authors also show that the elimination of EEs occurs through two di ff erent dynamical routes as a function of theforcing frequency and the strength of nonlinear damping. One is a transition from EEs to periodic oscillations, andanother is a transition from BOs to single-well oscillations intervened by EEs. The authors’ findings are supported byboth numerical and theoretical results, which include bifurcation diagram plots and Melnikov function estimates [45].It is remarked that the theoretically determined results are in good agreement with the numerically obtained results.In addition, the mechanism for the elimination of EEs is examined and the authors have found that the inclusion oflinear damping destroys the non-dissipative nature of the system, which attains a dissipative nature throughout theentire phase space of the system in the absence of external forcing. Consequently, the trajectories initiated anywherein the phase space follow a decaying solution, which is believed to be a key for the suppression of large-amplitudeoscillations.The remainder of this paper has been organized as follows: In Section 2, the authors study the dynamics of ananharmonic oscillator with and without nonlinear damping and demonstrate the non-trivial property of the coexistenceof dissipative and conservative nature of the nonlinearly damped anharmonic oscillator. Section 3 is devoted tothe study of the forced anharmonic oscillator with nonlinear damping in which the transition from double-well tosingle-well chaotic oscillations mediated by EEs and the response changes observed with respect to damping strengthvariation are presented. Control of EEs through the inclusion of linear damping into the system is illustrated in Section4. Following that, in the next section, the mechanism underlying suppression of EEs is examined. Finally, in Section6, the authors collect together their conclusions. 3 . Kaviya et al. / International Journal of Non-Linear Mechanics 00 (2020) 1–15
2. Dynamics of an unforced, anharmonic oscillator without and with nonlinear damping
In order to carry out the study and demonstrate the results obtained, first, the authors consider a simple prototypefor an anharmonic oscillator with cubic nonlinearity; that is,¨ x − γ x + β x = . (1)Here, the overdot denotes di ff erentiation with respect to time, γ is the coe ffi cient of the linear sti ff ness of the oscillator,and β is the coe ffi cient (strength) of the cubic sti ff ness nonlinearity. Equation (1) is said to preserve P -symmetry; thatis, x = − x . For the present numerical study, the authors have fixed the parameter values at γ = . β = . X = (0 ,
0) and X , = (cid:16) ± p γ/β, (cid:17) . Since the system has P -symmetricproperty, for the chosen parameter values, the system has one saddle equilibrium point at (0 ,
0) that is an unstable fixedpoint and two centers at ( ± , ff erent initial conditions are depicted as solid circles. The trajectories started frominitial conditions within the homoclinic orbit result in single-well periodic oscillations and the trajectories started fromoutside the homoclinic orbit experience large excursions to both potential-wells resulting in double-well oscillationsthat are also illustrated in Fig. 1(a).To understand the underlying dynamical mechanism, the authors have calculated the total energy of the system,which is given by, E = (cid:20) ˙ x + β x − γ x (cid:21) . (2)If one substitutes the initial values for x and ˙ x into Eq. (2), then E has negative values for the initial conditionschosen within the homoclinic orbit and the trajectories started from these initial conditions remain inside and result insingle-well periodic oscillations. Beyond this orbit, E ≥
0, and the trajectories move away from the homoclinic orbitresulting in double-well oscillations as shown in Fig. 1(a).Next, the authors introduce a position dependent damping, called here as nonlinear damping of the form α x ˙ x intothe system (1). The resulting system is given by¨ x + α x ˙ x − γ x + β x = , (3)which is of the Li´enard type ¨ x + f ( x ) ˙ x + g ( x ) =
0, where f ( x ) = α x and α is the nonlinear damping coe ffi cient, and g ( x ) = − γ x + β x in Eq. (3). The system (3) can be viewed as a cubic anharmonic oscillator (1) with nonstandardHamiltonian nature [46], or as a conservative nonlinear oscillator perturbed by the nonlinear damping α x ˙ x . For the pastseveral years, the invariance and integrability properties of this equation have been studied in detail [47, 48, 49, 50].When one includes the nonlinear damping term, the P -symmetry of the system (1) is broken instantly and Eq. (3) has PT –symmetric nature. That is, x = − x and t = − t .As in the anharmonic oscillator (1), the system (3) has three equilibrium points at X = (0 ,
0) and X , = (cid:16) ± p γ/β, (cid:17) . However, due to the PT – symmetric property, for the chosen parameter values the two centers ( X , ) inthe two potential-wells become stable focus ( X ) and unstable focus ( X ) in the corresponding positive and negativepotential-wells, respectively. Positions of the saddle, stable and unstable focus equilibrium points are indicated inFig. 1(b) with a square, an open circle, and a triangle, respectively.The trajectories are attracted to the stable focus in the positive potential-well when the initial conditions are chosenwithin the domain of attraction of this equilibrium point. For trajectories initiated outside this domain of attraction, theresulting motion is in the form of periodic oscillations. Based, on the choice of initial conditions, the system has eithera dissipative or a non-dissipative nature. The black closed loop in Fig. 1(b) is used to denote the homoclinic orbitthat separates regions with di ff erent types of motions. To be precise, one can consider the divergence of the vectorfield of system (3) expressed in a state-space form with the states being x and ˙ x . This divergence is equal to − α x ,4 . Kaviya et al. / International Journal of Non-Linear Mechanics 00 (2020) 1–15 γ = . β = .
