1 - Introduction
What do a gas pressure tank embarked on a rocket, a seismic fault and a busy market have in common? Recent research suggests that they can all be described in much the same basic physical terms: as self-organising systems which develop similar patterns over many scales, from the very small to the very large. And all three have the potential for extreme behaviour: rupture, quake or crash.
Similar characteristics are exhibited by other crises that often present fundamental societal impacts and range from large natural catastrophes such as volcanic eruptions, hurricanes and tornadoes, landslides, avalanches, lightning strikes, catastrophic events of environmental degradation, to the failure of engineering structures, social unrest leading to large-scale strikes and upheaval, economic drawdowns on national and global scales, regional power blackouts, traffic gridlock, diseases and epidemics, etc. Intense attention and efforts are devoted in the academic community, in goverment agencies and in the industries that are sensitive to or directly interested in these risks, to the understanding, assessment, mitigation and if possible prediction of these events.
Scientifically based catastrophe theories are usually based on simulations of scenarios from models. However, numerous sources of error exist, each of which may have a negative impact on the validity of the predictions based on the models. Some of the uncertainties are under control in the modelling process; they usually involve trade-offs between a more faithful description and manageable calculations. Other sources of errors are beyond control as they are inherent in the modeling methodology of the specific disciplines. The two known strategies for modelling are both limited in this respect: analytical theoretical predictions are out of reach for most complex problems. Brute force numerical resolution of the equations (when they are known) or of scenarios using supercomputers is reliable in the "center of the distribution'', i.e. in the regime far from the extremes where good statistics can be accumulated. Crises are extreme events that occur rarely, albeit with extraordinary impact, and are thus completely under-sampled and thus poorly constrained.
With colleagues from several relevant disciplines, we have developed a non-traditional approach to make scientific predictions of catastrophic events, based on the concepts and techniques of statistical and nonlinear physics. This approach provides a third line of attack bridging accross the two extreme strategies of analytical theory and brute force numerical simulations. Our modelling strategy uses bifurcation and catastrophe theory, dynamical critical phenomena and the renormalization group, nonlinear dynamical systems and the theory of partially (spontaneously or not) broken symmetries to direct the numerical resolution of more realistic models and to identify relevant signatures of impending catastrophes. This has been successfully applied to problems as varied as failures of engineering structures, stock market crashes and human parturition, with good potential for earthquakes. These case studies are discussed in some details below.
The outstanding scientific question that needs to be addressed to guide prediction is how large-scale patterns of catastrophic nature might evolve from a series of interactions on the smallest and increasingly larger scales, where the rules for the interactions are presumed identifiable and known. For instance, a typical report on an industrial catastrophe describes the improbable interplay between a succession of events. Each event has a small probability and limited impact in itself. However, their juxtaposition and chaining lead inexorably to the observed losses. A common denominator of the various examples of crises is that they emerge from a collective process: the repetitive actions of interactive nonlinear influences on many scales lead to a progressive build-up of large-scale correlations and ultimately to the crisis. In such systems, it has been found that the organization of spatial and temporal correlations does not stem, in general, from a nucleation phase diffusing accross the system. It results rather from a progressive and more global cooperative process occurring over the whole system by repetitive interactions.
For hundreds of years, science has proceeded on the notion that things can always be understood-and can only be understood-by breaking them down into smaller pieces, and by coming to know those pieces completely. Systems in critical states flout this principle. Important aspects of their behaviour cannot be captured knowing only the detailed properties of their component parts. The large scale behavior is more controlled by their cooperativity and scaling up of their interactions. This is the key idea underlying the four examples that illustrate this new approach to prediction: rupture of engineering structures, earthquakes, stock market crashes and human parturition.