Birth of the chaos theory

Poincaré and phase space

With the work of Laplace, the past and the future of the solar system could be calculated and the precision of this calculation depended on the capacity to know the initial conditions of the system, a real challenge for “geometricians,” as alluded to by d’Holbach and Le Verrier. Henri Poincaré developed another point of view, as follows: in order to study the evolution of a physical system over time, one has to construct a model based on a choice of laws of physics and to list the necessary and sufficient parameters that characterize the system (differential equations are often in the model). One can define the state of the system at a given moment, and the set of these system states is named phase space.

The phenomenon of sensitivity to initial conditions was discovered by Poincaré in his study of the the n-body problem, then by Jacques Hadamard using a mathematical model named geodesic flow, on a surface with a nonpositive curvature, called Hadamard’s billards. A century after Laplace, Poincaré indicated that randomness and determinism become somewhat compatible because of the long-erm unpredictability.

A very small cause, which eludes us, determines a considerable effect that we cannot fail to see, and so we say that this effect Is due to chance. If we knew exactly the laws of nature and the state of the universe at the initial moment, we could accurately predict the state of the same universe at a subsequent moment. But even If the natural laws no longer held any secrets for us, we could still only know the state approximately. If this enables us to predict the succeeding state to the same approximation, that is all we require, and we say that the phenomenon has been predicted, that It Is governed by laws. But this is not always so, and small differences in the initial conditions may generate very large differences in the final phenomena. A small error in the former will lead to an enormous error In the latter. Prediction then becomes impossible, and we have a random phenomenon.

This was the birth of chaos theory.

Rebirth of chaos theory

Edward Lorenz, from the Massachusetts Institute of Technology (MIT) is the official discoverer of chaos theory. He first observed the phenomenon as early as 1961 and, as a matter of irony, he discovered by chance what would be called later the chaos theory, in 1963, while making calculations with uncontrolled approximations aiming at predicting the weather. The anecdote is of interest: making the same calculation rounding with 3-digit rather than 6-digit numbers did not provide the same solutions; indeed, in nonlinear systems, multiplications during iterative processes amplify differences in an exponential manner. By the way, this occurs when using computers, due to the limitation of these machines which truncate numbers, and therefore the accuracy of calculations.

Lorenz considered, as did many mathematicians of his time, that a small variation at the start of a calculation would Induce a small difference In the result, of the order of magnitude of the initial variation. This was obviously not the case, and all scientists are now familiar with this fact. In order to explain how important sensitivity the to initial conditions was, Philip Merilees, the meteorologist who organized the 1972 conference session where Lorenz presented his result, chose himself the title of Lorenz’s talk, a title that became famous: “Predictability: does the flap of a butterfly’s wing in Brazil set off a tornado in Texas?”  This title has been cited and modified in many articles, as humorously reviewed by Nicolas Witkowski. Lorenz had rediscovered the chaotic behavior of a nonlinear system, that of the weather, but the term chaos theory was only later given to the phenomenon by the mathematician James A. Yorke, in 1975. Lorenz also gave a graphic description of his findings using his computer. The figure that appeared was his second discovery: the attractors.

FUNDAMENTAL CONCEPTS OF KNOT THEORY

A knot, succinctly, is an entwined circle. However, throughout this book we shall think of a knot as an entwined polygon in 3-dimensional space. The reason for this is that it allows us, with recourse to combinatorial topology, 1 to exclude wild knots. Close to the point P, in a sense we may take this to be a “limit” point, the knot starts to cluster together in a concertina fashion. Therefore, in the vicinity of such a point particular care needs to be taken with the nature of the knots. We shall not in this exposition apply or work within the constraints of such (wild) knots. In fact, since wild knots axe not that common, this will be the only reference to these kind of knots. Therefore, in order to avoid the above peculiarity, we shall assume, without exception, everything that follows is considered from the standpoint of combinatorial topology. As mentioned in the preface, our intention is to avoid as ,much as possible mathematical argot and to concentrate on the substance and application of knot theory. Infrequently, as above, it will be necessary, in order to underpin an assumption, to introduce such a piece of mathematical argot. Again, as mentioned in the preface, knowledge of such concepts will not usually be required to be able to understand what follows.

Definition. On a given knot K we may perform the following four operations.

(1) We may divide an edge, AB, in space of K into two edges, AC, CB, by placing a point C on the edge AB.

(1)’ [The converse of (1)] If AC and CB are two adjacent edges of K such that if C is erased AB becomes a straight line, then we may remove the point C.

(2) Suppose C is a point in space that does not lie on K. If the triangle ABC, formed by AB and C, does not intersect K, with the exception of the edge AB, then we may remove AB and add the two edges AC and CB.

(2)’ [The converse of (2)] If there exists in space a triangle ABC that contains two adjacent edges AC and CB of K, and this triangle does not intersect K, except at the edges AC and CB, then we may delete the two edges AC, CB and add the edge AB.

These four operations (1), (1)’, (2) and (2)’ are called the elementary knot moves.