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 Didier Sornette, Professor of Geophysics

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.

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.

2-PREDICTION OF RUPTURE IN COMPLEX SYSTEMS

2.1-NATURE OF THE PROBLEM The damage and fracture of materials are technologically of outstanding interest because of their economic and human cost. They cover a wide range of phenomena such as cracking of glass, aging of concrete, the failure of fiber networks in the formation of paper, and the breaking of a metal bar subject to an external load. Failures of composite systems are of upmost importance in the naval, aeronautics and space industries. By the term composite, we include both materials with constrasted microscopic structures and assemblages of macroscopic elements forming a super-structure. Chemical and nuclear plants suffer from cracking due to corrosion either of chemical or radioactive origin, aided by thermal and/or mechanical stress. More exotic but no less interesting phenomena include the fracture of old painting, the pattern formation of the cracks of drying mud in deserts, and rupture propagation in earthquake faults.

Despite the large amount of experimental data and the considerable effort that has been undertaken by material scientists, many questions about fracture and fatigue have not yet been answered. There is no comprehensive understanding of rupture phenomena, but only a partial classification in restricted and relatively simple situations. This lack of fundamental understanding is reflected in the absence of proper prediction methods for rupture and fatigue, that could be based on a suitable monitoring of the stressed system.

Many material ruptures occur by a ``one crack'' mechanism and a lot of effort is being devoted to the understanding, detection and prevention of the nucleation of the crack. Systems that do not fail by the ``one crack'' rupture mechanism are fiber composites, rocks, concrete under compression and materials with large distributed residual stresses. The common property shared by these systems is the existence of large inhomogeneities, that often limit the use of homogeneization theories for the elastic and more generally the mechanical properties. In these systems, failure may occur as the culmination of a progressive damage involving complex interactions between multiple defects and growing micro-cracks. In addition, other relaxation, creep, ductile, or plastic behaviors, possibly coupled with corrosion effects come into play. Many important practical applications involve the coupling between mechanical and chemical effects with the competition between several characteristic time scales. Application of stress may act as a catalyst of chemical reactions or, reciprocally, chemical reactions may lead to bond weakening and thus promote failure. A dramatic example is the aging of present aircrafts due to repeating loading in a corrosive environment [Committee on Aging of U.S. Air Force Aircraft,1997]. The interaction between multiple defects and the existence of several characteristic scales present a considerable challenge to the modelling and prediction of rupture.

2.2-THE ROLE OF HETEROGENEITY

In the early sixties, the Japanese seismologist K. Mogi noticed that the fracture process strongly depends on the degree of heterogeneity of materials: the more heterogeneous, the more warnings one gets; the more perfect, the more treacherous is the rupture. The failure of perfect crystals thus appears to be unpredictable while the fracture of dirty and deteriorated materials may be forecast. For once, complex systems could be simpler to apprehend! However, since its inception, this idea has not been much developed because it is hard to quantify the degrees of ``useful'' heterogeneity, which probably depend on other factors such as the nature of the stress field, the presence of water, etc. In our work on failure of mechanical systems, we have solved this paradox quantitatively using concepts inspired from statistical physics, a domain where complexity has long been studied as resulting from collective behavior. The idea is that, upon loading a heterogeneous material, single isolated microcracks appear and then, with the increase of load or time of loading, they both grow and multiply leading to an increase of the number of cracks. As a consequence, microcracks begin to merge until a ``critical density'' of cracks is reached at which the main fracture is formed. It is then expected that various physical quantities (acoustic emission, elastic, transport, electric properties, etc.) will vary. However, the nature of this variation depends on the heterogeneity. The new result is that there is a threshold that can be calculated: if disorder is too small, then the precursory signals are essentially absent and prediction is impossible. If heterogeneity is large, rupture is more continuous.

To obtain this insight, we used simple mechanical models of masses and springs with local stress transfer [Andersen et al., 1997]. This class of models does not claim precise realism but attempts rather to identify the different regimes of behavior. The scientific enterprise is paved with such reductionism that has worked surprisingly well. We were thus able to quantify how heterogeneity plays the role of a relevant field: systems with limited stress amplification exhibit a so-called tri-critical transition, from a Griffith-type abrupt rupture (first-order) regime to a progressive damage (critical) regime as the disorder increases. This effect was also demonstrated on a simple mean-field model of rupture, known as the democratic fiber bundle model. It is remarkable that the disorder is so relevant as to change the nature of rupture. In systems with long-range elasticity, the nature of the rupture process may not change qualitatively as above, but quantitatively: any disorder may be relevant in this case and make the rupture similar to a critical point; however, we have recently shown that the disorder controls the width of the critical region [Sornette and Andersen, 1998]. The smaller it is, the smaller will be the critical region, which may become too small to play any role in practice. For realistic systems, long-range correlations transported by the stress field around defects and cracks make the problem more subtle. Time dependence is expected to be a crucial aspect in the process of correlation building in these processes. As the damage increases, a new ``phase'' appears, where micro-cracks begin to merge leading to screening and other cooperative effects. Finally, the main fracture is formed leading to global failure. In simple intuitive terms, the failure of compositive systems may often be viewed as the result of a correlated percolation process. The challenge is to describe the transition from the damage and corrosion processes at the microscopic level to the macroscopic rupture.

