Introduction

Notation concerning Riemannian manifolds

We want now to replace the Euclidean space ℝn by a Riemannian manifold M and consider the possibility of having some kind of Sobolev inequalities. This brings in a whole new point of view. On Euclidean space, we could only discuss whether inequalities were true or not. In the more general setting of Riemannian manifolds, we can investigate the relations between various functional inequalities and the relations between these functional inequalities and the geometry of the manifold. We can search for necessary and/or sufficient conditions for a given Sobolev-type inequality to hold true. This leads to a better understanding of what information about M is encoded in various Sobolev-type inequalities.

Sobolev inequalities are useful when developing analysis on Riemannian manifolds, even more so than on Euclidean space, because other tools such as Fourier analysis are not available any more. This is particularly true when one studies large scale behavior of solutions of partial differential equations such as the Laplace and heat equations.

In the sequel, we will focus on complete, non-compact Riemannian manifolds. For compact manifolds, local Euclidean-type Sobolev inequalities are always satisfied and the interesting questions have to do with controlling the constants arising in these inequalities in geometric terms. We refer the interested reader to where this is discussed at length.

In previous chapters we discussed dynamical systems mainly from a geometrical or topological point of view. The geometrical approach is intuitively appealing and lends itself to suggestive graphical representations. Therefore, it has been tremendously successful in the study of low-dimensional systems: continuous-time systems with one, two or three variables; discrete-time systems with one or two variables. For higher-dimensional systems, however, the approach has encountered rather formidable obstacles and rigorous results and classifications are few. Thus, it is sometimes convenient to change perspective and adopt a different approach, based on the concept of measure, and aimed at the investigation of the statistical properties of ensembles of orbits. This requires the use and understanding of some basic notions and results, to which we devote this chapter. The ergodic theory of dynamical systems often parallels its geometric counterpart and many concepts discussed in chapters 3–8, such as invariant, indecomposable and attracting sets, attractors, and Lyapunov characteristic exponents will be reconsidered in a different light, thereby enhancing our understanding of them. We shall see that the ergodic approach is very powerful and effective for dealing with basic issues such as chaotic behaviour and predictability, and investigating the relationship between deterministic and stochastic systems.

From the point of view of ergodic theory, there is no essential difference between discrete- and continuous-time dynamical systems. Therefore, in what follows, we develop the discussion mostly in terms of maps, mentioning from time to time special problems occurring for flows.

In chapter 3 we discussed the behaviour of a dynamical system when it is displaced from its state of rest, or equilibrium, and, in particular, we studied the conditions under which the displaced system does not wander too far from equilibrium or even converges back to it as time goes by. For such cases, we call the equilibrium stable or asymptotically stable. But what happens if we perturb an unstable equilibrium?

For an autonomous linear system, if we exclude unlikely borderline cases such as centres, the answer to this question is straightforward: orbits will diverge without bound.

The situation is much more complicated and interesting for nonlinear systems. First of all, in this case we cannot speak of the equilibrium, unless we have established its uniqueness. Secondly, for nonlinear systems, stability is not necessarily global and if perturbations take the system outside the basin of attraction of a locally stable equilibrium, it will not converge back to it. Thirdly, besides convergence to a point and divergence to infinity, the asymptotic behaviour of nonlinear systems includes a wealth of possibilities of various degrees of complexity.

As we mentioned in chapter 1, closed-form solutions of nonlinear dynamical systems are generally not available, and consequently, exact analytical results are, and will presumably remain, severely limited. If we want to study interesting dynamical problems described by nonlinear differential or difference equations, we must change our orientation and adapt our goals to the available means.

In chapter 5 we studied bifurcations, that is, qualitative changes in the orbit structures of dynamical systems, which take place when parameters are varied. In chapter 6 we discussed chaos and provided a precise characterisation of chaotic dynamics. In this chapter we take up again the question of transition in a system's behaviour, with a view to understanding how complex dynamics and chaos appear as parameters change. This problem is often discussed under the label ‘routes to chaos’. The present state of the art does not permit us to define the prerequisites of chaotic behaviour with sufficient precision and generality, and we do not have a complete and exhaustive list of all such possible routes. In what follows we limit our investigation to a small number of ‘canonical’ transitions to chaos which are most commonly encountered in applications. In our discussion we omit many technical details and refer the reader to the relevant references.

