  The two paradigmatic nonlinear oscillatory models with parametric excitation
are studied. The authors provide theoretical evidence for the appearance of
extreme events (EEs) in those systems. First, the authors consider a well known
Lienard type oscillator that shows the emergence of EEs via two bifurcation
routes: Intermittency and period-doubling routes for two different critical
values of the excitation frequency. The authors also calculate the return time
of two successive EEs, defined as inter-event intervals, that follow
Poisson-like distribution, confirm the rarity of the events. Further, the total
energy of the Lienard oscillator is estimated to explain the mechanism for the
development of EEs. Next, the authors confirmed the emergence of EEs in a
parametrically excited microelectromechanical system. In this model, EEs occur
due to the appearance of stick-slip bifurcation near the discontinuous boundary
of the system. Since the parametric excitation is encountered in several
real-world engineering models, like macro and micromechanical oscillators, the
implications of the results presented in this paper are perhaps beneficial to
understand the development of EEs in such oscillatory systems.