2.3-SCALING, CRITICAL POINT AND RUPTURE PREDICTION

Motivated by the multi-scale nature of the second class of ruptures and analogies with the percolation model, physicists working in statistical physics started to suggest in the mid-eighties that rupture of sufficiently heterogeneous media would exhibits some universal properties, in a way maybe similar to critical phase transitions. The idea was to build on the knowledge accumulated in statistical physics on the so-called $N-$body problem and cooperative effects in order to describe multiple interactions between defects. However, most of the models were extremely naive and essentially all of them quasi-static with rather unrealistic loading rules. Some suggestive scaling laws were found to describe size effects and damage properties, but the relevance to real materials was not convincingly demonstrated with some exceptions. The interest of physicists for the modelling of rupture in heterogeneous media seems to have decreased since then except for a few groups.

In 1992, we proposed the first model of rupture with a realistic dynamical law for the evolution of damage, modelled as a space dependent damage variable, a realistic loading and with many growing interacting micro-cracks [Sornette and Vanneste, 1992]. We found that the total rate of damage, as measured for instance by the elastic energy released per unit time, increases as a power law of the time-to-failure on the approach to the global failure. In this model, rupture was indeed found to occur as the culmination of the progressive nucleation, growth and fusion between microcracks, leading to a fractal network, but the exponents were found to be non-universal and function of the damage law. This model has since then been found to describe correctly experiments on the electric breakdown of insulator-conducting composites [Lamaignere et al., 1996]. Another application is damage by electromigration of polycristalline metal films [Bradley and Wu, 1993].

In 1993, we extended these results by testing on real engineering composite structures the concept that failure in fiber composites may be described by a critical state, thus predicting that the rate of damage would exhibit a power law behavior [Anifrani et al., 1995]. This critical behavior may correspond to an acceleration of the rate of energy release or to a deceleration, depending on the nature and range of the stress transfer mechanism and on the loading procedure. We based our approach on a theory of many interacting elements called the renormalization group. The renormalization group can be thought of as a construction scheme or ``bottom-up'' approach to the design of large scale structures. Since then, other numerical simulations of statistical rupture models and controlled experiments have confirmed that, near the global failure point, the cumulative elastic energy released during fracturing of heterogeneous solids follows a power law behavior.

To get a qualitative understanding of the renormalization group theory, let us consider the usual way that composite or mechanical systems are designed (for engineering and industrial applications). This may be called the component system, or bottom-up design. First, it is necessary to thoroughly understand the properties and limitations of the materials to be used, and experimental tests are begun. With this knowledge, larger component parts are designed and tested individually. As deficiencies and design errors are noted they are corrected and verified with further testing. Since one tests only parts at a time these tests and modifications are not overly expensive. Finally one works up to the final design of the entire engine, to the necessary specifications. There is a good chance, by this time, that the global structure will generally succeed, or that any failures are easily isolated and analyzed because the failure modes, limitations of materials, etc., are well understood. There is a very good chance that the modifications to the system to get around the final difficulties are not very hard to make, for most of the serious problems have already been discovered and dealt with in the earlier, less expensive, stages of the process. The reliability and failure properties of such system is the result of a bottom-up approach of the reliability and failure properties of the constitutive elements, in other words calls for a hierarchical modelling. The renormalization group offers a general framework to formalize and calculate how a property or failure at a given scale may or may not cascade at higher levels.

Based on an extension of the usual solutions of the renormalization group and on explicit numerical and theoretical calculations, we were thus led to propose that the power law behavior of the time-to-failure analysis should be corrected for the presence of log-periodic modulations [Anifrani et al., 1995]. Since then, this method has been tested extensively during our continuing collaboration with the French Aerospace company Aerospatiale on pressure tanks made of kevlar-matrix and carbon-matrix composites embarked on the European Ariane 4 and 5 rockets. In a nutshell, the method consists in this application in recording acoustic emissions under constant stress rate and the acoustic emission energy as a function of stress is fitted by the above log-periodic critical theory. One of the parameter is the time of failure and the fit thus provides a ``prediction'' when the sample is not brought to failure in the first test. Improvements of the theory and of the fitting formula were applied to about 50 pressure-tanks. 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.

This example constitutes a remarkable example where rather abstract theoretical concepts borrowed from the rather esoteric field of statistical and nonlinear physics have been applied directly to a concrete industrial problem. This example is remarkable for another reason that we would like to relate.

2.4-DISCRETE SCALE INVARIANCE, COMPLEX EXPONENTS AND LOG-PERIODICITY

During our research on the acoustic emissions of the industrial pressure tank of the European Ariane rocket, we discovered the existence of log-periodic scaling in non-hierarchical systems. To fix ideas, consider the acoustic energy $E \sim (t_c - t)^{-\alpha}$ following a power law a function of time to failure. Suppose that there is in addition a log-periodic signal modulation $$ E \sim (t_c - t)^{-\alpha}~[1+ C\cos[2\pi {\log (t_c - t) \over \log \lambda}]]. $$ We see that the local maxima of the signal occur at $t_n$ such that the argument of the cosine is a multiple to $2\pi$, leading to a geometrical time series $t_c - t_n \sim \lambda^{-n}$ where $n$ is an integer. The oscillations are thus modulated in frequency with a geometric increase of the frequency on the approach to the critical point $t_c$. This apparent esoteric property turns out to be surprisingly general both experimentally and theoretically and we are probably only at the beginning of our understanding. From a formal point of view, log-periodicity can be shown to be 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.