5. Open triangles are used at ( ± ,
0) todenote elliptic equilibrium points and a square is used at (0,0) to denote a saddle point. The chosen initial conditions are indicated by filled circlesand the trajectories that are initiated from them are depicted with aqua (light gray) lines. The closed loop, which is shown in black, is a homoclinicorbit. (b) Phase portrait of unforced, anharmonic oscillator with nonlinear damping (3) and α = .
45. The filled circles within the homoclinic orbitrepresent initial conditions, from which the trajectories that follow have a dissipative nature. The filled triangles represent initial conditions, fromwhich the trajectories that follow have a non-dissipative nature. The corresponding motions are periodic oscillations. which is positive, negative, or zero depending on the location in state space. For a positive value of α , the system isdissipative when x >
0, conservative when x =
0, and neither conservative nor dissipative when x <
0. The nonlineardamping term in Eq. (3) acts as an energy dissipating term as well as an energy adding term, which can give rise toself-sustained oscillations. This position-dependent phenomenon enables one to understand the controlling aspects ofcertain biological and chemical oscillations [51].One can also examine the coexistence of dissipative and non-dissipative nature of the system (3) in terms of thetotal energy of the system [31]. One can write down the total energy for the system given by Eq. (3) as E = (cid:20) ˙ x + α ˙ x (cid:16) x − γβ (cid:17) + β (cid:16) x − γβ (cid:17) (cid:21) × e α Ω tan − α ˙ x + β (cid:18) x − γβ (cid:19) Ω ˙ x − (cid:18) γ β (cid:19) e απ Ω , (4)where Ω = p β − α . If one substitutes the initial conditions for ( x , ˙ x ) into Eq. (4), then, for some initial conditions, E has negative values and for those locations in the phase space the system has a dissipative nature. On the otherhand, the system has a non-dissipative nature, when the total energy of the system E ≥
0. In particular, in Fig. 1(b),the total energy of the system has negative values inside the homoclinic orbit. For phase space locations outside thehomoclinic orbit, one has positive E values.It is recalled that with the addition of the nonlinear damping term α x ˙ x in Eq. (1), the P – symmetry of theanharmonic oscillator is broken instantly and transformed into a PT – symmetry. Consequently, the fixed point inthe negative potential-well turns into a source and repels trajectories started within the homoclinic orbit. The repelledtrajectories are attracted by the equilibrium point in the positive well that acts as a sink. When one quasi-staticallyincreases the strength of nonlinear damping α , apparently, the dissipation is quicker. However, it is really intriguing tounderstand the dynamics of the system when it exhibits chaotic behavior and how the symmetry breaking influencesthe chaotic dynamics of this system with respect to the variation in the nonlinear damping strength. To investigatethis, in the next section, the authors include an external periodic force into the system (3) and study the developmentof new dynamical states that emerge with respect to the nonlinear damping.
3. Dynamics of the forced anharmonic oscillator with nonlinear damping
With the inclusion of an external periodic force of the form F sin( ω t ), Eq. (3) can be rewritten as,¨ x + α x ˙ x − γ x + β x = F sin( ω t ) , (5)5 . Kaviya et al. / International Journal of Non-Linear Mechanics 00 (2020) 1–15 x from Eq. (5) illustrating the occurrence ofdouble-well oscillations (DWOs), BOs, EEs and single-well chaos as a function of α ∈ [0 , .
