This volume is intended as an introductory textbook on the theory of chaos and is addressed to physicists and engineers who wish to be acquainted with this new and exciting science associated with non-linear deterministic systems. Mathe­matics are, of course, a pre-requisite tool in such a study, and we did not shirk the task of discussing complex mathematical issues while preferring, in general, through inclination and training to focus the attention of the reader on a physical understanding of phenomena.

Of course, we have to admit that a number of distinguished textbooks incorporat­ing chaos as a primary or secondary subject have appeared in the last few years. In particular, we draw the reader's attention to the treatises of Moon, Thompson & Stewart, Kreuzer, Berge, Pomeau & Vidal, Schuster and Nicolis & Prigogine, to mention only six recent texts that are not primarily mathematical. The excel­lent book of F. C. Moon - which has been followed by an expanded exposition - is mainly concerned with experimental techniques and offers interesting insights into the chaotic response of mechanical systems. The book of J. M. T. Thompson and H. B. Stewart is, as to be expected, brilliantly written with a broad outlook on the subject mainly directed at mechanical systems and structures but also including asides into subjects like the Ray 1 eig h- B e n ar d convection and the Lorenz system of equations. E. Kreuzer's compact book is addressed to non-linear oscillations and mechanical systems; it is based on a sound mathematical foundation and displays a deep knowledge of the dynamic response of non-linear systems. The book by P. Berge, Y. Pomeau and C. Vidal contains inter alia an extensive investigation of the transitions to chaos and their experimental verification on the example of the Rayleigh-Benard convection. H. G. Schuster's monograph is an outstanding exposition of chaotic manifestations in non-linear physical systems written with precision and economy for theoretical physicists of advanced training. The last of the aforementioned books, that of G. Nicolis and I. Prigogine, offers a profound and comprehensible study of the dynamics of non-linear systems far from thermo­dynamic equilibrium. In addition, we should like to mention the monographs of Hermann Haken on synergetics which - on a mathematically demanding basis - deal with the systematic investigation of structure formation in open dissipative systems.

In this array of textbooks, we now submit our text in the hope that aspiring physicists and engineers will find it of value in their efforts to understand and apply the complex theory of chaos. We have endeavoured not only to expound the general theory as far as possible, but also to include a broad range of physical subjects like fluid mechanics, Rayleigh-Bénard convection, biomechanics, astronomy, physical chemistry and other mechanical and electrical systems represented by the Duffing and van der Pol equations; see our descriptive account in Chapter 1.

Once chaotic manifestations were first perceived consciously in meteorology, the world of classical science as it existed 30 years ago seemed to fade away. Physicists and mechanicians, guided by the epoch-making writings of Kepler and Newton, were unconsciously influenced over centuries to such an extent that they were aware only of regular motions - whether linear or non-linear - and thus were not capable of perceiving irregular phenomena. As early as the turn of the cen­tury, Henri Poincaré had indeed drawn attention to the possibility of irregular behaviour in deterministic systems; in the absence of computers, however, this could not be registered directly. While recognising the great contributions of an Osborne Reynolds, we were thus not in a position to penetrate the mysteries of turbulence in a flow or in the atmosphere and in the oceans and hence to compre­hend irregularities in natural phenomena. For three centuries, research had been unconsciously directed at regularity.

However, in the early 1970s, an élite of researchers in Europe and the United States initiated a concentrated effort to clear a path of understanding through all these disorders. This group of adventurous and unconventional scientists included physicists, biologists, chemists and mathematicians, all attempting to seek links between different kinds of irregularities in animate and inanimate nature. These irregularities are to be found in the dynamics of our heartbeat and in explosive variations in certain wildlife populations as well as in the turbulence of a flow and the erratic motion of a meteor. Economists were prompted to investigate the theory of economic cycles. All these phenomena and a multitude of others, such as forks of lightning, were observed with curiosity and analysed.

At the same time, mathematicians such as Vladimir Igorevich Arnol'd made new fundamental contributions to our knowledge of local and global bifurcations in non-linear dynamics. Unavoidably, scientists re-discovered the pioneering work of Henri Poincaré. All these efforts could not have been conceived and realised without the revolution in science and engineering generated by the explosively growing availability and capacity of electronic computers which began a few years after World War II.