During the third century BC, Euclid and his students introduced the concept of space dimension, which can take positive integer values equal to the number of independent directions. For instance, a line has dimension one and we live in a space of dimension three and a spacetime of dimension four. During the second half of the nineteen century and the twentieth century, the notion of dimensions was generalized to fractional values. The word ``fractal'' was coined by Mandelbrot to describe sets consisting of parts similar to the whole, and which can be described by a fractional dimension. This generalization of the notion of a dimension from integers to real numbers reflects the conceptual jump from translational invariance to continous scale invariance. Science progresses by analogies and generalization, thus allowing to embody in simpler concepts an increasing broad phenomenology. Here, there is a further generalization of the notion of dimension, according to which the dimensions or exponents are taken from the set of complex numbers. This generalization captures the interesting and rich phenomenology of systems exhibiting discrete scale invariance, a weaker form of scale invariance symmetry, associated with log-periodic corrections to scaling. Discrete scale invariance is a weaker kind of scale invariance according to which the system or the observable obeys scale invariance only for specific choices of magnifications, which form in general an infinite but countable set of values. This property can be seen to encode also the concept of lacunarity of the fractal structure.

Encouraged by our observation of log-periodicity in rupture phenomena, we started to investigate whether similar signatures could be observed in other systems. And looking more closely, we were led to find them in many systems in which they had been previously unsuspected. These structures have long been known as possible from the formal solutions of renormalization group equations in the seventies but were rejected as physically irrelevant. They were studied in the eighties in a rather academic context of special artificial hierarchical geometrical systems. Our work led us to realize that discrete scale invariance and its associated complex exponents and log-periodicity may appear ``spontaneously'' in natural systems, i.e. without the need for a pre-existing hierarchy. Examples that we have documented [Sornette, 1998] are diffusion-limited-aggregation clusters, rupture in heterogeneous systems, earthquakes, animals (a generalization of percolation) among many other systems. Complex scaling could also be relevant to turbulence, to the physics of disordered systems, as well as to the description of out-of-equilibrium dynamical systems. Some of the physical mechanisms at the origin of these structures are now better understood. General considerations using the framework of field theories, the framework to describe fundamental particle physics and condensed matter systems, show that they should constitute the rule rather than the exception, similarly to the realization that chaotic (non-integrable) dynamical systems are more general that regular (integrable) ones. In addition to a fascinating physical relevance of this abstract notion of complex dimensions, the even more important aspect in our point of view is that discrete scale invariance and its signatures may provide new insights in the underlying mechanisms of scale invariance and be very useful for prediction purposes.

3-TOWARDS A PREDICTION OF EARTHQUAKES?

An important effort is carried out world-wide in the hope that, maybe sometimes in the future, the grail of useful earthquake prediction will be attained. Among others, the research comprises continuous observations of crustal movement and geodetic surveys, seismic observations, geoelectric-geomagnetic observations, geochemical and groundwater measurements. The seismological community has been criticized in the past by promising results using various prediction techniques (e.g. anomalous seismic wave propagations, dilatancy diffusion, Mogi donuts, pattern recognition algorithms, etc.) that have not delivered to the expected level. The need for a reassessment of the physical processes has been recognized and more fundamental studies are persued on crustal structures in seismogenic zones, historical earthquakes, active faults, laboratory fracture experiments, earthquake source processes, etc.

There is even now an opinion gaining momemtum that earthquakes could be inherently unpredictable [Geller et al., 1997]. The argument is that past failures and recent theories suggest fundamental obstacles to prediction. It is then proposed that the emphasis be placed on basic research in earthquake science, real-time seismic warning systems, and long-term probabilistic earthquake hazard studies. It is true that useful predictions are not available at present and seem hard to get in the near future but would it not be a little presomptuous to claim that prediction is impossible? Many passed examples in the development of Science have taught us that unexpected discoveries can modify completely what was previously considered possible or not. In the context of earthquakes, the problem is made more complex by the societal implication of prediction with, in particular, the question of where to direct in an optimal way the limited available ressources.

We here focus on the scientific problem and describe a new direction that suggests reason for optimism. Recall that an earthquake is triggered when a mechanical instability occurs and a fracture (the sudden slip of a fault) appears in a part of the earth crust. The earth crust is in general complex (in composition, strength, faulting) and groundwater may play an important role. How can then one expect to unravel this complexity and achieve a useful degree of prediction? We need to understand the nature of the organization of the crust, then the characteristic properties of large earthquakes and the nature of signatures that could be used for prediction.

3.1-THE LARGE SCALE SELF-ORGANIZATION OF THE CRUST

Seismicity is characterized by an extraordinary rich phenomenology and variability which makes very difficult the development of a coherent explanatory and predictive framework. In the late eighties, we and other groups independently proposed that the concept of self-organized criticality (SOC) could provide a plausible framework. Apart from the rationalization that it provides for the Gutenberg-Richter law for earthquakes, the power law fault length distribution and for the fractal geometry of sets of earthquake epicenters and fault patterns, it has not been exploited until recently to advance our understanding on the crust organization and about the very rich and subtle properties found in tectonics and seismology.