5] with F = . ω = . H s . In the inset, the presence ofEEs is depicted. x . x . -2 0 2 4 6 7 8 x . -2 0 2 4 6 7 8 x . t × C oun t s (i) × C oun t s (j) × C oun t s (k) × C oun t s x (l) × -2 0 2-2.5 0 2.5 x . (e)-2 0 2 4-2.5 0 2.5 x . (f)-2 0 2 4-2.5 0 2.5 x . (g)-2 0 2 4-2.5 0 2.5 x . x (h) (a)(b)(c)(d) Figure 3. Time evolution, phase portrait, and counts of the events that occurred in both potential-wells of the system (5). (a), (e) and (i): Double-well chaotic oscillations for α =
0. (b), (f) and (j): BOs for α = .
44. (c), (g) and (k): EEs for α = .
45. (d), (h) and (l): Single-well bounded chaosfor α = . . Kaviya et al. / International Journal of Non-Linear Mechanics 00 (2020) 1–15 here F and ω are the amplitude and frequency of the sinusoidal excitation, respectively. The integrable property andbifurcation structures of the system (3) have been previously studied in detail for F = F , x axis with respect tothe forcing amplitude F [31]. Due to this oscillation of the equilibrium points, the dissipative and non-dissipativeregions of the system are also oscillating in time. Consequently, the dissipative region is enlarged in the phasespace. This oscillation of the fixed points is governed by stretching and folding actions, which eventually lead thesystem to chaotic behavior when the initial conditions are chosen from the dissipative region and for suitable valuesof F and ω . For the current study, the amplitude of the external force is fixed at F = . α = ω , the system (5) exhibits double-well chaos. Here, theauthors mention double-well chaos in the sense that the system jumps alternatively between the two potential-wellsand averagely spends equal time in them. When the nonlinear damping is included, the system has an unstable focusin the negative potential-well and the trajectories are repelled by this equilibrium point. Consequently, time spent bythe system in the negative potential-well gets reduced, and for most of the time, the system oscillates only in the rightpotential-well for su ffi ciently large values of α . If one looks at this situation in the time domain, the system exhibitbounded small-amplitude (single-well) chaotic oscillations at most of the times and travels to the next potential-wellintermittently, which produces large-amplitude (double-well) oscillations. This type of oscillation is generally knownas bursting-like oscillations, during which the system exhibits coexisting large-amplitude oscillations alternating withthe small-amplitude oscillations. The system exhibits BOs for a range of nonlinear damping parameter values. Toclassify BOs from double-well oscillations, the authors have calculated the total time (T) spent by the system in theright and left potential-wells, namely T R and T L , respectively, and estimated the ratio T ratio = T L / T R . For T ratio > α further, the large-amplitude chaotic bursts are found to occur occasionally (with the ratio of T ratio < H s is used to distinguish it from the other dynamical states. Further, at acritical value of α , the large-amplitude oscillations are suddenly reduced and the system exhibit single-well chaos withthe ratio of T ratio =
0, which then eventually leads to periodic oscillations (via reverse period-doubling bifurcation)for larger values of α . One can note that when α has negative values, then the trajectories are attracted into the leftpotential-well.To verify this transition, the authors have numerically calculated the one-parameter bifurcation diagram of thesystem by plotting the maxima of the dynamical variable ˙ x of the system (5) as a function of α , as shown in Fig. 2.In the range of α ∈ [0 , . α = x is also calculated and plotted in Figs. 3(i). One candiscern that the number of events (oscillations) occurring in the two potential-wells are equal with the ratio of T ratio = α ∈ (0 . , . α = .
44. Compared to the double-well oscillations, in BOs, the number of maxima in the left potential-well is slightly reduced with T ratio = α ∈ (0 . , . α for which the EEs occur is highlighted in dark gray and plotted separately as an inset in Fig. 2. Thetime evolution and phase portrait plots are depicted in Figs. 3(c) and 3(g), respectively, for α = .
45. These plotsconfirm the occasional occurrence of large-amplitude oscillations. The authors also emphasize here that some of thelarge-amplitude oscillations are higher than the threshold H s (horizontal line), which are qualified as EEs, whereasin Figs. 3(a) and 3(b) the threshold H s is larger than the system amplitude. Furthermore, in Fig. 3(k), the authorsshow that the counts of maxima in the left potential-well are drastically reduced, while that the number of events inthe right potential-well is increased with the ratio of T ratio = α > . α = .
