The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors

Suitable for upper-level undergraduates and graduate students, this text presents an extensive analysis of Lorenz equations that comprises the most widely studied parameter ranges. Starting with general remarks and descriptions of those simple properties of the equations that can be deduced mathematically, the text examines the birfurcation associated with a homoclinic origin, describes the parameter range possessing a well-understood strange attractor in a whole interval of parameter values, and explores the results of some simple numerical experiments within a parameter range of period doubling. Using a combination of different numerical techniques and a careful theoretical analysis of the changes in the behavior of the unstable manifold of the origin, the text demonstrates a change in behavior from strange attractor to period doublings, thereby exhibiting new properties of the Lorenz equations.
Successive chapters justify earlier methods, with descriptions of periodic orbits and other trajectories using sequences of symbols; outline an analysis of the observable behavior when a parameter becomes enlarged; and conclude with a brief summary of the preceding material, a discussion of alternative approaches to Lorenz equations, and extensive appendixes.