Roughly speaking, SOC refers to the spontaneous organization of a system driven from the outside in a dynamical statistical stationary state, which is characterized by self-similar distributions of event sizes and fractal geometrical properties. SOC refers to the class of phenomena occurring in slowly driven out-of-equilibrium systems made of many interactive components, which possess the following fundamental properties :

-a highly non-linear behavior, namely essentially a threshold response,

-a very slow driving rate ,

-a globally stationary regime, characterized by stationary statistical properties, and

-power distributions of event sizes and fractal geometrical properties.

The crust obeys these four conditions:

-the threshold response is associated with the stick-slip instability of solid friction or to a rupture threshold thought to characterize the behavior of a fault upon increasing applied stress;

-The slow driving rate is that of the slow tectonic deformations thought to be exerted at the borders of a given tectonic plate by the neighboring plates and at its base by the underlying lower crust and mantle.

-The stationarity condition ensures that the system is not in a transient phase, and distinguished the long-term organization of faulting in the crust from, for instance, irreversible rupture of a sample in the laboratory.

-The power laws and fractal properties reflect the notion of scale invariance, namely measurements at one scale are related to measurements at another scale by a normalization involving a power of the ratio of the two scales. These properties are important and interesting because they characterize systems with many relevant scales and long-range interactions as probably exist in the crust.

We have recently tested the usefulness of the SOC hypothesis by measuring its predictive and explanatory power outside the range of observations that have helped defined it. We thus explored how the SOC concept can help understand the observed earthquake clustering on relatively narrow fault domains and the phenomenon of induced seismicity by human activity such as water impoundment in artificial lakes, gas and ore extraction. We found that both pore pressure changes and mass transfers leading to incremental deviatoric stresses of less than 10 atmospheric pressure are sufficient to trigger seismic instabilities in the uppermost crust with magnitude ranging up to $7$ in otherwise historically aseismic areas. We argued that these observations are in accord with the SOC hypothesis as they show that a significant fraction of the crust is not far from instability and can thus be made unstable by minute perturbations. The properties of induced seismicity and their rationalization in terms of the SOC concept provide further evidence that potential seismic hazards extend over a much larger area than that where earthquakes are frequent.

3.2-LARGE EARTHQUAKES

There is a series of surprising and somewhat controversial studies showing that many large earthquakes have been preceded by an increase in the number of intermediate sized events. The relation between these intermediate sized events and the subsequent main event has only recently been recognized because the precursory events occur over such a large area that they do not fit prior definitions of foreshocks [Jones and Molnar, 1979]. In particular, the $11$ earthquakes in California with magnitudes greater than $6.8$ in the last century are associated with an increase of precursory intermediate magnitude earthquakes measured in a running time window of five years [Knopoff et al., 1996]. What is strange about the result is that the precursory pattern occured with distances of the order of $300$ to $500 ~km$ from the futur epicenter, i.e. at distances up to ten times larger that the size of the futur earthquake rupture. Furthermore, the increased intermediate magnitude activity switched off rapidly after a big earthquake in about half of the cases. This implies that stress changes due to an earthquake of rupture dimension as small as $35 ~km$ can influence the stress distribution to distances more than ten times its size. This result defies usual models.

This observation is not isolated. There is mounting evidence that the rate of occurrence of intermediate earthquakes increases in the tens of years preceding a major event. Sykes and Jaume [1990] present evidence that the occurrence of events in the range $5.0-5.9$ accelerated in the tens of years preceding the large San Francisco bay area quakes in 1989, 1906, and 1868 and the Desert Hot Springs earthquake in 1948. Lindh [1990] points out references to similar increases in intermediate seismicity before the large 1857 earthquake in southern California and before the 1707 Kwanto and the 1923 Tokyo earthquakes in Japan. More recently, Jones (1992, 1994) has documented a similar increase in intermediate activity over the past $8$ years in southern California. This increase in activity is limited to events in excess of $M = 5.0$; no increase in activity is apparent when all events M > 4.0 are considered. Ellsworth et al. [1981] also reported that the increase in activity was also limited to events larger than $M = 5$ prior to the 1989 Loma Prieta earthquake in the San Francisco Bay area. Bufe and Varnes [1993] analyse the increase in activity which preceded the 1989 Loma Prieta earthquake in the San Francisco Bay area while Bufe et al. [1994] document a current increase in seismicity in several segments of the aleutian arc.

Recently, we have investigated more quantitatively these observations and asked what is the law, if any, controlling the increase of the precursory activity [Bowman et al., 1998]. 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 (Benioff strain) 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 [Bowman et al., 1998]. An empirical relation between the logarithm of the critical region radius (R) and the magnitude of the final event (M) was found, such that log R is proportional to 0.5 M, suggesting that the largest probable event in a given region scales with the size of the regional fault network.

To rationalize these observations, it is natural to invoke again the importance of heterogeneity. Recall that Mogi showed experimentally on a variety of materials that, the larger the disorder, the stronger and more useful are the precursors to rupture. For a long time, the Japanese research effort for earthquake prediction and risk assessment was based on this very idea [Mogi, 1974]. Can our previous results obtained for engineering rupture phenomena be applied to earthquakes?

