《14-The-Lorenz-System_2016_Differential-Equations-Dynamical-Systems-and-an-Introduction-to-Chaos》.pdf

14 The Lorenz System So far, in all of the differential equations we have studied, we have not encoun- tered any “chaos.” The reason is simple: The linear systems of the first few chapters always have straightforward, predictable behavior. (OK, we may see solutions wrap densely around a torus as in the oscillators of Chapter 6, but this is not chaos.) Also, for the nonlinear planar systems of the last few chap- ´ ters, the Poincare–Bendixson Theorem completely eliminates any possibility of chaotic behavior. So, to find chaotic behavior, we need to look at nonlinear, higher-dimensional systems. In this chapter we investigate the system that is, without doubt, the most famous of all chaotic differential equations, the Lorenz system from meteorol- ogy. First formulated in 1963 by E. N. Lorenz as a vastly oversimplified model of atmospheric convection, this system possesses what has come to be known as a strange attractor. Before the Lorenz model started making headlines, the only types of stable attractors known in differential equations were equilibria and closed orbits. The Lorenz system truly opened up new horizons in all areas of science and engineering, as many of the phenomena present in the Lorenz system have later been found in all of the areas we have previously investigated (biology, circuit theory, mechanics, and elsewhere). In the ensuing nearly 50 years, much progress has been made in the study of chaotic systems. Be forewarned, however, that the analysis of the chaotic behavior of particular systems, such as the Lorenz system, is usually extremely difficult. Most of the chaotic behavior that is readily understandable arises from geometric models for particular differential equations, rather than from Differential Equations, Dynamical Systems, and an Introduction to Chaos. DOI: 10.1016/B978-0-12-382010-5.00014-2 c 2013 Elsevier Inc. All right