A decade later, in the mid-Seventies, the group of scientists working on çhaos had established itself as an exponentially growing co-fraternity which was re-shaping the concepts of modern science nolens volens. We have now reached the stage at which, in nearly every major university of scientific standing, researchers apply themselves to the manifestations of chaos, irrespective of their formal specialisa­tion. Indeed, at Los Alamos, a special centre for the study of non-linear systems was created and coordinates work on chaos and related manifestations.

Inevitably, the study of chaos has generated new advanced techniques of applying computers and the refined graphic facilities of modern hardware. In this way, we can view displays of delicate, highly imaginative textures which illustrate complex­ity in a formerly unexpected way. This new science involves the study of fractals, bifurcations, periodicities and intermittency. All these manifestations inspire us to a new understanding of the concept of motion. In all our observations of the world, we now continuously discover manifestations of chaos as, for example, in the rising and quivering column of cigarette smoke which suddenly breaks out into a wild disorder. Similar phenomena may be seen if we look at the complex oscillatory response of a flag fluttering or snapping back and forth in the wind. Observing a dripping tap, we note a transformation from a steady pattern to a random one.

Chaos, in fact, is noticed today - thanks to the revolutionary findings of Ed­ward Lorenz in 1963 who was the first of all modern scientists to comprehend chaotic evolutions and developed a simplified model of chaotic manifestation in the weather. Indeed, in this weather model, he noted the extreme sensitivity of the response arising from small changes in the initial conditions and mentioned the so-called butterfly effect first. But chaos is also contained latently in the response of an airplane in flight and arises inter alia through turbulent boundary layers and separation effects as well as other chaotic manifestations. If a dense stream of cars chokes a motorway, this is also an example of a chaotic response. Irrespective of the medium in which these chaotic outbursts take place, the behaviour obeys certain common general rules. To underline the broad panorama of chaotic man­ifestations, we refer in the last section of the book to chaotic phenomena in our solar system and discuss in particular the practically certain chaotic response of Hyperion, one of the outer moons of Saturn.

Our incidental remarks are intended to demonstrate that chaos poses problems in most realms of present-day research that do not fit into the traditional patterns of scientific thinking. In contrast, the imaginative study of chaos allows us to discover the universal characteristics of the response of complex non-linear systems. The first chaos researchers who initiated this discipline shared certain preoccupations. They were, for example, fascinated by patterns, especially those that emerge and repeat themselves at the same time on different scales. To the initiated researcher, odd questions like how long the jagged coast of Great Britain is became part of a fundamental enquiry. These early scientists had a talent for exploring complexity and studying jagged edges and sudden leaps. Inevitably, these apostles in chaos speculated about determinism and free will and about the nature of conscious intelligence.

The most profound thinkers of the new science asserted, and still assert today, that 20th century science will be remembered for three great scientific philosophical concepts: relativity, quantum mechanics and chaos. We go one step further and think that the exploration of chaos will determine the mainstream of scientific discovery in the 21st century and shape the evolution of physics, mechanics and also chemistry; naturally, this will also affect engineering. In this way, the preachers of the new science also believe that chaos erodes principles of Newton's physics. As one eminent physicist puts it: relativity eliminated the Newtonian illusion of absolute space and time; the quantum theory eliminated the Newtonian dream of a controllable procedure of measurement and, ultimately, chaos eliminated the

Laplacian phantasy (or pipedream) of deterministic predictability. Of these three scientific revolutions, the revolution in chaos applies to the whole universe as we comprehend and observe it and affects us through manifestations on a human scale.

For all the brilliant achievements of great physicists, we have to ask ourselves today how this edifice of physics could have evolved over so many years into a phantastic mission without providing means of answering some of the most fun­damental questions about nature. How does life begin and what is the mystery of turbulence? In a universe ruled by entropy and drawing inexorably towards greater and greater disorder, how does order establish itself? Moreover, objects of everyday life like fluids and non-linear mechanical systems came to be consid­ered as so ordinary that physicists could assume that they were well understood, at least by engineers. Until thirty years or so ago, the reason for our ignorance of such systems was caused by the absence of computer and graphic equipment. In the meantime, experimental and numerical computation has proved that our ignorance was profound.

