Influence of dissipation on extreme oscillations of a forced anharmonic oscillator

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 this symmetric 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 (xẋ) into the system. Then, the parity symmetry of the system is broken instantly and the stability of the two elliptic points is altered to result in stable focus and unstable focus in the positive and negative potential-wells, respectively. Consequently, the system is dual-natured and is either non-dissipative or dissipative, depending on location in the phase space. The total energy of the system is used to explain this dual nature of the system. Furthermore, when one includes a periodic external forcing with suitable parameter values into the nonlinearly damped anharmonic oscillator system and starts to increase the damping strength, the parity symmetry of the system is not broken right away, 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 if one incorporates linear damping into the system. Hence, it is believed that a novel means has been identified for controlling the EEs that occur in forced anharmonic oscillator system with nonlinear damping. The numerically calculated results are in good agreement with the theoretically obtained results on the basis of Melnikov’s function. Further, it is demonstrated that when one includes linear damping into the system, this system has a dissipative nature throughout the entire phase space of the system. This is believed to be the key to the elimination of EEs.