Using nonlinear dynamics to predict Old Faithful

Keywords--Logis t ics map, Geyser eruptions, Nonlinear dynamics, Old Faithful. 1. I N T R O D U C T I O N A N D P R O B L E M M O T I V A T I O N Old Faithful, the most famous geyser in Yellowstone National Park, is known for the regularity and predictability of its eruptions. Predictability, however, has many gradations. Although it is possible to predict an eruption after observing the previous one, there is currently no way to predict more than one eruption into the future, to base predictions on solely eruption intervals, or to make a daily forecast of Old Faithful's behavior. The current prediction algorithm for Old Faithful is based on observation and involves a correlation between the duration of an eruption and the time until the next. A longer eruption duration results in a longer interval between eruptions, and a shorter duration results in a shorter interval. This correspondence was first noted in 1938, though the values in the algorithm have been updated. The predictions have an accuracy of approximately plus or minus ten minutes [1]. Mathematicians and statisticians have been interested in the problem of predicting Old Faithful using techniques from their fields for some time. Analyzing eruption data from Old Faithful is a common exercise in many statistics courses. Statisticians commonly note the bimodal distribution of the data (for example, see [2]). Since a given eruption interval tends to be either "long" or "short", a histogram of eruption intervals would show two clusters of interval frequency, one near The author would like to thank K. M. Short for initiating and encouraging this work. It would not have been possible without his input and support. 0895-7177/05/$ see front matter (~) 2005 Elsevier Ltd. All rights reserved. Typeset by ~4j~4S-TEX doi: 10.1016/j.mcm.2004.07.013