Diagnosing the structures of attractors

Abstract

We have seen briefly in the previous chapter how tori and so-called "strange attractors" in phase space may provide the underlying mathematical representations of the processes involved in the creation of turbulence. Stationary and temporally periodic solutions, which we have covered in depth in Chapters 2 to 14, provide the simplest examples of attractors--the phase space limits of model trajectories--and their properties are relatively easy to describe analytically. However, with the need to consider more complicated temporal flows when modeling the transition to turbulence, it becomes essential to define quantitative measures for characterizing and categorizing the various types of attractors. In this chapter we introduce the Lyapunov exponents (Oseledec, 1968), the Lyapunov dimension (Kaplan and Yorke, 1979), and the correlation dimension (Grassberger and Procaccia, 1983a,b). These quantities give information about the attractor structure, solution transitions, and predictability characteristics of nonlinear models. We investigate how these quantitative measures of attractor dynamics might be helpful in distinguishing different types of chaotic attractors, and thereby possibly distinguishing some of the different routes to chaos. Moreover, because the seventh modeling principle suggests that a model must have sufficient complexity to represent a phenomenon in adequate detail, these quantitative measures may help us evaluate the adequacy of a particular model. Specifically, we illustrate the utility of the Lyapunov and correlation dimensions in characterizing attractors in several low-order spectral models, including the Lorenz (1963) model, the seven-coefficient generalized Lorenz system of Chang and Shirer (1984), and a related eleven-coefficient model.