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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.