46. For larger values of α , eventually, the system givesrise to periodic oscillations via reverse period-doubling bifurcation. The authors wish to point out here that the above7 . Kaviya et al. / International Journal of Non-Linear Mechanics 00 (2020) 1–15 α ) for two di ff erent values of ξ with F = . ω = . ξ = .
01. (b) Existence of periodic oscillations and single-well chaos withoutthe occurrence of EEs for ξ = .
06. Red (dark gray) continuous line represents the threshold H s . mentioned dynamical states arise only if one chooses the initial conditions inside the dissipative region. If one choosesthe initial conditions outside the dissipative region, the system exhibits quasi-periodic oscillations [27, 31].Therefore, when one incorporates the nonlinear damping in the forced anharmonic oscillator and increases thedamping strength, the P –symmetry of the system is broken instantly, and the system has PT – symmetry, whichalters the stability of the equilibrium points. Consequently, the trajectories approaching the left potential-well arerepelled by the unstable fixed point and attracted to the stable equilibrium point in the right potential-well. In otherwords, the total time spent by the system in the left (right) potential-well is decreased (increased) when one increasesthe damping strength, which manifests a transition from double-well to bursting-like oscillations and then to single-well chaos via EEs. Through this study, the authors have elucidated the origin and emerging mechanism of EEs ina forced anharmonic oscillator in the presence of nonlinear damping and external forcing. With regard to control ofEEs, the studies carried out in the literature have been quite limited and narrow in scope. Therefore, this needs furtherattention. In this vein, in the next section, the authors introduce linear damping in Eq. (5) and study the impact of iton EEs in the system response.
4. Influence of linear damping on extreme events in forced anharmonic oscillator with nonlinear damping
After including a linear damping term ( ξ ˙ x ) in Eq. (5), the resulting system is of the form¨ x + α x ˙ x + ξ ˙ x − γ x + β x = F sin( ω t ) , (6)where ξ is the strength of the linear damping, which is positively valued. If one considers the divergence of the vectorfield of system (3) expressed in state-space form with the states being x , ˙ x , and θ = ω t , then the divergence is equalto − ( α x + ξ ). It is clear from this expression that through a choice of an appropriately large enough value of ξ , thedivergence can always be negative. This means that the flow can be dissipative throughout the phase space. In keepingwith this, if one increases the value of ξ from zero, it is found that the large-amplitude oscillations are completelyeliminated from the system dynamics even for small values of ξ and only small-amplitude single-well chaos is feasiblein the system (6). To confirm this phenomenon, the authors have plotted the one-parameter bifurcation diagram forthe response of the system (6) with respect to α for two di ff erent fixed values of ξ . Figs. 4(a) and 4(b) are for ξ = . ξ = .
06, respectively. The continuous line in Figs. 4(a) and 4(b) represents the threshold H s . In the absence oflinear damping, the system exhibit di ff erent dynamical states as shown in Fig. 2. With the inclusion of linear dampingwith ξ = .
01, for low values of α , the system exhibits double-well periodic oscillations and the system exhibits BOsin the range of α ∈ [0 . , . H s appears, as evident from Fig. 4(a). Upon increasing the nonlinear damping strengthin the range of α ∈ (0 . , . H s . For further increase in α , the systemresponse is in the form of single-well chaotic oscillations. When the authors increase the linear damping strength to8 . Kaviya et al. / International Journal of Non-Linear Mechanics 00 (2020) 1–15 ω ) for di ff erent values of ξ , α = . F = .
2. (a) ξ =
0; there are two di ff erent routes for the emergence of EEs. (b) ξ = ξ = ξ = ω . Insets are included to depict the period-doubling sequences, which lead to single-well chaos when decreasing ω . ξ = .
