Influence of dissipation on extreme oscillations of a forced anharmonic oscillator
Introduction
Forced and damped nonlinear oscillators have been considered as paradigms for mimicking the dynamics of various physical and engineering systems such as Josephson junctions, electrical circuits, optical systems, macromechanical and microelectromechanical oscillators, and so on [1], [2], [3], [4], [5]. Damping, which is used to model loss of energy due to friction and viscous forces, is ubiquitous in many mechanical systems and this characteristic influences the performance of the oscillators in different ways. Most oscillatory systems are subject to different damping combinations and each one of them has different effects on the considered dynamical systems. In general, it is common to use a linear damping model for describing damping or dissipation experienced by a system. However, in many oscillatory systems such as microelectromechanial and nanoelectromechanical oscillators, nonlinear damping is found to play a significant role. For example, in nanoelectromechanical systems made from carbon nanotubes and graphene, damping is found to strongly depend on the amplitude of motion, and the damping force is nonlinear in nature [6]. These systems are being used for mass and force sensing applications [7], [8]. Researchers have exploited the nonlinear nature of damping in these systems to improve the figures of merit for both nanotube and graphene resonators. In Ref. [9], the ion steady-state motion is well described by the Duffing oscillator model with an additional nonlinear damping term. Both the linear damping and nonlinear damping can be tuned with the laser-cooling parameters helping one to investigate the mechanical noise squeezing in laser cooling. Recently, the influence of nonlinear damping on the motion of a nanobeam resonator was studied and it was found that nonlinear damping can have a significant impact on the dynamics of micromechanical systems [10]. In fluid mechanics, linearly forced isotropic turbulence can be described by an anharmonic oscillator model with nonlinear damping [11]. From a dynamics viewpoint, it has been shown that nonlinear damping can be used to suppress chaos in oscillatory systems [12], [13], [14]. The stability of responses of nonlinearly damped, hard and soft Duffing oscillators have also been analyzed [15], [16]. In addition, the effect of nonlinear damping in forced Duffing 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 forcing or a parametric excitation [2], [23], [24], [25]. Specifically, the development of mixed-mode oscillations and extreme events (EEs) have been recently reported in systems influenced by nonlinear damping [26], [27], [28], [29], [30]. The rare and recurrent occurrence of large-amplitude events in system variables with heavy tails in the probability distribution is a signature of EEs. Examples of EEs that occur in natural and engineering systems include rogue waves in optical systems and oceans, epidemics, large-scale power black-outs in electrical power grids, harmful algal blooms in marine ecosystems, jamming in computer and transportation networks, stock market crashes, and epileptic seizures [31], [32], [33], [34], [35], [36], [37], [38], [39]. Similar statistical behaviors of the appearance of sudden changes in the system variables have been noticed in many dynamical systems governed by nonlinear equations with nonlinear damping [27], [29], [30]. However, an understanding of the occurrence of EEs in such systems is still being developed and the significance of nonlinear damping for the development of EEs has not received careful attention. Furthermore, an understanding of the mechanism that triggers EEs in dynamical systems is crucial for developing strategies to control such events. Although this is out of reach in natural systems, it may certainly be possible in several engineering systems, such as power grid networks, mechanical systems, optical systems, and so on. In these systems, one can design control techniques to avoid the emergence of EEs. In line with this, control of EEs in dynamical systems has been recently investigated [29], [34], [40], [41], [42]. However, the studies carried out in this direction are quite limited and a systematic study on control of EEs is still in the early stages of research. Motivated by the above, in this paper, the authors investigate the dynamics 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 by the 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 ()-symmetry is preserved. Due to this symmetric nature, the system has two neutrally stable elliptic equilibrium points in both positive and negative potential-wells. Therefore, the system has a conservative nature in the entire phase space and can 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 that the system motions are single-well periodic oscillations when the initial conditions are chosen from the region where the total energy is negatively valued. On the contrary, the system motions are in the form of double-well periodic oscillations 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 into the anharmonic oscillator equation and investigate the system dynamics with respect to the nonlinear damping parameter. Due to the inclusion of nonlinear damping term, the symmetry of the system is broken instantly and the system has a parity and time-reversal () - symmetry; this alters the stability of the equilibrium points. For the positive values of , the equilibrium point in the negative potential-well becomes a source and repels nearby trajectories. The repelled trajectories are attracted by the fixed point in the right potential-well that acts as a sink. Therefore, the system has a dissipative nature in some regions of the phase space in which the trajectories are damped and attracted to the right potential-well. At the same time, the system has non-dissipative dynamics in other areas of phase space where the trajectories 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 of the system.