If rupture of a laboratory sample is the well-defined conclusion of the loading history, the same cannot be said for earthquakes. The problem is that it is not clear how to reconcile this idea with both the nature of the dynamical rupture propagation and the large scale and time organization of the crust previously discussed, that rather suggest a succession of complex coupled irregular cycles, apparently quite different from the critical point picture in which a large earthquake is the culmination of a preparatory stage. We have recently found a way out of this conundrum by studying a simple numerical model of earthquakes on a hierarchical fault structure driven at a slow average uniform rate, taking into account the crust heterogeneity and the existence of relaxation processes [Huang et al., 1998]. We observe that, while the system self-organizes at large time scales according to the expected statistical characteristics, such a the Gutenberg-Richter law for earthquake magnitude frequency, most of the large earthquakes have precursors occuring over time scales of tens of years and over distances of hundreds of kilometers. This type of behavior is documented in earthquake catalogs as we have shown, but its interpretation leads to considerable difficulty as it is hard to understand how $1-10 km$ ruptures can be related over distance of $100 ~km$. Within the critical view point, these intermediate earthquakes are both ''witnesses'' and ''actors'' of the building-up of correlations. These precursors produce an energy release, which when measured as a time-to-failure process, is quite consistent with a power law behavior. In addition, the statistical average (over many large earthquakes) of the correlation length, measured as the maximum size of the precursors, also increases as a power law of the time to the large earthquake. These two properties qualify a critical behavior. From the point of view of self-organized criticality, this is surprising news: the individual large earthquakes do not lose their ``identity'' because they belong to the large scale and long time collective behavior of the tectonic plate.

3.3-LOG-PERIODICITY?

We must add a third and last touch to the picture, which uses the concept of discrete scale invariance, its associated complex exponents and log-periodicity, as discussed above. In the presence of the frozen nature of the disorder together with stress amplification effects, we showed that the critical behavior of rupture is described by complex exponents, in other words, the measurable physical quantities can exhibit a power law behavior (real part of the exponents) with superimposed log-periodic oscillations (due to the imaginary part of the exponents). Physically, this stems from a spontaneous organization on a fractal fault system with ``discrete scale invariance''. The practical upshot is that the log-periodic undulations may help in ``synchronizing'' a better fit to the data. In the above numerical model, most of the large earthquakes whose period is of the order of a century can be predicted in this way $4$ years in advance with a precision better than a year. For the real earth, we do not know yet as several difficulties hinder a practical implementation, such as the definition of the relevant space-time domain. A few encouraging results have been obtained but much remains to test these ideas systematically, especially using the methodology presented above to detect the regional domain of critical maturation before a large earthquake {Sornette and Sammis, 1995].

While encouraging and suggestive, extreme caution should be exercized before even proposing that this method is useful for predictive purpose but the theory is beautiful in its self-consistency and, even if probably inacurate in details, it may provide a useful guideline for the future.

4-PREDICTING FINANCIAL CRASHES?

Stock market crashes are momentous financial events that are fascinating to academics and practitioners alike. Within the efficient markets literature, only the revelation of a dramatic piece of information can cause a crash, yet in reality even the most thorough {\em post-mortem} analyses are typically inconclusive as to what this piece of information might have been. For traders, the fear of a crash is a perpetual source of stress, and the onset of the event itself always ruins the lives of some of them, not to mention the impact on the economy.

A few years ago, we advanced the hypothesis [Sornette et al., 1996] 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 here: in mathematical terms, complex dynamical 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. Given the puzzling and violent nature of stock market crashes, it is worth investigating whether there could possibly be a link.

In doing so, we have found three major points. First, it is entirely possible to build a dynamic model of the stock market exhibiting well- defined critical points that lies within the strict confines of rational expectations, a landmark of economic theory, and is also intuitively appealing. We stress the importance of using the framework of rational expectation in contrast to many other recent attempts. When you invest your money in the stock market, in general you do not do it at random but try somehow to optimize your strategy with your limited amount of information and knowledge. The usual criticism addressed to theories abandoning the rational behavior condition is that the universe of conceivable irrational behavior patterns is much larger than the set of rational patterns. Thus, it is sometimes claimed that allowing for irrationality opens a Pandora's box of ad hoc stories that have little out-of-sample predictive powers. To deserve consideration, a theory should be parsimonious, explain a range of anomalous patterns in different contexts, and generate new empirical implications.

Second, we find that the mathematical properties of a dynamic system going through a critical point are largely independent of the specific model posited, much more so in fact than ``regular'' (non-critical) behavior, therefore our key predictions should be relatively robust to model misspecification.

Third, these predictions are strongly borne out in 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. We also discovered in a systematic testing procedure a signature of near-critical behavior that culminated in a two weeks interval in May 1962 where the stock market declined by $12\%$. The fact that we ``discovered'' the ``slow crash'' of 1962 without prior knowledge of it just be trying to fit our theory is a reassuring sign about the integrity of the method. Analysis of more recent data showed a clear maturation towards a critical instability that can be tentatively 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 question: has human nature changed?

