Influence of time-delay feedback on extreme events in a forced Liénard system

Abstract

A periodically forced Li\'enard system is capable of generating frequent large-amplitude chaotic bursts for a range of system and external forcing parameter values which are known as mixed mode oscillations. Particularly, if these large chaotic bursts occur infrequently and randomly, then they are characterized as extreme events. We present a numerical study of the effect of self-time-delay feedback on these extreme events in this system and interestingly find that extreme events can be completely eliminated from the system dynamics, even for smaller values of delay feedback strength. We show here that the number of extreme events is reduced, and the probability of the occurrence of high-amplitude events is transformed from a long-tailed statistics to the localized structure, as a function of the feedback strength corroborates our results. Further, we show that the autonomous Li\'enard system loses its conservative nature when delay feedback is added and only a dissipative nature remains in the entire phase space, which is the underlying mechanism behind the elimination of such large events. Further, we have revealed a type of delay-induced damping behavior, named anomalous damping, in which the amplitude of the oscillations suddenly vanishes when the total energy of the system becomes zero.