Furthermore, when one considers an external periodic forcing of the anharmonic oscillator, in the absence of nonlinear damping, the system preserves symmetry and the system motions are manifested as double-well chaotic oscillations for certain values of amplitude and frequency of the external forcing. As earlier mentioned, the inclusion of nonlinear damping makes the system asymmetric and the unstable focus in the left potential-well does not attract system trajectories. Therefore, the number of trajectories traveling into the left potential-well is gradually reduced as a function of the nonlinear damping strength. The system exhibits large-amplitude oscillations that are alternated with small-amplitude oscillations. These oscillations are named as bursting-like oscillations (BOs). Specifically, for a certain range of the nonlinear damping parameter, the large-amplitude (double-well) oscillations occur sporadically and recurrently with a highly unpredictable nature. These rarely occurring large-amplitude oscillations are characterized as EEs. To differentiate EEs from other dynamical states, the threshold has been numerically estimated [27]. Here, is the time-averaged peak value of one of the system variables and stands for the mean standard deviation. In other words, the threshold height is equal to the time-averaged mean value of the peak plus eight times the standard deviation derived for a long run with the iterations of time units (after leaving out transients). During the occurrence of EEs the large-amplitude oscillations occur occasionally. Therefore, the peaks are larger than the threshold . By contrast, for the other dynamical states, the average peak value () is quite high. Hence, becomes higher than the large peaks. Finally, the system exhibits single-well bounded chaotic oscillations when one increases the damping strength above the threshold value. In a nutshell, the authors have found that by including the nonlinear damping term into the forced anharmonic oscillator system, the –symmetry is not broken instantaneously. 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, with respect to variation in the nonlinear damping parameter.
In accordance with the goal of suppressing large-amplitude oscillations and to identify means to control EEs, a linear damping term is included in the forced anharmonic oscillator along with nonlinear damping and the authors examine the responses of the resulting dynamical system. Interestingly, it is found that the large-amplitude oscillations are 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 different dynamical routes as a function of the forcing frequency and the strength of nonlinear damping. One is a transition from EEs to periodic oscillations, and another is a transition from BOs to single-well oscillations intervened by EEs. The authors’ findings are supported by both numerical and theoretical results, which include bifurcation diagram plots and Melnikov function estimates [43]. 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 of linear damping destroys the non-dissipative nature of the system, which attains a dissipative nature throughout the entire phase space of the system in the absence of external forcing. Consequently, the trajectories initiated anywhere in the phase space follow a decaying solution, which is believed to be a key for the suppression of large-amplitude oscillations.
The remainder of this paper has been organized as follows: In Section 2, the authors study the dynamics of an anharmonic oscillator with and without nonlinear damping and demonstrate the non-trivial property of the coexistence of dissipative and conservative nature of the nonlinearly damped anharmonic oscillator. Section 3 is devoted to the study of the forced anharmonic oscillator with nonlinear damping in which the transition from double-well to single-well chaotic oscillations mediated by EEs and the response changes observed with respect to damping strength variation are presented. Control of EEs through the inclusion of linear damping into the system is illustrated in Section 4. Following that, in the next section, the mechanism underlying suppression of EEs is examined. Finally, in Section 6, the authors collect together their conclusions.