06, the large-amplitude oscillations are completely eradicated and the system exhibits periodic and single-wellchaos with respect to variation in α . This situation is illustrated in Fig. 4(b) where the transition from double-wellperiodic state to single-well chaos occurs without EEs.Accordingly, the large-amplitude oscillations are completely eliminated when one includes the linear dampinginto the forced anharmonic oscillator with nonlinear damping. It is also of interest to study how the large oscillationsare removed from the system for a fixed value of α . To this end, the value of α is fixed as constant and the authorsstudy the system dynamics by varying the forcing frequency ω and linear damping strength ξ . In the absence of lineardamping ( ξ =
0) in Eq. 5, large chaotic bursts occur via two distinct dynamical routes, namely, the intermittency andperiod-doubling routes as one see the Fig. 5(a) from left to right by increasing the forcing frequency ( ω ) from lowervalues and observe the figure from right to left by decreasing ω from higher values, respectively. This dynamics isdepicted in Fig. 5(a), in which the maxima of the system variable ( ˙ x ) is plotted with respect to the forcing frequency( ω ) for ξ =
0. In this figure, if one moves from left to right by increasing ω , the system undergoes a sudden transitionfrom periodicity to large-amplitude chaotic bursting via the intermittency route at ω = . ω , the periodic attractorbifurcates into bounded single-well chaos via the period-doubling bifurcation sequence, which can be seen from theinset of Fig. 5(a). When decreasing ω further, the chaotic attractor slowly increases in size and suddenly at ω = . ω ∈ [0 . , . ξ in Eq. (6), the occurrence of large-amplitude oscillations in the ω parameter region issignificantly reduced. This case is illustrated in Figs. 5(b) to 5(d) for di ff erent values of ξ . In the absence of ξ , large-amplitude oscillations occur in the range of ω ∈ [0 . , . ξ = . ω ∈ [0 . , . ξ = .
02, the region of large-amplitude oscillations is further reduced and occurs only in a small portion of ω ∈ [0 . , . . Kaviya et al. / International Journal of Non-Linear Mechanics 00 (2020) 1–15 -0.02 0 0.02 0.4 0.6 0.8 M ( ω ) ω ξ = 0.0ξ = 0.015ξ = 0.02ξ = 0.05 ξ ∆ n ∆ t Figure 6. Melnikov function M ( ω ) variation with respect to the excitation frequency ω for di ff erent ξ values. One can note that the window overthis function is positive is reduced as ξ is increased. Fig. 5(c). Finally, for ξ = .
06, the system exhibits only periodic oscillations, which is shown in Fig. 5(d). Fromthese results, one can confirm that the large-amplitude oscillations are completely removed from the system when oneincludes and increases the linear damping strength.The reduction of the excitation frequency window over which chaos is possible in the system (6) is analyticallyconfirmed by computing the Melnikov function [45]. This function is an analytical tool that can be used to studythe global behavior of the system. Specifically, this function provides a procedure for analyzing and estimating whenchaotic behavior is expected in the system. In order to be able to carry out the Melnikov analysis, one needs toconsider the external forcing ( F ) and the nonlinear damping ( α ) terms in Eq. (6) as small perturbations. The Melnikovfunction associated with Eq. (6) is given by M ( ω ) = √ β (cid:18) √ πβ F ΩΛ − γ (cid:18) π √ αγ + βξ q γβ (cid:19)(cid:19) β , (7)where Λ = sech (cid:18) πω √ γ (cid:19) . When M ( ω ) is positive, then the system can exhibit chaotic dynamics for the correspondingparameter values. In Fig. 6, the authors have plotted the Melnikov function M ( ω ) with respect to the forcing frequency ω for di ff erent values of ξ (same values as those used to generate the results of Fig. 5). It is noted that M ( ω ) is positivevalued in a certain range of ω for ξ = ξ = M ( ω ) < ω ,which is in conformity with the authors’ numerical findings.It is noted here that the analytically calculated chaotic region in terms of the ω parameter does not exactly matchwith the numerically obtained results given in Fig. 5, since it has been assumed for the Melnikov function calculationthat the external forcing and the nonlinear damping terms in Eq. (6) are small perturbations. However, the widths ofthe chaotic regions obtained both numerical and theoretical calculations are in good agreement. To validate this, theauthors have calculated the di ff erence ∆ = ( ω − ω ) for ξ = ∆ ξ = ( ω ξ − ω ξ ) for di ff erent values of ξ , where ω ,ξ and ω ,ξ are the critical values of the forcing frequency at which chaos emerged via the period-doubling bifurcationand intermittency routes, respectively. The ratio ∆ = ∆ ξ / ∆ is estimated for both numerically obtained data ( ∆ n ) fromthe maximal Lyapunov exponent of Eq. (6) and theoretically obtained results ( ∆ t ) from the Melnikov function (7).The comparison between these results is shown in the inset of Fig. 6. There is good agreement with each other.From Fig. 5, the authors found that the EEs emerged via two di ff erent routes with respect to variation