1.6. Analytic ODE and Their Solutions 31 are closely related and equally as important, such as having a pos- itive “Lyapounov Exponent”, the existence of so-called “strange at- tractors”, “homoclinic tangles”, and “horseshoe maps”. These latter concepts are quite technical, and we will not attempt to define or describe them here (but see the references below). In recent years chaos theory and the related areas of dynamical systems and nonlinear science, have been the focus of enormous ex- citement and enthusiasm, giving rise to a large and still rapidly grow- ing literature consisting of literally hundreds of books, some technical and specialized and others directed at the lay public. Two of the best nontechnical expositions are David Ruelle’s “Chance and Chaos” and James Gleick’s “Chaos: Making a New Science”. For an excellent introduction at a more mathematically sophisticated level see the col- lection of articles in “Chaos and Fractals: The Mathematics Behind the Computer Graphics”, edited by Robert Devaney and Linda Keen. Other technical treatment we can recommend are Steven Strogatz’ “Nonlinear Dynamics and Chaos”, Hubbard and West’s “Differential Equations: A Dynamical Systems Approach”, Robert Devaney’s “A First Course in Chaotic Dynamical Systems”, and Tom Mullin’s “The Nature of Chaos”. 1.6. Analytic ODE and Their Solutions Until now we have worked entirely in the real domain, but we can equally well consider complex-valued differential equations. Of course we should be precise about how to interpret this concept, and in fact there are several different interpretations with different levels of interest and sophistication. Using the most superficial generalization, it seems as if there is nothing really new—since we can identify C with R2, a smooth vector field on Cn is just a smooth vector field on R2n. But even here there are some advantages in using a complex approach. Recall the important two-dimensional real linear system dx dt = −y, dy dt = x, mentioned earlier, that arises when we reduce the harmonic oscillator equation d2x dt2 = −x to a first-order system. We saw that if we regard R2 as C and write z = x + iy as usual, then our system becomes dz dt = iz, so the solution with initial condition
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