As the revolution in chaos evolves in a cascade of new surprises, leading physicists find it quite natural to return without embarrassment to problems immediately related to our human nature. They are just as happy studying puffed clouds as galaxies. Leading journals publish articles on the strange dynamics of a ball bouncing on a table as well as on esoteric quantum physics. The simplest non­linear systems - and practically all systems in the real world are non-linear - seem to generate extremely difficult problems of predictability. At the same time, order may suddenly succeed chaos and vice versa. In most systems, we can observe the cohabitation of chaos and order. We are looking at a new kind of science with which we expect to bridge the gulf between knowledge of what one single item can do - e.g. a water molecule or one cell of a heart tissue - and what an assembly of millions of them can create in co-operation.

In the past, physicists traditionally deduced from complex results a hypothesis on complex causes. When they observed a random relationship between what goes into a system and what emerges from it, they assumed that they would have to introduce randomness into a realistic theory by adding the effects of noise or error. In contrast, the modern study of chaos was generated by the realisation in the early 1960s that quite simple, but still non-linear deterministic mathematical equations could generate responses every bit as surprising as the phantasmagoric turbulence of a waterfall. Small differences in initial conditions in the input are quickly magnified and produce large differences in the output. Thanks to Lorenz, we now assign this phenomenon to an extremely sensitive dependence on initial conditions. To take the classic example of weather, such effects may be expressed with a mocking smile as part of the butterfly effect. Thus, a butterfly stirring the air in Beijing might generate stormy weather the following month in New York.

We conclude our prologue by offering five very brief illustrations of the revolution that shaped the new theory of chaos.

Edward Lorenz was the first inspired scientist who, through his numerical computational experiments, perceived the essence of chaos. He worked as a meteorologist in 1961 at the MIT on an ungainly primitive computer known by thé impressive name of Royal McBee. By selecting a simple model of the Rayleigh-Bénard convection in a layer of air activated by a difference of temperature, he proposed to represent its complex response by an innocuous system of three non-linear ordinary differential equations. With this model, he hoped to study the problem of weather forecasting. He used a primitive printing device to produce a graph of the direction and speed of wind. In this way, he made the epoch-making discovery that the smallest difference in the initial conditions may produce diverging courses and patterns which drift apart until they bear no resemblance. This was the clue to the butterfly effect and the ultimate realisation of the unpredictability of weather over longer periods of time. Lorenz's findings were revolutionary and initiated the search for the nature of chaos. Lorenz also discovered a partial view of the corresponding strange attractor in the phase space, but did not name it in this way. Edward Lorenz's work and its influence on the sciences can be illustrated by a quotation from the New Testament:

'I6ov t\\mov Trvp t\\mt]v vXtju àvéïKTeLV

Kaivq ôta'd'ijKri, 'E-klotoXt) 'lotKujfiov Hi, 5

Behold, how great a matter a little fire kindleth.

General Epistle of James, Hi, 5

Mitchell Feigenbaum joined the Los Alamos National Laboratory in 1974. Feigenbaum brought to Los Alamos a conviction that the understanding of non-linear problems was practically non-existent. One of his first investi­gations concerned a most elementary logistic map with a quadratic form depending on a single parameter. He discovered that this primitive math­ematical system produced not only the expected steady answers but also, through a cascade of bifurcations, periodic and also doubly and higher pe­riodic responses leading to chaotic manifestations beyond certain values of the parameter. But this chaos was again interrupted by windows of regular response. Feigenbaum went a step further and proved the universality of his findings which apply with a surprising similarity to different and more complex mathematical expressions.

