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# Prediction of Catastrophes

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**Essay selected as one of the ten finalist to the James S. McDonnell, Centennial Fellowships (november 1998)**

**Abstract:** We propose that catastrophes, as they occur in various disciplines, have similarities both in the failure of standard models and the way that systems evolve towards them. We present a non-traditional general methodology for the 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 across the two standard strategies of analytical theory and brute force numerical simulations. It has been successfully applied to problems as varied as failures of engineering structures, stock market crashes and human parturition, with potential for earthquakes.

In the problem of failure of engineering structures, we propose that heterogenous systems fail by exhibiting a critical behavior, characterized by the presence of log-periodic patterns. This prediction has been tested extensively during our continuing collaboration with the French Aerospace company Aerospatiale on gas pressure tanks embarked on the European Ariane rockets. Our theory was applied to about 50 pressure-tanks and the results indicate that a precision of a few percent in the determination of the stress at rupture is obtained using acoustic emission recorded 20% below the stress at rupture. These successes have warranted an international patent and the selection of this non-destructive evalution technique as the routine qualifying procedure in the industrial fabrication process.

It was during our research on the acoustic emissions of the industrial pressure tank of the European Ariane rocket that we discovered the existence of log-periodic scaling in non-hierarchical structures. Log-periodicity in a power law self-similar signal means that there are superimposed oscillations modulated in frequency with a geometric increase of the frequency on the approach to the critical point. This apparent esoteric property turns out to be surprisingly general both experimentally and theoretically and we are probably only at the beginning of its understanding. From a formal point of view, log-periodicity is nothing but the concrete expression of the fact that exponents or more generally dimensions can be "complex'', i.e. belong to these numbers which when squared give negative values. The practical upshot is that the log-periodic undulations may help in "synchronizing'' a better fit to the data.

Inspired by our previous consideration of the critical nature of rupture and extending it to seismicity, we have invented a systematic procedure to test for the existence of critical behavior and to identify the region approaching criticality, based on a comparison of the observed cumulative energy release and the accelerating seismicity predicted by theory. This method has been used to find the critical region before all earthquakes along the Californian San Andreas system since 1950 with M > 6.5. The statistical significance of our results was assessed by performing the same procedure on a large number of randomly generated synthetic catalogs. The null hypothesis, that the observed acceleration in all these earthquakes could result from spurious patterns generated by our procedure in purely random catalogs, was rejected with 99.5% confidence. The application of the critical theory and its log-periodic signatures is presently investigated vigorously to test its range of validity.

In the context of economy, we describe our hypothesis that stock market crashes are caused by the slow buildup of powerful subterranean forces that come together in one critical instant. The use of the word "critical'' is not purely literary: in mathematical terms, complex dynamic systems such as the stock market can go through so-called "critical'' points, defined as the explosion to infinity of a normally well-behaved quantity. As a matter of fact, as far as nonlinear dynamic systems go, the existence of critical points may be the rule rather than the exception. This led us to develop models and theoretical formulas that have been tested successfully on the U.S. ~stock market crashes of 1929 and 1987: indeed it is possible to identify clear signatures of near-critical behavior many years before the crashes and use them to "predict'' (out of sample) the date where the system will go critical, which happens to coincide very closely with the realized crash date. Our theory has also been used to analyze more recent stock market data leading to a clear signature of an impending critical instability that could be associated to the turmoil of the US stock market at the end of october 1997. It may come as a surprise that the same theory is applied to epochs so much different in terms of speed of communications and connectivity as 1929 and 1997. It may be that what our theory addresses is the fundamental question: has human nature changed?

Parturition is the act of giving birth. While not usually considered as catastrophic, it is arguably the major event in a life (apart from its termination) and it is interesting that our theoretical approach extends to this situation. Indeed, notwithstanding the large number of investigations on the factors that could trigger parturition in higher mammals (monkeys and humans), we still do not have a clear signature in any of the measured variables. In collaboration with a team of obstetricians, we have proposed a new framework which allows us to rationalize the various laboratory and clinical observations on the maturation, the triggering mechanisms of parturition, the existence of various abnormal patterns as well as the effect of external stimulations of various kinds. Within the proposed mathematical model, parturition is seen as a "critical'' instability or phase transition from a state of quietness, characterized by a weak incoherent activity of the uterus in its various parts as a function of time (state of activity of many small incoherent intermittent oscillators), to a state of globally coherent contractions where the uterus functions as a single macroscopic oscillator leading to the expulsion of the baby. A number of new predictions and suggestions for improvements in medical care is currently been tested.

Let us finally mention possible extensions of the theory for future research on prediction of societal breakdowns, terrorism, large scale epidemics, and of the vulnerability of civilisations.