As we near the end of our story, the reader will now appreciate that there are many steps in the process of reducing the Navier–Stokes equations to a low-dimensional model for the dynamics of coherent structures. Some of these involve purely mathematical issues, but most require an interplay among physical considerations, judgement, and mathematical tractibility. While our development of a general strategy for constructing low-dimensional models has been based on theoretical developments such as the POD and dynamical systems methods, the general theory is still sketchy and, in specific applications, many details remain unresolved.

The mathematical techniques we have drawn on lie primarily in probability and dynamical systems theory. In this closing chapter we review some aspects of the reduction process and attempt to put them into context. Some prospects for rigor in the reduction process are also mentioned. This is by no means a comprehensive review or discussion of future work; instead, we have chosen to highlight a few recent applications of dynamical and probabilistic ideas to illustrate lines along which a general theory might be further developed.

We start by discussing some desirable properties for low-dimensional models, and criteria by which they might be judged. We then outline in Section 12.2 an a priori short-term tracking estimate which describes, in a probabilistic context, how rapidly typical solutions of the model equations are expected to diverge from those of the full Navier–Stokes equations restricted to the model domain.

In the preceding eight chapters we have developed our basic tools and techniques. In this chapter and the next we shall illustrate their use in the derivation and analysis of low-dimensional models of the wall region of a turbulent boundary layer. First, the Navier–Stokes equations are rewritten in a form that highlights the dynamics of the coherent structures (CS) and their interaction with the mean flow. To do this, both the neglected (high) wavenumber modes and the mean flow must be modelled, unlike a large eddy simulation (LES), in which only the neglected high modes are modelled. Second, using physical considerations, we select a family of empirical subspaces upon which to project the equations. Galerkin projection is then carried out. In doing this, we restrict ourselves to a small physical flow domain, and so the response of the (quasi)local mean flow to the coherent structures must also be modelled. This chapter describes each step of the process in some detail, drawing on material presented in Chapters 2, 3, and 4. After deriving the family of low-dimensional models, in the last three sections we discuss in more depth the validity of assumptions used in their derivation. In Chapter 10 we shall describe the use of the dynamical systems ideas, presented in Chapters 5 through 8, in the analysis of these models, and interpret their solutions in terms of the dynamical behavior of the fluid flow.

The irregular and unpredictable time evolution of many nonlinear systems has been dubbed ‘chaos.’ It occurs in mechanical oscillators such as pendula or vibrating objects, in rotating or heated fluids, in laser cavities, and in some chemical reactions. Its central characteristic is that the system does not repeat its past behavior (even approximately). Periodic and chaotic behavior are contrasted in Figure 1.1. Yet, despite their lack of regularity, chaotic dynamical systems follow deterministic equations such as those derived from Newton's second law.

The unique character of chaotic dynamics may be seen most clearly by imagining the system to be started twice, but from slightly different initial conditions. We can think of this small initial difference as resulting from measurement error, for example. For nonchaotic systems this uncertainty leads only to an error in prediction that grows linearly with time. For chaotic systems, on the other hand, the error grows exponentially in time, so that the state of the system is essentially unknown after a very short time. This phenomenon, which occurs only when the governing equations are nonlinear, is known as sensitivity to initial conditions. Henri Poincare (1854–1912), a prominent mathematician and theoretical astronomer who studied dynamical systems, was the first to recognize this phenomenon. He described it as follows: ‘…it may happen that small differences in the initial conditions produce very great ones in the final phenomena. A small error in the former will produce an enormous error in the latter.

The remarkable fact that determinism does not imply either regular behavior or predictability has had a major impact on many fields of science, engineering, and mathematics. The discovery of chaos changes our understanding of the foundations of physics, and has many practical applications as well. This subject sheds new light on the workings of lasers, fluids, mechanical structures, chemical reactions, earthquakes, neural networks, and biological rhythms.

Interest in chaos (or more generally, nonlinear dynamics) grew rapidly after 1963, when Lorenz published his numerical work on a simplified model of convection and discussed its implications for weather prediction. The research literature has exploded, and many books on chaotic dynamics have appeared. The first edition of this book was the first work aimed at a level at once more accessible than graduate texts, and yet more suitable for nonspecialists, including undergraduates in science, than various popular books on chaos. It has been used by scientists and students wishing to have a true introduction, and as a text or text supplement for courses in mathematics, physics, and engineering. These include short courses on chaos, classical mechanics, or modern physics. Some of the material can be included in an introductory course in physics, engineering, or differential equations.

Chaotic dynamics: an introduction introduces chaotic dynamics through the study of the driven pendulum, a simple system whose nonlinear properties are often ignored in teaching mathematics and physics.

In this chapter we discuss three mathematical constructs that are generally useful in the study of dynamical systems: phase space, the Poincaré section, and power spectra. Phase space is the mathematical space of the dynamical variables of a system. The Poincaré section is a ‘snapshot’ of the motion in the phase space, taken at regular time intervals. The power spectrum is computed using Fourier analysis to display the frequency composition of the time variation of the dynamical variables.

Phase space

The phase space of a dynamical system is a mathematical space with orthogonal coordinate directions representing each of the variables needed to specify the instantaneous state of the system. For example, the state of a particle moving in one dimension is specified by its position (x) and velocity (v); hence its phase space is a plane. On the other hand, a particle moving in three dimensions would have a six-dimensional phase space with three position and three velocity directions. A phase space may be constructed in several different ways. For example, momenta can be used instead of velocities.

Let us focus the discussion on the pendulum and begin with the familiar simple pendulum in the small amplitude approximation where the restoring term, sinθ, is taken as θ. (Recall that the equations are written in dimensionless form for simplicity, with time measured in units of the inverse of the natural frequency.)

This appendix provides listings which may be used in their present or modified versions for exercises in the text. The listings are in the language True BASICtm and their implementation requires the use of the True BASICtm Language System together with the True BASICtm toolkits for scientific and 3D graphics. These are available from True BASIC Inc., 12 Commerce Ave., West Lebanon, NH 03784, USA. The language system may be used to run or modify the programs. Hardware requirements include an IBM compatible machine with 512K of memory, and a Hercules, CGA, EGA, or VGA graphics adaptor. A math coprocessor is highly desirable. These programs and some others are also available bound in an executable, menu-driven, self-contained software package from one of the authors (GLB) as indicated under ‘diskette order information’.

The programs are of three types: those which solve the differential equations of the pendulum, those which iterate discrete maps, and those which analyse experimental data. (The last group requires the data file found on the above mentioned software package.) Because the Runge–Kutta algorithm is complex, the computer processing time for differential equations solutions is much longer than for the iteration of maps. The reader is warned that a few of the text diagrams took many hours to produce even with a 486 IBM compatible machine.

The computer exercises of Chapter 2 are based upon the programs, PENDULUM, POINCARE, and EXPFFT. PENDULUM, Listing 1, provides a two-dimensional phase plane representation of the pendulum. The differential equation is that of the nonlinear, damped, driven pendulum.