xx Historical Prologue dependence on initial conditions or, in more common language, the butterfly effect. Lorenz used the latter term because he interpreted the phenomenon to mean that a butterfly flapping its wings in Australia today could affect the weather in the United States a month later. We describe more of his work in Chapter 7. It was not until the 1970s that Lorenz’s work became known to the more theoretical mathematical community. Since that time, much effort has gone into showing that Lorenz’s basic ideas about these equations were correct. Recently, Warwick Tucker has shown, using a computer-assisted proof, that this system not only has sensitive dependence on initial conditions, but also has what is called a “chaotic attractor”. (See Chapter 7.) About the same time as Lorenz, Ueda discovered that a periodically forced Van der Pol system (or other nonlinear oscillator) has what is now called a chaotic attractor. Systems of this type are also discussed in Chapter 7. (For a later publication by Ueda, see also [Ued92].) Starting about 1970 and still continuing, there have been many other numer- ical studies of nonlinear equations using computers. Some of these studies were introduced as simple examples of certain phenomena. (See the discussion of the R¨ ossler Attractor given in Section 7.4.) Others were models for specific situations in science, engineering, or other fields in which nonlinear differential equations are used for modeling. The book [Enn97] by Enns and McGuire presents many com- puter programs for investigation of nonlinear functions and differential equations that arise in physics and other scientific disciplines. In sum, the last 40 years of the twentieth century saw the growing importance of nonlinearity in describing physical situations. Many of the ideas initiated by Poincar´ e a century ago are now much better understood in terms of the mathematics involved and the way in which they can be applied. One of the main contributions of the modern theory of dynamical systems to these applied fields has been the idea that erratic and complicated behavior can result from simple situations. Just because the outcome is chaotic, the basic environment does not need to contain stochastic or random perturbations. The simple forces themselves can cause chaotic outcomes. There are three books of a nontechnical nature that discuss the history of the development of “chaos theory”: the best seller Chaos: Making a New Science by James Gleick [Gle87], Does God Play Dice?, The Mathematics of Chaos by Ian Stewart [Ste89], and Celestial Encounters by Florin Diacu and Philip Holmes [Dia96]. Stewart’s book puts a greater emphasis on the role of mathematicians in the development of the subject, while Gleick’s book stresses the work of researchers making the connections with applications. Thus, the perspective of Stewart’s book is closer to the one of this book, but Gleick’s book is accessible to a broader audience and is more popular. The book by Diacu and Holmes has a good treatment of Poincar´ e’s contribution and the developments in celestial mechanics up to today.

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