The Butterfly Effect Introduction

THE BUTTERFLY EFFECT Étienne GHYS CNRS-UMPA ENS Lyon etienne.ghys@ens-lyon.fr It is very unusual for a mathematical idea to disseminate into the society at large. An interesting example is chaos theory, popularized by Lorenz’s butterfly effect: “does the flap of a butterfly’s wings in Brazil set off a tornado in Texas?” A tiny cause can generate big consequences! Can one adequately summarize chaos theory in such a simple minded way? Are mathematicians responsible for the inadequate transmission of their theories outside of their own community? What is the precise message that Lorenz wanted to convey? Some of the main characters of the history of chaos were indeed concerned with the problem of communicating their ideas to other scientists or non-scientists. I’ll try to discuss their successes and failures. The education of future mathematicians should include specific training to teach them how to explain mathematics outside their community. This is more and more necessary due to the increasing complexity of mathematics. A necessity and a challenge! INTRODUCTION In 1972, the meteorologist Edward Lorenz gave a talk at the 139th meeting of the American Association for the Advancement of Science entitled “Does the flap of a butterfly’s wings in Brazil set off a tornado in Texas?”. Forty years later, a google search “butterfly effect” generates ten million answers. Surprisingly most answers are not related to mathematics or physics and one can find the most improbable websites related to movies, music, popular books, video games, religion, philosophy and even Marxism! It is very unusual that a mathematical idea can disseminate into the general society. One could mention Thom's catastrophe theory in the 1970's, or Mandelbrot's fractals in the 1980's, but these theories remained confined to the scientifically oriented population. On the contrary, chaos theory, often presented through the butterfly effect, did penetrate the nonscientific population at a very large scale. Unfortunately, this wide diffusion was accompanied with an oversimplification of the main original ideas and one has to admit that the transmission procedure from scientists to nonscientists was a failure. As an example, the successful book The butterfly effect by Andy Andrews “reveals the secret of how you can live a life of permanent purpose” and “shows how your everyday actions can make a difference for generations to come” which is not exactly the message of the founding fathers of chaos theory! In Spielberg's movie Jurassic Park, Jeff Goldblum introduces himself as a “chaotician” and tries (unsuccessfully) to explain the butterfly effect and unpredictability to Ghys Abcde+3 ICME-12, 2012 the charming Laura Dern; the message is scientifically more accurate but misses the main point. If chaos theory only claimed that the future is unpredictable, would it deserve the name “theory”? After all, it is well known that “Prediction is very difficult, especially the future!” 1 . A scientific theory cannot be limited to negative statements and one would be disappointed if Lorenz's message only contained this well known fact. The purpose of this talk is twofold. On the one hand, I would like to give a very elementary presentation of chaos theory, as a mathematical theory, and to give some general overview on the current research activity in this domain with an emphasis on the role of the so-called physical measures. On the other hand, I would like to analyze the historical process of the development of the theory, its successes and failures, focusing in particular on the transmission of ideas between mathematics and physics, or from Science to the general public. This case study might give us some hints to improve the communication of mathematical ideas outside mathematics or scientific circles. The gap between mathematicians and the general population has never been so wide. This may be due to the increasing complexity of mathematics or to the decreasing interest of the population for Science. I believe that the mathematical community has the responsibility of building bridges. A BRIEF HISTORY OF CHAOS FROM NEWTON TO LORENZ Determinism One of the main pillars of Science is determinism: the possibility of prediction. This is of course not due to a single person but one should probably emphasize the fundamental role of Newton. As he was laying the foundations of differential calculus and unraveling the laws of mechanics, he was offering by the same token a tool enabling predictions. Given a mechanical system, be it the solar system or the collection of molecules in my room, one can write down a differential equation governing the motion. If one knows the present position and velocity of the system, one should simply solve a differential equation in order to determine the future. Of course, solving a differential equation is not always a simple matter but this implies at least the principle of determinism: the present situation determines the future. Laplace summarized this wonderfully in his 1814 “Essai philosophique sur les probabilités” (Laplace, 