The Impact of Chaos on Science and Society (UNU, 1997, 415 pages)
 4. The impact of chaos on mathematics
 (introduction...) Abstract I. Introduction: A historical view II. The Lorenz attractor III. The Feigenbaum bifurcation IV. Hydrodynamic turbulence V. Ergodic theory of differentiable dynamical systems: Axiom A systems VI. Ergodic theory of differentiable dynamical systems: General systems VII. Quadratic maps of the interval and the Hénon attractor VIII. Zeta functions IX. Conclusion References

### I. Introduction: A historical view

One can hardly discuss the impact of chaos on mathematics and ignore the impact of mathematics on chaos. Looking at the problem more carefully, one realizes that there is indeed a rich history of interactions between the mathematical theory of dynamical systems, and its applications in the field of chaos. Such a situation prevails rather generally for the relations of mathematics with its applications, but it is particularly striking here because chaos is, basically, a mathematical phenomenon. It will therefore be worth discussing briefly the history of the complex relations between the mathematics of dynamical systems theory, and the analysis of chaos in physics and other sciences.

We start with some definitions. A dynamical system is simply a time evolution defined by a differential equation

(1)

in the case of a continuous time, or by an equation of the form

xt+1=f(xt) (2)

in the case of a discrete time t. The variable x belongs to some finite or infinite dimensional space M. Notice that the time does not explicitly occur in the right-hand sides of (1) and (2): we consider autonomous systems. There is thus a (nonlinear) time evolution operator ft such that x(t)=ftx(0). In the discrete time case, ft is simply the tth iterate of f. We have

f0=identity, ft·fs=ft+s

We shall assume that F in (1) or f in (2) is differentiable. This looks like a modest technical assumption, but it is in fact the key to interesting developments.

Consider the change dx(t) corresponding to a change dx(0) in initial condition. We have

Typically - and we shall not be more precise at this point - both dx(t) and f/x behave exponentially with time, i.e.

dx(t) ~ elt.

When l<0, we have stability, when l>0 we have sensitive dependence on initial condition. It is clear how this can happen at an unstable fixed point, for instance, but what is more surprising is that sensitive dependence on initial condition can occur for (almost) all initial condition. This prevalence of sensitivity is what we now call chaos.

Chaos, as we have just defined it, was discovered by J. Hadamard at the end of the nineteenth century in a special (Hamiltonian) dynamical system called the geodesic flow on a manifold of negative curvature [1]. Hadamard immediately understood the philosophical importance of his result: an arbitrarily small uncertainty on the initial condition entails a large uncertainty on the predicted state of the system after a sufficiently long time. Other scientists (P. Duhem, H. Poincaré) also understood the importance of the phenomenon discovered by Hadamard, and Poincaré [2] discusses the relevance of sensitive dependence on initial condition to the dynamics of a hard sphere system, and to weather predictions.

The early discovery of chaos had however no lasting influence on physics. The new ideas were forgotten and had to be rediscovered again, much later and independently. On the mathematical side, however, the work of Hadamard and Poincaré led to uninterrupted progress up to the present day, with contributions of such people as Kolmogorov, Smale, and many more. Incidentally, an essential step in the mathematical development of dynamical systems theory was the creation of ergodic theory, for which ideas originating in physics were important.

The time evolution of chaotic systems is typically complicated and irregular-looking. Indeed, a regular time evolution is predictable and therefore not chaotic. When the interest for complicated and irregular time evolutions developed among physicists in the 1970s, to give what is now called the theory of chaos, all kinds of new scientific tools existed that had not been available to Poincaré. One such tool is the electronic computer, which allowed Lorenz [3] to compute in 1963 a chaotic time evolution, and visualize it in the form of what we now call a strange attractor. Other tools were mathematical, like ergodic theory. Finally, there were new experimental tools permitting for instance a detailed study of the onset of hydrodynamic turbulence. It is this experimental study that showed that hydrodynamical tubulence is chaotic, as we would now say, corresponding to the claim of Ruelle and Takens [4], in 1971, that it is described by strange attractors.

It must be said at this point that, however insightful and brilliant, the physical ideas of Poincaré on chaos were at the level of scientific philosophy. Because new tools are available, our present understanding of chaos in physics is at the level of quantitative science. And attempted applications of chaos to biology, economics, etc., which are now at the level of scientific philosophy, may later reach the level of quantitative science.

I have stressed up to now the idea that the study of chaos consisted in applying available mathematics to the understanding of natural phenomena. But there have also been important influences in the opposite direction: from the physics of chaos to mathematics. New mathematical tools were created because they were needed to understand chaos, or new ideas coming from chaos were found to have surprisingly important mathematical content, and were then developed for their own sake. In fact, the limit between physical and mathematical ideas is often unclear and the same people have been seen sometimes functioning as physicists, sometimes as mathematicians.

In what follows I shall give a list of some mathematical developments where the ideas of chaos played a significant role. The list does not attempt at being exhaustive, and the scope and nature of the different items is rather different.