Historical Prologue xix The application of these ideas to physical systems really never stopped. One of these applications, which has been studied since earliest times, is the descrip- tion and determination of the motion of the planets and stars. The study of the mathematical model for such motion is called celestial mechanics, and involves a finite number of bodies moving under the effects of gravitational attraction given by the Newtonian laws. Birkhoff, Siegel, Kolmogorov, Arnold, Moser, Herman, and many others investigated the ideas of stability and found complicated behavior for systems arising in celestial mechanics and other such physical systems, which could be described by what are called Hamiltonian differential equations. (These equations preserve energy and can be expressed in terms of partial derivatives of the energy function.) K. Sitnikov in [Sit60] introduced a situation in which three masses interacting by Newtonian attraction can exhibit chaotic oscillations. Later, Alekseev showed that this could be understood in terms of a “Smale horseshoe”, [Ale68a], [Ale68b], and [Ale69]. The book by Moser, [Mos73], made this result available to many researchers and did much to further the applications of horse- shoes to other physical situations. In the 1971 paper [Rue71] introducing strange attractors, Ruelle and Takens indicated how the ideas in nonlinear dynamics could be used to explain how turbulence developed in fluid flow. Further connections were made to physics, including the periodic doubling route to chaos discovered by Feigenbaum, [Fei78], and independently by P. Coullet and C. Tresser, [Cou78]. Relating to a completely different physical situation, starting with the work of Belousov and Zhabotinsky in the 1950s, certain mathematical models of chemical reactions that exhibit chaotic behavior were discovered. They discovered some systems of differential equations that not only did not tend to an equilibrium, but also did not even exhibit predictable oscillations. Eventually, this bizarre situation was understood in terms of chaos and strange attractors. In the early 1920s, A.J. Lotka and V. Volterra independently showed how dif- ferential equations could be used to model the interaction of two populations of species, [Lot25] and [Vol31]. In the early 1970s, May showed how chaotic out- comes could arise in population dynamics. In the monograph [May75], he showed how simple nonlinear models could provide “mathematical metaphors for broad classes of phenomena.” Starting in the 1970s, applications of nonlinear dynamics to mathematical models in biology have become widespread. The undergraduate books by Murray [Mur89] and Taubes [Tau01] afford good introductions to bio- logical situations in which both oscillatory and chaotic differential equations arise. The books by Kaplan and Glass [Kap95] and Strogatz [Str94] include a large number of other applications. Another phenomenon that has had a great impact on the study of nonlinear differential equations is the use of computers to find numerical solutions. There has certainly been much work done on deriving the most eﬃcient algorithms for carrying out this study. Although we do discuss some of the simplest of these, our focus is more on the use of computer simulations to find the properties of solutions. E. Lorenz made an important contribution in 1963 when he used a computer to study nonlinear equations motivated by the turbulence of motion of the atmosphere. He discovered that a small change in initial conditions leads to very different outcomes in a relatively short time this property is called sensitive

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