Are sequences of volcanic eruptions deterministically chaotic?
Authors: Sornette A. , Dubois J. , Cheminée J. L. , Sornette D. 
Published: February 2008

In order to address the question posed in the title, two tests have been applied to the sequence of repose periods between eruptions in two volcanoes (Piton de la Fournaise in La Réunion Island and Mauna Loa and Kilauea in Hawaii). In the first test a phase space is constructed from the sequence of repose periods, and the dimension D of the attractor is computed as a function of the dimension d of the embedding phase space. D(d) is found to saturate to a constant value around 2 (La Réunion) and around 4 (Hawaii) for large values of d. This signals the existence of a few degrees of freedom (of order 2 and 4), respectively, of the dynamical system underlying the time evolution of volcanic eruptions. Following this first observation we have then attempted to explain the deterministic dynamic by constructing a so‐called “return map” of the repose periods. This is done by extracting a subsequence from the full sequence of repose periods between eruptions. Different schemes for obtaining these subsequences, corresponding to various Poincaré sections, have been examined. One particular method is inspired by the specific Poincaré section, consisting of selecting the maxima of one of the variables, which is used to analyze the Rössler and Lorenz attractors in the nonlinear physics literature. The return maps are found to be in reasonable agreement with what one should expect from a deterministic dynamical system spoiled, however, by noise. The analysis is not perfect because of additional fluctuations which are, however, not strong enough to blur completely the relevant information. These return maps are compared to those obtained from a set of repose periods artificially synthesized with a pseudorandom number generator possessing the same distribution. However, the information provided by the return maps is not sufficient to determine the dynamical equations underlying the eruptions. Therefore a general procedure to test further the determinism of the data set, which generalizes the construction of the return map to arbitrary dimension, is presented. A three‐dimensional representation of the attractor (for the Piton de la Fournaise volcano) is given and compared with the same representation for a set of repose periods artificially synthesized with a pseudorandom number generator. We find an attractor which presents a global structure reminiscent of the Rössler attractor band in the Duffing system. In contrast, the reconstructed “attractor” using a pseudorandom generator does not exhibit a coherent structure. Adding together these different pieces of evidence, we conclude that volcanic eruptions might well be controlled by deterministic dynamics of low dimensions. While chaos places a fundamental limit on long‐term prediction, it suggests the very interesting possibility of making efficient nonlinear short‐term forecasting of new eruptions from the previous series of eruptions. We have not attempted to develop a nonlinear forecasting procedure from previous eruptions because, in order to be efficient, one needs first to identify as precisely as possible the geometrical structure of the reconstructed attractor. At present, the results do not have the necessary quality to carry out this project. We think that in the future, important improvements could be achieved by monitoring and recording a complete set of data related to the evolution of volcanoes.