in the forcingfrequency. It is also of interest to understand how the elimination of such large events occurred as a function of lineardamping strength ( ξ ) in these two di ff erent routes. To this end, the authors have again plotted the bifurcation diagramsfor the responses of Eq. (6) with respect to variation in ξ for ω = . ω = . ξ = H s .After including the linear damping and increasing its strength for su ffi ciently large values, the system undergoes atransition from EEs to periodic oscillations. On the other hand, for ω = . . Kaviya et al. / International Journal of Non-Linear Mechanics 00 (2020) 1–15 ξ ). (a) ω = . ω = . linear damping, the system exhibits BOs. When one increases the damping strength, for large values, EEs still occurin the system. As one further increases ξ , the system response is led to single-well chaotic oscillations. Hence, theauthors believe that they have identified that the elimination of EEs can occur through two distinct routes dependingon the values of ω and ξ .To identify the global dynamical behavior of the system and to understand the elimination of large events in a largeparameter plane, the authors have numerically studied the two-parameter space by varying the forcing frequency ( ω )and linear damping strength ( ξ ), as illustrated in Fig. 8. The regions corresponding to EEs, BOs, single-well chaoticmotions, and periodic motions are shown in this figure. To separate the BOs and EEs, the authors use the threshold H s . When the peak response values are larger than H s , the corresponding motion is labelled as an EE. On the otherhand, when the peak response values are smaller than H s , the corresponding motion is labelled as a MMO. The largestLyapunov exponent of the system is calculated to distinguish chaos from periodic oscillations. From Fig. 8, one canclearly note that there are two distinct routes to the elimination of large-amplitude chaotic oscillations with respectto the parameter ξ . It is also worth noting here that when one increases the strength of linear damping, the transitionfrom EEs to periodic oscillations occurs over a narrow region of ω , whereas the transition from EEs to single-wellchaos occurs over a wide range of the ω plane which is also evident from Fig. 8.Based on the results presented here, it is believed that the authors have identified that the large-amplitude oscilla-tions can be completely eliminated from the nonlinearly damped and forced anharmonic oscillator with an appropriatestrength of linear damping. In the next section, the possible mechanism for the elimination of large-amplitude oscil-lations is examined.
5. A mechanism for controlling extreme events
The inclusion of linear damping into Eq. (6) is found to transform the non-dissipative nature of the system into adissipative one. This occurs throughout the phase space. To illustrate this, the authors have calculated the change inthe total energy of the system (6) in the presence of linear damping and nonlinear damping without external forcing( F = (cid:16) dEdt (cid:17) of Eq. 6 can be written as dEdt = − e α Ω tan − α ˙ x + β (cid:18) x + γβ (cid:19) Ω ˙ x × ( ξ ˙ x ) < , (8)where dEdt < ff erent initial conditions (inside and outside of the homoclinic orbit)for ξ = . ff erent initial conditionsslowly converge towards the stable focus. Nevertheless, without the linear damping, the trajectories started outside11 . Kaviya et al. / International Journal of Non-Linear Mechanics 00 (2020) 1–15 ω ) and linear damping strength ( ξ ). In the light gray colored region, EEs occur. BOs occur in the dark gray coloredregion. The black-colored region corresponds to single-well bounded chaos and the region in white is where periodic oscillations occur.Figure 9. Phase portrait of Eq. (6) with F =
0. Evolutions of trajectories that emerge from di ff erent initial conditions are illustrated in green (lightgray) lines. The chosen initial conditions are marked with white filled circles. The rate of change of energy dEdt is depicted with a gray shade. Onecan note that the rate is negative valued, confirming that the system (6) has a decaying solution or a dissipative behavior in the whole phase space. . Kaviya et al. / International Journal of Non-Linear Mechanics 00 (2020) 1–15 the homoclinic orbit have neutrally stable periodic orbits, as depicted in Fig. 1(b). The authors have also plotted therate of change of energy (cid:16) dEdt (cid:17) in Fig. 9 as a gray shaded. It is pointed out that if one includes the external periodicforcing with suitable values of F and ω , then the system exhibits chaotic oscillations which are confined within thesingle-well without having any large excursions (double-well oscillations). As mentioned earlier, the divergence of thevector field of the forced system can be kept negative through an appropriate choice of the linear damping strength ξ .From these observations, one can realize that the inclusion of linear damping can modify the system to be dissipativethroughout the whole phase space. This is found to be a key to the elimination of large-amplitude oscillations such asBOs and EEs.
