We observe extremely large amplitude intermittent spikings in a dynamical variable of a periodically forced Liénard-type oscillator and characterize them as extreme events, which are rare, but recurrent and larger in amplitude than a threshold. The extreme events occur via two processes, an interior crisis and intermittency. The probability of occurrence of the events shows a long-tail distribution in both the cases. We provide evidence of the extreme events in an experiment using an electronic analog circuit of the Liénard oscillator that shows good agreement with our numerical results.

Intermittent large amplitude events are seen in the temporal evolution of a state variable of many dynamical systems. Such intermittent large events suddenly start appearing in dynamical systems at a critical value of a system parameter and continues for a range of parameter values. Three important processes of instabilities, namely, interior crisis, Pomeau–Manneville intermittency, and the breakdown of quasiperiodic motion, are most common as observed in many systems that lead to such occasional and rare transitions to large amplitude spiking events. We characterize these occasional large events as extreme events if they are larger than a statistically defined significant height. We present two exemplary systems, a single system and a coupled system, to illustrate how the instabilities work to originate extreme events and they manifest as non-trivial dynamical events. We illustrate the dynamical and statistical properties of such events.

We study extreme and critical events in the forced Liénard systems with charge control memristor. It has been found that the system exhibits hidden attractors either in the absence or presence of an external sinusoidal force. We give evidence that these attractors play a crucial role in the appearance of critical events. We attempt to explain the mechanism leading to the emergence of catastrophic transitions. Finally, we present that the observed critical transitions are typical for memristor based models and understanding of them gives some insight on how to avoid these types of devastating events at the time of the device fabrication process.

We report the existence of bursting oscillations and mixed-mode oscillations in a Liénard system when it is driven externally by a sinusoidal force. The bursting oscillations transit from a periodic phase to spiking trains through chaotic windows, as the control parameter is varied. The mixed-mode oscillations appear via an alternate sequence of periodic and chaotic states, as well as Farey sequences. The primary and their associated secondary mixed-mode oscillations are detected for the suitable choices of system parameters. Additionally, the system is also found to possess multistability nature. Our investigations involve numerical simulations as well as real time hardware experiments using a simple analog electronic circuit. The experimental observations are in conformation with numerical results.