xviii Historical Prologue continued the work of Poincar´ e and found many different types of long-term limiting behavior, including the α- and ω-limit sets introduced in Sections 4.1 and 11.1. His work resulted in the book [Bir27] from which the term “dynamical systems” comes. During the first half of the twentieth century, much work was carried out on nonlinear oscillators, that is, equations modeling a collection of springs (or other physical forces such as electrical forces) for which the restoring force depends non- linearly on the displacement from equilibrium. The stability of fixed points was studied by several people including Lyapunov. (See Sections 4.5 and 5.3.) The ex- istence of a periodic orbit for certain self-excited systems was discovered by Van der Pol. (See Section 6.3.) Andronov and Pontryagin showed that a system of differen- tial equations was structurally stable near an attracting fixed point, [And37] (i.e., the solutions for a small perturbation of the differential equation could be matched with the solutions for the original equations). Other people carried out research on nonlinear differential equations, including Bendixson, Cartwright, Bogoliubov, Krylov, Littlewood, Levinson, and Lefschetz. The types of solutions that could be analyzed were the ones which settled down to either (1) an equilibrium state (no motion), (2) periodic motion (such as the first approximations of the motion of the planets), or (3) quasiperiodic solutions which are combinations of several periodic terms with incommensurate frequencies. See Section 2.2.4. By 1950, Cartwright, Littlewood, and Levinson showed that a certain forced nonlinear oscillator had in- finitely many different periods that is, there were infinitely many different initial conditions for the same system of equations, each of which resulted in periodic mo- tion in which the period was a multiple of the forcing frequency, but different initial conditions had different periods. This example contained a type of complexity not previously seen. In the 1960s, Stephen Smale returned to using the topological and geometric perspective initiated by Poincar´ e to understand the properties of differential equa- tions. He wrote a very influential survey article [Sma67] in 1967. In particular, Smale’s “horseshoe” put the results of Cartwright, Littlewood, and Levinson in a general framework and extended their results to show that they were what was later called chaotic. A group of mathematicians worked in the United States and Europe to flesh out his ideas. At the same time, there was a group of mathematicians in Moscow lead by Anosov and Sinai investigating similar ideas. (Anosov generalized the work of Hadamard to geodesics on negatively curved manifolds with variable curvature.) The word “chaos” itself was introduced by T.Y. Li and J. Yorke in 1975 to designate systems that have aperiodic behavior more complicated than equilibrium, periodic, or quasiperiodic motion. (See [Li,75].) A related concept introduced by Ruelle and Takens was a strange attractor. It emphasized more the complicated geometry or topology of the attractor in phase space, than the com- plicated nature of the motion itself. See [Rue71]. The theoretical work by these mathematicians supplied many of the ideas and approaches that were later used in more applied situations in physics, celestial mechanics, chemistry, biology, and other fields.
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