4.1-THEORY OF CRASHES WITH RATIONAL AGENTS

Consider a single asset that pays no dividends and for simplicity, we ignore the interest rate, risk aversion, information asymmetry, and the market-clearing condition. In this dramatically stylized framework, rational expectations are simply equivalent to the familiar hypothesis that the present price of the asset is equal to its expectation over the future conditional on information revealed up to the present.

Suppose that we introduce an exogenous probability of crash. Then, simple calculations show that the higher the probability of a crash, the faster the price must increase (conditional on having no crash) in order to satisfy the no-arbitrage condition (i.e. that there are no ``free lunch''). Intuitively, investors must be compensated by the chance of a higher return in order to be induced to hold an asset that might crash. This is the only effect that we wish to capture in this part of the model. This effect is fairly standard, and it was pointed out earlier in a closely related model of bubbles and crashes under rational expectations by Blanchard (1979, top of p.389). It may go against the naive preconception that price is adversely affected by the probability of the crash, but our result is the only one consistent with rational expectations.

A few additional points deserve careful attention. First, the crash is modelled as an exogenous event: nobody knows exactly when it could happen, which is why rational traders cannot earn abnormal profits by anticipating it. Second, the probability of a crash itself is an exogenous variable that must come from outside this model. There is no feedback loop whereby prices would in turn affect either the arrival or the probability of a crash. This may not sound totally satisfactory, but it is hard to see how else to obtain crashes in a rational expectations model: if rational agents could somehow trigger the arrival of a crash they would choose never to do so, and if they could control the probability of a crash they would always choose it to be zero. In our model, the crash is a random event whose probability is driven by external forces, and {\em once this probability is given} it is rationally reflected into prices. If the other alternatives are to give up rational expectations or to give up studying crashes, we prefer to stick with our approach.

4.2-THE CRASH

We now explain how to obtain a crash in terms of reasonable models of micro-level agent behavior. We start by a discussion in naive terms. A crash happens when a large group of agents place sell order simultaneously. This group of agents must create enough of an imbalance in the order book for market makers to be unable to absorb the other side without lowering prices substantially. One curious fact is that the agents in this group typically do not know each other. They did not convene a meeting and decide to provoke a crash. Nor do they take orders from a leader. In fact, most of the time, these agents disagree with one another, and submit roughly as many buy orders as sell orders (these are all the times when a crash {\em does not} happen). The key question is: by what mechanism did they suddenly manage to organize a coordinated sell-off?

We propose the following answer: all the traders in the world are organized into a network (of family, friends, colleagues, etc) and they influence each other {\em locally} through this network. Specifically, if I am directly connected with $k$ nearest neighbors, then there are only two forces that influence my opinion: (a) the opinions of these $k$ people; and (b) an idiosyncratic signal that I alone receive. Our working assumption here is that agents tend to {\em imitate} the opinions of their nearest neighbors, not contradict them. It is easy to see that force (a) will tend to create order, while force (b) will tend to create disorder. The main story that we are telling in here is the fight between order and disorder. As far as asset prices are concerned, a crash happens when order wins (everybody has the same opinion: selling), and normal times are when disorder wins (buyers and sellers disagree with each other and roughly balance each other out). We must stress that this is exactly the opposite of the popular characterization of crashes as times of chaos.

Our answer has the advantage that it does not require a global coordination mechanism: we will show that macro-level coordination can arise from micro-level imitation. Furthermore, it relies on a somewhat realistic model of how agents form opinions. It also makes it easier to accept that crashes can happen for no rational reason. If selling were a decision that everybody reached independently from one another just by reading the newspaper, either we would be able to identify unequivocally the triggering news after the fact (and for the crashes of 1929 and 1987 this was not the case), or we would have to assume that everybody becomes irrational in exactly the same way at exactly the same time (which is distasteful). By contrast, our reductionist model puts the blame for the crash simply on the tendency for agents to imitate their nearest neighbors. We do not ask why agents are influenced by their neighbors within a network: since it is a well- documented fact, we take it as a primitive assumption rather than as the conclusion of some other model of behavior. Presumably some justification for these imitative tendencies can be found in evolutionary psychology. Note, however, that there is no information in our model, therefore what determines the state of an agent is pure noise.

The output of the model is a quantity termed the susceptibility which measures the sensitivity of the average state to a perturbation. The susceptibility has a second interpretation as (a constant times) the variance of the average opinion around its expectation of zero caused by the random idiosyncratic shocks . Another related interpretation is that, if you consider two agents and you force the first one to be in a certain state, the impact that your intervention will have on the second agent will be proportional to the susceptibility. For these reasons, we believe that the susceptibility correctly measures the ability of the system of agents to agree on an opinion. If we interpret the two states in a manner relevant to asset pricing, it is precisely the emergence of this global synchronization from local imitation that can cause a crash. Thus, we will characterize the behavior of the susceptibility, and we will posit that the hazard rate of crash follows a similar process. We do not want to assume a one-to-one mapping between hazard rate and susceptibility because there are many other quantities that provide a measure of the degree of coordination of the overall system, such as the correlation length (i.e.~the distance at which imitation propagates) and the other moments of the fluctuations of the average opinion.