1814): “We ought then to consider the present state of the universe as the effect of its previous state and as the cause of that which is to follow. An intelligence that, at a given instant, could comprehend all the forces by which nature is animated and the respective situation of the beings that make it up, if moreover it were vast enough to submit these data to analysis, would encompass in the same formula the movements of the greatest bodies of the universe and those of the lightest atoms. For such an intelligence nothing would be uncertain, and the future, like the past, would be open to its eyes.” 1 See www.peterpatau.com/2006/12/bohr-leads-berra-but-yogi-closing-gap.html for an interesting discussion of the origin of this quotation. Ghys ICME-12, 2012 abcde+2 The fact that this quotation comes from a book on probability theory shows that Laplace's view on determinism was far from naïve (Kahane, 2008). We lack the “vast intelligence” and we are forced to use probabilities in order to understand dynamical systems. Sensitivity to initial conditions In his little book “Matter and Motion” published in 1876, Maxwell insists on the sensitivity to initial conditions in physical phenomena (Maxwell, 1876): “There is a maxim which is often quoted, that ‘The same causes will always produce the same effects.’ To make this maxim intelligible we must define what we mean by the same causes and the same effects, since it is manifest that no event ever happens more that once, so that the causes and effects cannot be the same in all respects. [...] There is another maxim which must not be confounded with that quoted at the beginning of this article, which asserts ‘That like causes produce like effects’. This is only true when small variations in the initial circumstances produce only small variations in the final state of the system. In a great many physical phenomena this condition is satisfied; but there are other cases in which a small initial variation may produce a great change in the final state of the system, as when the displacement of the ‘points’ causes a railway train to run into another instead of keeping its proper course.” Notice that Maxwell seems to believe that “in great many cases” there is no sensitivity to initial conditions. The question of the frequency of chaos in nature is still at the heart of current research. Note also that Maxwell did not really describe what we would call chaos today. Indeed, if one drops a rock from the top of a mountain, it is clear that the valley where it will end its course can be sensitive to a small variation of the initial position but it is equally clear that the motion cannot be called “chaotic” in any sense of the word: the rock simply goes downwards and eventually stops. Fear for chaos It is usually asserted that chaos was “discovered” by Poincaré in his famous memoir on the 3-body problem (Poincaré, 1890). His role is without doubt very important, but maybe not as much as is often claimed. He was not the first to discover sensitivity to initial conditions. However, he certainly realized that some mechanical motions are very intricate, in a way that Maxwell had not imagined. Nevertheless chaos theory cannot be limited to the statement that the dynamics is complicated: any reasonable theory must provide methods allowing some kind of understanding. The following famous quotation of Poincaré illustrates his despair when confronted by the complication of dynamics (Poincaré, 1890): “When we try to represent the figure formed by these two curves and their infinitely many intersections, each corresponding to a doubly asymptotic solution, these intersections form a type of trellis, tissue, or grid with infinitely fine mesh. Neither of the two curves must ever cut across itself again, but it must bend back upon itself in a very complex manner in order to cut across all of the meshes in the grid an infinite number of times. The complexity of this figure is striking, and I shall not even try to draw it. Nothing is more suitable for providing us with an idea of the complex nature of the three-body problem, and of all the problems of dynamics in general […].” Ghys Abcde+3 ICME-12, 2012 One should mention that ten years earlier Poincaré had written a fundamental memoir “Sur les courbes définies par des équations différentielles” laying the foundations of the qualitative theory of dynamical systems (Poincaré, 1881). In this paper, he had analyzed in great detail the behavior of the trajectories of a vector field in the plane, i.e. of the solutions of an ordinary differential equation in dimension 2. One of his main results — the Poincaré-Bendixson theorem — implied that such trajectories are very well behaved and converge to an equilibrium point or to a periodic trajectory (or to a so-called “graphic”): nothing chaotic in dimension 2! In his 1890 paper, he was dealing with differential equations in dimension 3 and he must have been puzzled — and scared — when he realized the complexity of the picture. Taming chaos In 1898 Hadamard wrote a fundamental paper on the dynamical behavior of geodesics on