Didier Sornette

Several authors have proposed discrete renormalization group models of earthquakes, viewing them as a kind of dynamical critical phenomena. Here, we propose that the assumed discrete scale invariance stems from the irreversible and intermittent nature of rupture which ensures a breakdown of translational invariance. As a consequence, we show that the renormalization group entails complex critical exponents, describing log-periodic corrections to the leading scaling behavior. We use the mathemati

We present an analysis of the time behavior of the $S\&P500$ (Standard and Poors) New York stock exchange index before and after the October 1987 market crash and identify precursory patterns as well as aftershock signatures and characteristic oscillations of relaxation. Combined, they all suggest a picture of a kind of dynamical critical point, with characteristic log-periodic signatures, similar to what has been found recently for earthquakes. These observations are confirmed on other smaller

Multifractality is ubiquitously observed in complex natural and socioeconomic systems. Multifractal analysis provides powerful tools to understand the complex nonlinear nature of time series in diverse fields. Inspired by its striking analogy with hydrodynamic turbulence, from which the idea of multifractality originated, multifractal analysis of financial markets has bloomed, forming one of the main directions of econophysics. We review the multifractal analysis methods and multifractal models

We propose that catastrophic events are "outliers" with statistically different properties than the rest of the population and result from mechanisms involving amplifying critical cascades. We describe a unifying approach for modeling and predicting these catastrophic events or "ruptures," that is, sudden transitions from a quiescent state to a crisis. Such ruptures involve interactions between structures at many different scales. Applications and the potential for prediction are discussed in re

Levy and Solomon have found that random multiplicative processes wt =λ1λ2...λt (with λj > 0) lead, in the presence of a boundary constraint, to a distribution P(wt) in the form of a power law wt-(1+μ). We provide a simple exact physically intuitive derivation of this result based on a random walk analogy and show the following: 1) the result applies to the asymptotic (t→∞) distribution of wt and should be distinguished from the central limit theorem which is a statement on the asymptotic distrib

The scientific study of complex systems has transformed a wide range of disciplines in recent years, enabling researchers in both the natural and social sciences to model and predict phenomena as diverse as earthquakes, global warming, demographic patterns, financial crises, and the failure of materials. This book applies the author's experience in these areas to propose a simple, powerful, and general theory of how, why, and when stock markets crash. Most attempts to explain market failures see

Rank‐ordering statistics provide a perspective on the rare, largest elements of a population, whereas the statistics of cumulative distributions are dominated by the more numerous small events. The exponent of a power law distribution can be determined with good accuracy by rank‐ordering statistics from the observation of only a few tens of the largest events. Using analytical results and synthetic tests, we quantify the systematic and the random errors. We also study the case of a distribution