Super extreme events and dragon kings in a memristive coupled Rulkov maps

Unforeseeable rare events, referred to as extreme events (EEs), can occur in various real-world contexts. When comparing EEs, the super EEs and dragon kings are the most harmful and occur occasionally in both natural and artificial systems. In this study, we consider a discrete system of Rulkov maps coupled via a tangential form of the memristive function to investigate the existence of both super EEs and dragon king events. Primarily, bifurcation analysis and Lyapunov exponents demonstrate that the transition to hyper-chaos occurs when the control parameter is increased to larger values. Following that, we analyze the appearance of super EEs by setting the parameter in the region of hyper-chaos, and we perform the critical amplitude threshold and probability density function to illustrate such occurrences. Furthermore, we use statistical characterizations such as the complementary cumulative distribution function (CCDF) and the DK test to delineate the emergence of the dragon kings. Finally, we perform a two-parameter analysis to acquire a complete picture of the occurrence of the observed events.