Hi. Let us next consider some of the finer points of the work of the highly gifted Benoit Mandelbrot with particular reference to the Mandelbrot set. This set is possibly the most complex object existing in mathematics. Benoit Mandelbrot initiated his search in 1979 on a generalisation of a certain class of shapes known as Julia sets. These were originally invented by two distinguished French mathematicians, Gaston Julia and Pierre Fatou, dur­ing World War I in France. Had these two researchers possessed computers and graphics, they would no doubt have been co-discoverers of chaos. The French mathematician Adrien Douady described the Julia sets as follows: "You obtain an incredible variety of Julia sets: some are a fatty cloud, oth­ers are a skinny bush of brambles, some look like meandering sparks which float in the air after a firework has gone off. One has the shape of a rabbit but lots of them have sea-horse tails".

In 1979, Mandelbrot ingeniously created one image in a complex plane that could serve as a dictionary or catalogue of all Julia sets. Scientists like Julia, Fatou, Hubbard, Barnsley and Mandelbrot invented novel rules on how to construct extravagant geometrical shapes that are now called fractals and are ruled by the principle of self-similarity. They produce strange ethereal pictures of great beauty which are seemingly dotted with separate, solitary islands, all giving the impression of a cosmos different from our own. The rules of its construction instruct us how to deduce from the image on one scale the corresponding pictogram on the next level of magnification as it appears to us through a microscope. The two fine mathematicians Douady and Hubbard applied an imaginative chain of subtle mathematics to prove that the aforementioned solitary molecules are, in fact, connected with the mainland by a delicate filigree. It may interest the reader to be informed that Peitgen and Richter, one a mathematician and the other a physicist, have devoted years to the propagation of Mandelbrot sets and their presen­tation as a new philosophy of art.

Another important research topic associated with the evolution of the chaos theory is that of attractors to which trajectories in the phase space are im­pelled. For example, a point or a limit cycle is such an important character­istic entity in phase space. This kind of problem - this time in connection with the mystery of turbulent flow - attracted the attention of two distin­guished Belgian mathematicians, David Ruelle and Floris Takens. Aston­ishingly enough, they were unaware of the revolutionary findings of Edward Lorenz in 1963 and his partial presentation of the then unnamed strange attractor. Ruelle and Takens wanted to check the assertion of Landau that turbulence is generated by an infinite sequence of Hopf bifurcations. Using a very demanding mathematical argumentation in association with Poincare sections, they proved that the assertion of Landau must be erroneous since his scheme does not produce stretching and folding of the trajectories in the phase space and does not reflect a high sensitivity to initial conditions, both of which are expected characteristic features of a turbulent transition. Moreover, the authors construed the first complete strange attractor and named it so appropriately.

Let us ultimately remark on another exciting topic, that of the dimen­sion of fractal shapes. In a sense, the degree of irregularity of such shapes corresponds to the ability of such a structure to take up space. Thus, a one- dimensional straight line does not fill any space at all. But a Koch curve - a kind of idealised snowflake - ruled by a fractal construction principle and possessing an infinite length but enclosing a finite area, has no integer dimension. In this way, its dimension exceeds the dimension one of a line but is less than the dimension two of an area. Mandelbrot determined this fractal dimension as 1.2618.

The concept of fractal dimension as expressed today by a number of alternative definitions has now taken hold of physics and non-linear system theory. Thus, we now know that the strange attractor of Lorenz has the dimension 2.06. Fractal non-integer dimensions now appertain to the theory of chaos. They have also taken hold of geo-physicists who have to describe the infinite complexity of the surface of our earth. This and many other developments brought the acceptance of this non-Euclidian and fractal geometry as a tool for solving problems.

Before closing the prologue, let us address our attention and veneration to one of the greatest scientists of this century, Henri Poincaré, who applied his intellectual power with phenomenal success to two topics: topology and dynamical systems. At the turn of the century, Poincaré was probably the last great mathematician to apply geometric imagination to the laws of motion of our physical world. He was the first to presage the mathematical concept of chaos. His writings and in particular his monumental work "Les méthodes nouvelles de la mécanique céleste" contain more than a profound hint at some kind of unpredictability, nearly as severe as that which Lorenz discovered. He was one of the most brilliant scientists of this century. Following his death, topology flourished, but the modern theory of dynamical systems languished. With this reverence to an inspiring inventive spirit, we conclude the prologue.