Statistical properties of homoclinic bursting: An approach using infinite-modal maps toward predicting extreme events

Predicting extreme events in nonlinear dynamical systems is challenging due to a limited understanding of their statistical properties. This study numerically and theoretically investigates the statistical properties of infinite-modal maps arising from homoclinic bursting to predict extreme events. The numerical investigation presents bifurcation diagrams, Lyapunov exponents, height probability distributions, and interevent interval probability distributions for infinite-modal maps. The theoretical analysis derives analytical formulae for these statistical properties using a randomization theory of infinite-modal maps. Utilizing the derived analytical formulae, a parameter estimation method for infinite-modal maps is also proposed, which enables practical application of the theoretical results. Finally, this study demonstrates the applicability of this method for non-stationary data with time-dependent parameters. These findings provide a foundation for predicting extreme events based on their mechanisms.