  Dynamics of a periodically forced anharmonic oscillator (AO) with cubic
nonlinearity, linear damping, and nonlinear damping, is studied. To begin with,
the authors examine the dynamics of an AO. 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 nonlinear damping into the system. Then,
the 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. Furthermore, when one includes a periodic external
forcing with suitable parameter values into the nonlinearly damped AO system
and starts to increase the damping strength, the 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 extreme events (EEs).
Furthermore, it is found that the large-amplitude oscillations developed in the
system are completely eliminated if one incorporates linear damping into the
system. 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.