We argue that these properties are very robust to model misspecification. We claim that models of crash that combine the following features:

-A system of noise traders who are influenced by their neighbors; -Local imitation propagating spontaneously into global cooperation;

-Global coperation among noise traders causing crash;

-Prices related to the properties of this system;

-System parameters evolving slowly through time; would display the same characteristics as ours, namely prices following a power law in the neighborhood of some critical date, with either a real or complex critical exponent. What all models in this class would have in common is that the crash is most likely when the locally imitative system goes through a {\em critical} point.

In physics, critical points are widely considered to be the most interesting properties of complex systems. A system goes critical when local influences propagate over long distances and the average state of the system becomes exquisitely sensitive to a small perturbation. Another characteristic is that critical systems are self-similar across scales: in our example, at the critical point, an ocean of traders who are mostly bearish may have within it several islands of traders who are mostly bullish, each of which in turns surrounds lakes of bearish traders with islets of bullish traders; the progression continues all the way down to the smallest possible scale: a single trader [Wilson, 1979]. Intuitively speaking, critical self-similarity is why local imitation cascades through the scales into global coordination.

Because of scale invariance, the behavior of a system near its critical point must be represented by a power law (with real or complex critical exponent): it is the only family of functions that are homogenous, i.e.~they remain unchanged (up to scalar multiplication) when their argument gets rescaled by a constant. In general, physicists study critical points by forming equations to describe the behavior of the system across different scales, and by analyzing the mathematical properties of these equations. This is known as {\em renormalization group theory} (Wilson, 1979) as already discussed. Before renormalization group theory, the fact that a system's critical behavior had to be correctly described at all scales simultaneously prevented standard approximation methods from giving satisfactory results. But renormalization group theory turned this liability into an asset by building its solution precisely on the self-similarity of the system across scales. Let us add that, in spite of its conceptual elegance, this method is nonetheless mathematically challenging.

For our purposes, however, it is sufficient to keep in mind that the key idea proposed here is the following: the massive and unpredictable sell-off occuring during stock market crashes comes from local imitation cascading through the scales into global cooperation when a complex system approaches its critical point. Regardless of the particular way in which this idea is implemented, it will generate the same universal implications.

Strictly speaking, these equations are approximations valid only in the neighborhood of the critical point. We have proposed a more general formula with additional degrees of freedom to better capture behavior away from the critical point. The specific way in which these degrees of freedom are introduced is based on a finer analysis of the renormalization group theory that is equivalent to including the next term in a systematic expansion around the critical point and introduce a log-periodic component to the market price behavior.



4.3-EXTENDED EFFICIENCY AND SYSTEMIC INSTABILITY

Our main point is that the market anticipates the crash in a subtle self-organized and cooperative fashion, hence releasing precursory ``fingerprints'' observable in the stock market prices. In other words, this implies that market prices contain information on impending crashes. If the traders were to learn how to decipher and use this information, they would act on it and on the knowledge that others act on it and the crashes would probably not happen. Our results suggest a weaker form of the ``weak efficient market hypothesis'' [Fama, 1991], according to which the market prices contain, in addition to the information generally available to all, subtle informations formed by the global market that most or all individual traders have not yet learned to decipher and use. Instead of the usual interpretation of the efficient market hypothesis in which traders extract and incorporate consciously (by their action) all informations contained in the market prices, it may be that the market as a whole can exhibit an ``emergent'' behavior not shared by any of its constituant. In other words, we have in mind the process of the emergence of intelligent behaviors at a macroscopic scale that individuals at the microscopic scale have not idea of. This process has been discussed in biology for instance in animal populations such as ant colonies or in connection with the emergence of conciousness [Anderson et al., 1988; Holland, 1992]. The usual efficient market hypothesis will be recovered in this context when the traders learn how to extract this novel collective information and act on it.

Most previous models proposed for crashes have pondered the possible mechanisms to explain the collapse of the price at very short time scales. Here in contrast, we propose that the underlying cause of the crash must be searched years before it in the progressive accelerating ascent of the market price, the speculative bubble, reflecting an increasing built-up of the market cooperativity. From that point of view, the specific manner by which prices collapsed is not of real importance since, according to the concept of the critical point, any small disturbance or process may have triggered the instability, once ripe. The intrinsic divergence of the sensitivity and the growing instability of the market close to a critical point might explain why attempts to unravel the local origin of the crash have been so diverse. Essentially all would work once the system is ripe. Our view is that the crash has an endegeneous origin and that exogeneous shocks only serve as triggering factors. We propose that the origin of the crash is much more subtle and is constructed progressively by the market as a whole. In this sense, this could be termed a systemic instability. This understanding offers ways to act to mitigate the build-up of conditions favorable to crashes.

5-PREDICTING HUMAN PARTURITION

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. This is maybe not so surprising in view of the commonalities with the previous examples.

Can we predict parturition? Notwithstanding the large number of investigations on the factors that could trigger parturition in superior mammals (monkeys and humans), we still do not have a clear signature in any of the measured variables. This is to be contrasted to the situation for inferior mammals such as cats, cows, etc, for which the secretion of a specific hormone can be linked unambiguously to the triggering of parturition.