6. Conclusions
To close the article, it is stated that the authors have studied the dynamics of a forced anharmonic oscillator byincluding nonlinear damping and linear damping terms. The unforced anharmonic oscillator has P -symmetry. Thismeans that the equilibrium points in both positive and negative potential-wells are identical. The system exhibit single-well periodic oscillations if one chooses the initial conditions very near to the neutrally stable elliptic points (in boththe wells), whereas the system exhibit double-well periodic oscillations when one chooses the initial conditions awayfrom the equilibrium points. These di ff erent motions can be distinguished by using the total energy of the system, asdiscussed here.Furthermore, the author have studied the e ff ect of nonlinear damping in the anharmonic oscillator by including thenonlinear damping term x ˙ x and found that the system becomes PT -symmetric nature due to the presence of nonlineardamping term. Hence, the stable nature of the equilibrium points is changed and the two neutrally stable equilibriumpoints become unstable and stable focus in the negative and positive potential-wells, respectively. This system isshown to be capable of generating two distinct dynamical behaviors (dissipative and non-dissipative) depending onthe initial conditions. The total energy of the system has been derived to illustrate the dual nature of the system.Additionally, the authors have shown that under the influence of an external periodic force, without the nonlineardamping term, the system exhibits double-well chaotic behavior for certain values of amplitude and frequency ofthe forcing. If one includes the nonlinear damping and increases the damping strength, for large values, the systemundergoes a transition from double-well chaotic oscillations to single-well chaos mediated by EEs.To influence the dissipation characteristics of the system, the linear damping is included into the forced anharmonicoscillator along with nonlinear damping. This inclusion if found to help in completely eliminating large-amplitudeevents from the system dynamics. In the control parameter space spanned by the forcing frequency and the strength oflinear damping, the authors have identified that the elimination of such large-amplitude oscillations occur through twodistinct routes, one, a transition from EEs to periodic oscillations, and another, a transition from BOs to single-wellchaos intervened by EEs. These results have been supported both numerically by plotting the bifurcation diagrams andanalytically by calculating the Melnikov function. The analytically determined results are found to agree well withthe numerically obtained results. The mechanism for the elimination of EEs has also been examined with numericalstudies and the findings are in good agreement with the authors’ analytically obtained results. By including the lineardamping, the authors illustrate that one can realize a system with a dissipative nature throughout the phase space. Thisis found to be a key for the suppression of EEs. It is emphasized that the results shown in this work are robust tochanges in the system and forcing parameters.
7. Acknowledgment
B. Kaviya acknowledges SASTRA Deemed University for providing Teaching Assistantship. The work of R.Suresh is supported by the SERB-DST Fast Track scheme for Young Scientist under Grant No. YSS / / / / EMR-II and B. Balachandran gratefully acknowledges the partial support received for this work through the U.S. Na-tional Science Foundation Grant No. CMMI1854532 13 . Kaviya et al. / International Journal of Non-Linear Mechanics 00 (2020) 1–15 References [1] Guckenheimer J, Holmes PJ. Nonlinear oscillations, dynamical systems and bifurcation of vector fields. New York: Springer; 1983.[2] Kovacic I, Brennan MJ. The Du ffi ng Equation: Nonlinear Oscillators and their Behaviour. London: Wiley; 2011.