Modeling of recurrent threshold crossings due to noise with long memory

Abstract

This thesis addresses the recurrent threshold crossing behavior of long-time correlated noise. The behavior of long-time correlated noise like f / 1 , 5 . 1 / 1 f , and 2 / 1 f can be associated with the behavior of many phenomena in nature, so it is of interest to study the behavior of this noise. Our method of modeling their recurring behavior relies on setting a particular threshold level for a particular level of noise and observing how frequently the noise crosses the threshold level. We also add a periodic drive to the noise which enables it to cross the threshold level easily when it is at peak, and vice versa. This technique provides a model for the changing seasons that occur during every year. We also compare the recurrence behavior of threshold crossings from our computer simulations with theoretical results from the Rice formula. We have related the recurrence of these threshold crossings with the recurrence of natural disasters. Therefore we are providing a model to predict the recurrence of a natural disaster once that disaster has previously occurred. From our results, we conclude that once a natural disaster has occurred, there is a high probability of its recurrence in a short time, and this probability gradually decreases with time.