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 Liénard 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, confirming the rarity of the events. Further, the total energy of the Liénard 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 a 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.
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