[3] Lakshmanan M, Murali K. Chaos in Nonlinear Oscillators: Synchronization and Control. World Scientific. Singapore: 1996.[4] Ji JC, Leung AYT. Bifurcation control of a parametrically excited Du ffi ng system. Nonlinear Dyn 2002;27:411-17.[5] Li H, Preidikman S, Balachandran B, Mote Jr CD. Nonlinear free and forced oscillations of piezoelectric microresonators. J. Micromechanicsand Microengineering 2006;16:356-67.[6] Eichler A, Moser J, Chaste J, Zrdojek M, Wilson-Rae I, Bachtold A. Nonlinear damping in mechanical resonators made from carbon nan-otubes and graphene. Nat. Nanotechnol 2011;6(339).[7] Ekinci KL, Yang YT, Roukes ML. Ultimate limits of inertial mass sensing based upon nanoelectromechanical systems. J. Appl. Phys2004;95:2682-9.[8] Papariello L, Zilberberg O, Eichler A, Chitra A. Ultrasensitive hysteretic force sensing with parametric nonlinear oscillators. Phys. Rev. E2016;94(022201).[9] Akerman N, Kotler S, Glickman Y, Dallal Y, Keselman A, Ozeri R. Single-ion nonlinear mechanical oscillator. Phys. Rev. E2010;82(061402(R)).[10] Zaitsev S, Shtempluck O, Buks E, Gottlieb O. Nonlinear damping in a micromechanical oscillator. Nonlinear Dyn 2012;67:859-83.[11] Ran Z. One exactly soluble model in isotropic turbulence. Appl. Fluid Mech 2009;5:41-67.[12] Siewe MS, Cao H, Sanju´an MAF. E ff ect of nonlinear dissipation on the basin boundaries of a driven two-well RayleighDu ffi ng oscillator.Chaos Solitons Fractals 2009;39:1092-99.[13] Miwadinou CH, Monwanou AV, Chabi Orou JB. E ff ect of Nonlinear Dissipation on the Basin Boundaries of a Driven Two-Well ModifiedRayleighDu ffi ng Oscillator. Int. J. Bifurc. Chaos 2015;25(1550024).[14] Ravindra B, Mallik AK. Chaotic response of a harmonically excited mass on an isolator with non-linear sti ff ness and damping characteristics.J. Sound Vib 1995;182:345-53.[15] Ravindra B, Mallik AK. Role of nonlinear dissipation in soft Du ffi ng oscillators. Phys. Rev. E 1994;49:4950-54.[16] Ravindra B, Mallik AK. Stability analysis of a nonlinearly damped Du ffi ng oscillator. J. Sound Vib 1994;171:708-16.[17] Bikdash MU, Balachandran B, Nayfeh AH. Melnikov analysis for a ship with general roll damping, Nonlinear Dyn 1994;6:101-24.[18] Nayfeh AH, Mook DT. Nonlinear Oscillations. New York: Wiley (Wiley Classics Library); 1995.[19] Almog R, Zaitsev S, Shtempluck O, Buks E. Noise Squeezing in a Nanomechanical Du ffi ng Resonator. Phys. Rev. Lett 2007;98(078103).[20] Baltanas JP, Trueba JL, Sanju´an MAF. Energy dissipation in a nonlinearly damped Du ffi ng oscillator. Physica D 2001;159:22-34.[21] Sanju´an MAF. The e ff ect of nonlinear damping on the universal escape oscillator. Int. J. Bifurc. Chaos 1999;9:735-44.[22] Jing XJ, Lang, ZQ. Frequency domain analysis of a dimensionless cubic nonlinear damping system subject to harmonic input. Nonlinear Dyn2009;58:469-85.[23] Leuch A, Papariello L, Zilberberg O, Degen CL, Chitra R, Eichler A. Parametric Symmetry Breaking in a Nonlinear Resonator. Phys. Rev.Lett 2016;117(214101).[24] Lifshitz R, Cross MC. Nonlinear Dynamics of Nanomechanical and Micromechanical Resonators. New York: Wiley; 2008.[25] Patidar V, Sharma A, Purohit G. Dynamical behaviour of parametrically driven Du ffi ng and externally driven HelmholtzDu ffi ng oscillatorsunder nonlinear dissipation. Nonlinear Dyn 2016;83:375-88.[26] Kingston KL, Tamilmaran K. Bursting oscillations and mixed-mode oscillations in driven Li´enard system. Int. J. Bifurc. Chaos2017;27(1730025).[27] Kingston KL, Tamilmaran K, Pal P, Feudel U, Dana SK. Extreme events in the forced Li´enard system. Phys. Rev. E 2017;96(052204).[28] Kingston KL, Suresh K, Tamilmaran K. Mixed-mode oscillations in memristor emulator based Li´enard system. AIP Conference Proceedings2018;1942(060008).[29] Chandrasekar VK, Senthilvelan M, Lakshmanan M. On the general solution for the modified Emden-type equation ... x + α x ˙ x + β x =
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