Section snippets
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 prototype for an anharmonic oscillator with cubic nonlinearity; that is, Here, the overdot denotes differentiation with respect to time, is the coefficient of the linear stiffness of the oscillator, and is the coefficient (strength) of the cubic stiffness nonlinearity. Eq. (1) is said to preserve -symmetry; that is, . For the present numerical study, the authors have
Dynamics of the forced anharmonic oscillator with nonlinear damping
With the inclusion of an external periodic force of the form , Eq. (3) can be rewritten as, here and are the amplitude and frequency of the sinusoidal excitation, respectively. The integrable property and bifurcation structures of the system (3) have been previously studied in detail for [50], [51]. When in Eq, (5), the symmetry of the equilibrium points is broken and they start moving along the axis with respect to the forcing amplitude [29]. Due
Influence of linear damping on extreme events in forced anharmonic oscillator with nonlinear damping
After including a linear damping term () in Eq. (5), the resulting system is of the form where is the strength of the linear damping, which is positively valued. If one considers the divergence of the vector field of system (3) expressed in state-space form with the states being , , and , then the divergence is equal to . It is clear from this expression that through a choice of an appropriately large enough value of , the divergence can always
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 a dissipative one. This occurs throughout the phase space. To illustrate this, the authors have calculated the change in the total energy of the system (6) in the presence of linear damping and nonlinear damping without external forcing (). The rate of change of energy of Eq. (6) can be written as where indicates that the system
Conclusions
To close the article, it is stated that the authors have studied the dynamics of a forced anharmonic oscillator by including nonlinear damping and linear damping terms. The unforced anharmonic oscillator has -symmetry. This means 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 both the wells), whereas the
CRediT authorship contribution statement
B. Kaviya: Methodology, Formal analysis, Investigation, Software, Writing - original draft. R. Suresh: Conceptualization, Validation, Formal analysis, Writing - original draft, Writing - review & editing, Supervision, Funding acquisition. V.K. Chandrasekar: Conceptualization, Methodology, Formal analysis, Writing - original draft, Writing - review & editing, Funding acquisition. B. Balachandran: Conceptualization, Writing - original draft, Writing - review & editing, Funding acquisition.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
B. Kaviya acknowledges SASTRA Deemed University for providing Teaching Assistantship. The work of R. Suresh is supported by the SERB-DST Fast Track, India scheme for Young Scientist under Grant No. YSS/2015/001645. The work of V. K. Chandrasekar forms a part of a research project sponsored by the CSIR EMR, India Grant No. 03(1444)/18/ EMR-II and B. Balachandran gratefully acknowledges the partial support received for this work through the U.S. National Science Foundation Grant No. CMMI1854532.
References (51)
- et al.
Effect of nonlinear dissipation on the basin boundaries of a driven two-well Rayleigh–Duffing oscillator
Chaos Solitons Fractals
(2009) - et al.
Chaotic response of a harmonically excited mass on an isolator with non-linear stiffness and damping characteristics
J. Sound Vib.
(1995) - et al.
Stability analysis of a nonlinearly damped Duffing oscillator
J. Sound Vib.
(1994) - et al.
Energy dissipation in a nonlinearly damped Duffing oscillator
Physica D
(2001) - et al.
Rogue waves: new forms enabled by GPU computing
Phys. Lett. A
(2014) - et al.
Liénard-type chemical oscillator
Eur. Phys. J. B
(2014) - et al.
Nonlinear Oscillations, Dynamical Systems and Bifurcation of Vector Fields
(1983) - et al.
The Duffing Equation: Nonlinear Oscillators and their Behaviour
(2011) - et al.
Chaos in Nonlinear Oscillators: Synchronization and Control
(1996) - et al.
Bifurcation control of a parametrically excited Duffing system
Nonlinear Dynam.
(2002)
Nonlinear free and forced oscillations of piezoelectric microresonators
J. Micromech. Microeng.
Nonlinear damping in mechanical resonators made from carbon nanotubes and graphene
Nat. Nanotechnol.
Ultimate limits of inertial mass sensing based upon nanoelectromechanical systems
J. Appl. Phys.
Ultrasensitive hysteretic force sensing with parametric nonlinear oscillators
Phys. Rev. E
Single-ion nonlinear mechanical oscillator
Phys. Rev. E
Nonlinear damping in a micromechanical oscillator
Nonlinear Dynam.
One exactly soluble model in isotropic turbulence
Appl. Fluid Mech.
Effect of nonlinear dissipation on the basin boundaries of a driven two-well modified Rayleigh–Duffing oscillator
Int. J. Bifurcation Chaos
Role of nonlinear dissipation in soft Duffing oscillators
Phys. Rev. E
Melnikov analysis for a ship with general roll damping
Nonlinear Dynam.
Nonlinear Oscillations
Noise squeezing in a nanomechanical Duffing resonator
Phys. Rev. Lett.
The effect of nonlinear damping on the universal escape oscillator
Int. J. Bifurcation Chaos
Frequency domain analysis of a dimensionless cubic nonlinear damping system subject to harmonic input
Nonlinear Dynam.
Parametric symmetry breaking in a nonlinear resonator
Phys. Rev. Lett.
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