Knowledge of precursors and predictors of human parturition would be important both for our understanding of the controlling mechanisms and for practical use for detection and diagnostic of various abnormalities of birth process. They involve a multitude of genetic, metabolic, nutritional, hormonal and environmental factors. Present research is however hindered by the lack of a clear recognized correlation between the time evolution of these various variables with the initiation of parturition.

5.1-CRITICAL THEORY OF PARTURITION

In collaboration with a team of obstetricians, we have proposed [Sornette et al., 1994] a coherent logical 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. Our approach gives a number of new predictions and suggests a strategy for future research and clinical studies, which present interesting potentials for improvements in predicting methods and in describing various prenatal abnormal situations.

We have proposed to view the occurrence of parturition as an instability, in which the control parameter is a maturity parameter (MP), roughly proportional to time, and the order parameter is the amplitude of the coherent global uterine activity in the parturition regime. This idea is summarized by the concept of a so-called supercritical ``bifurcation''. This simple view is in apparent contradiction with the extreme complexity of the fetus-mother system, which can be addressed at several levels of descriptions, starting at the highest level from the mother, the fetus and their coupling through the placenta. For example, in the mother, the myometrium plays an important role in pregnancy, maturation and onset of labor. It is now well-established that the human myometrium is an heterogeneous tissue formed of several layers which differ in their embryological origin and which exhibit quite different histological and pharmalogical properties. In the uterine corpus, one must distinguish the outer (longitudinal) and the inner (circular) layers. These two layers composed mainly of smooth muscle cells are separated by an intermediate layer containing a large amount of vascular and connective tissues, but poor in smooth muscle cells. The inner and outer muscle layers have different patterns of contractility and differ in their response and sensitivity to contractile and relaxant agents. This is just an example of the complexity which goes on down to the molecular level, with the action of many substances providing positive and negative feedbacks evolving as a function of maturation. The basis of our simple theory relies on many recent works in a variety of domains (mathematics, hydrodynamics, optics, chemistry, biology, etc) which have shown that a lot of complex systems consisting of many nonlinear coupled subsystems or components may self-organize and exhibit coherent behavior of a macroscopic scale in time and/or space, in suitable conditions. The Rayleigh-B\'enard fluid convection experiment is one of the simplest paradigm for this type of behavior. The coherent behavior appears generically when the coupling between the different components becomes strong enough to trigger or synchronize the initially incoherent subsystems. There are many observations in human parturition where an increasing ``coupling'' is associated with maturation of the fetus leading to the cooperative synchronized action of all muscle fibers of the uterus characteristic of labor.

5.2-PREDICTIONS

Perhaps, the most vivid illustration of the increasing coupling as maturation increases is provided by monitoring the uterine activity, using standard external tocographic techniques. Away from term, the muscle contractions during gestation are generally weak and characterized by local bursts of activity both in time and space. Increasing uterine activity is observed when the term is approaching, culminating in a complete modification of behavior where regular globally coherent contractions reflects the spatial and time coherence of all the muscles constituting the uterus. The transition between the premature regime and the parturition regime at maturity is characterized by a systematic tendency to increasing uterine activity, both in amplitude, duration of the bursts and spatial extension of the activated uterine domains. The susceptibility of the fetus-mother system (to influence the uterine response) to external perturbations or stimulations seems to increase notably on the approach of parturition, since important modifications and reactions of the uterus may result from relatively small stimulii from the mother or fetus.

The main prediction is that, on the approach to the critical instability, one expects a characteristic increase of the fluctuations of uterine activity. Other quantities that could be measured and which are related to the uterine activity are expected to present a similar behavior. The cooperative nature of maturation and parturition proposed here rationalizes the present inability to establish unequivocally predictive parameters of the biochemical events preceeding myometrical activity and/or cervical ripening involved in preterm labor. Our theory suggests a precise experimental methodology in order to obtain an early diagnosis, essential for the efficient treatment of prematurity, which still constitutes the major cause of neonatal morbidity and mortality. In particular, monitoring muscle tremors or vibrations as a function of time of muscle fibers of the uterus would provide quantitative tests of the theory with respect to the spatio-temporal build-up of contractile fluctuations. Our theory also correctly accounts for the observations that external factors affecting the mother such as heavy work and psychologic stress are able to modify the maturity of the uterus measured by the progressive modification of the cervix and more frequent uterine contractions. These external factors, in addition to produce direct contraction stimulations, could also be able to modify the post-maturity parameter and control the susceptibility of the fetus-mother system to small influences which can trigger the change from discordant contrations to concordant contractions of a premature or post-mature labor.

We note finally that the whole policy for the description of risk factors has been based on an implicit and unformalized hypothesis of a critical transition which is explicited in our theoretical framework. The prevention program for preterm deliveries [Papiernik et al., 1984] was also based on the hypothesis of such a critical transition and the understanding that a small reduction of a triggering factor could be enough to prevent the uterus from beginning its critical phase of activity. The high susceptibility of the fetus-mother system to various factors is also at the origin of the fact that the conventional system of calculation of the risk factors does not explain the real success of the prevention which has been observed [Papiernik et al., 1984] . Effectively applied in France, our system, which is based on this idea of a critical transition, was able to reduce significantly the rate of preterm births for all french women measured on Haguenau population of pregnant women from 1971 to 1982, or on randomized samples of all french births.

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