Period-doubling route to chaos

Although the period-doubling route to chaos could be discussed in a rather general framework (cf. Eckmann 1981, pp. 648–9), here we shall treat it in the context of noninvertible one-dimensional maps, because they provide an interesting topic per se and are by far the most common type of dynamical system encountered in applications. Before describing the period-doubling scenario in detail, we discuss some general results covering a broad class of one-dimensional maps.

Dynamics is the study of the movement through time of variables such as heartbeat, temperature, species population, voltage, production, employment, prices and so forth.

This is often achieved by means of equations linking the values of variables at different, uniformly spaced instants of time, i.e., difference equations, or by systems relating the values of variables to their time derivatives, i.e., ordinary differential equations. Dynamical phenomena can also be investigated by other types of mathematical representations, such as partial differential equations, lattice maps or cellular automata. In this book, however, we shall concentrate on the study of systems of difference and differential equations and their dynamical behaviour.

In the following chapters we shall occasionally use models drawn from economics to illustrate the main concepts and methods. However, in general, the mathematical properties of equations will be discussed independently of their applications.

A static problem

To provide a first, broad idea of the problems posed by dynamic vis-à-vis static analysis, we shall now introduce an elementary model that could be labelled as ‘supply-demand-price interaction in a single market’. Our model considers the quantities supplied and demanded of a single good, defined as functions of a single variable, its price, p. In economic parlance, this would be called partial analysis since the effect of prices and quantities determined in the markets of all other goods is neglected. It is assumed that the demand function D(p) is decreasing in p (the lower the price, the greater the amount that people wish to buy), while the supply function S(p) is increasing in p (the higher the price, the greater the amount that people wish to supply).

Invariant, attracting sets and attractors with a structure more complicated than that of periodic or quasiperiodic sets are called chaotic. Before providing precise mathematical definitions of the properties of chaotic systems, let us first try to describe them in a broad, nonrigorous manner. We say that a discrete- or continuous-time dynamical system is chaotic if its typical orbits are aperiodic, bounded and such that nearby orbits separate fast in time. Chaotic orbits never converge to a stable fixed or periodic point, but exhibit sustained instability, while remaining forever in a bounded region of the state space. They are, as it were, trapped unstable orbits. To give an idea of these properties, in figure 6.1 we plot a few iterations of a chaotic discrete-time map on the interval, with slightly different initial values. The two trajectories remain close for the first few iterations, after which they separate quickly and have thereafter seemingly uncorrelated evolutions. Time series resulting from numerical simulations of such trajectories look random even though they are generated by deterministic systems, that is, systems that do not include any random variables. Also, the statistical analysis of deterministic chaotic series by means of certain linear techniques, such as estimated autocorrelation functions and power spectra, yields results similar to those determined for random series. Both cases are characterised by a rapidly decaying autocorrelation function and a broadband power spectrum.

remark 6.1 Notice that one or another (but not all) of the properties listed above can be found in nonchaotic systems as well.

Over the years we have had the rare opportunity to teach small classes of intelligent and strongly motivated economics students who found nonlinear dynamics inspiring and wanted to know more. This book began as an attempt to organise our own ideas on the subject and give the students a fairly comprehensive but reasonably short introduction to the relevant theory. Cambridge University Press thought that the results of our efforts might have a more general audience.

The theory of nonlinear dynamical systems is technically difficult and includes complementary ideas and methods from many different fields of mathematics. Moreover, as is often the case for a relatively new and fast growing area of research, coordination between the different parts of the theory is still incomplete, in spite of several excellent monographs on the subject. Certain books focus on the geometrical or topological aspects of dynamical systems, others emphasise their ergodic or probabilistic properties. Even a cursory perusal of some of these books will show very significant differences not only in the choice of content, but also in the characterisations of some fundamental concepts. (This is notoriously the case for the concept of attractor.)

For all these reasons, any introduction to this beautiful and intellectually challenging subject encounters substantial difficulties, especially for non-mathematicians, as are the authors and the intended readers of this book. We shall be satisfied if the book were to serve as an access to the basic concepts of nonlinear dynamics and thereby stimulate interest on the part of students and researchers, in the physical as well as the social sciences, with a basic mathematical background and a good deal of intellectual curiosity.