We begin the detailed study of chaotic behaviour with dissipative systems. We consider permanently chaotic dynamics (cf. Section 1.2.1), and we start our investigations within the framework of a simple ‘model’ map, the baker map. The most important quantities characteristic of chaos will be introduced via this example. The simplicity of the map makes the exact treatment of numerous chaos properties possible, an exceptional feature in the world of chaotic processes. Next we turn to the investigation of a physical system, the kicked oscillator, with different kicking amplitudes. These functions will be chosen in such a way that, in the first case, the attractor is similar to that of the baker map. In the second, the attractor has a different structure and exhibits a general property of chaotic attractors: it appears to be a single continuous curve. The special form of the amplitude function continues to make its exact construction possible. This is no longer so, however, with the third choice, representing a typical chaotic system. The parameter dependence of chaotic systems will also be discussed within the class of kicked oscillators. Based on all these examples, we summarise the most important properties of chaos, first of all at the level of maps. As measures of irregularity, unpredictability and complex phase space structures, we introduce the concepts of topological entropy, Lyapunov exponents and the fractal dimension of chaotic attractors, respectively. Special emphasis will be given to the presentation and characterisation of the natural distribution of chaotic attractors.

The simplest motion occurs in one-dimensional systems subjected to time-independent forces. The most important characteristics of regular (non-chaotic) behaviour will be demonstrated by means of such motion, but this also provides an opportunity for us to formulate some general features. The overview starts with the investigation of the dynamics around unstable and stable equilibrium states, where the essentials already appear in a linear approximation. Outside of a small neighbourhood of the equilibrium state, however, non-linear behaviour is usually present, which manifests itself, for example, in the co-existence of several stable and unstable states, or in the emergence of such states as the parameters change. We monitor the motion in phase space and become acquainted with the geometrical structures characteristic of regular motion. The unstable states, and the curves emanated from such hyperbolic points, the stable and unstable manifolds, play the most important role since they form, so to say, the skeleton of all possible motion. In the presence of friction, trajectories converge to the attractors of the phase space. For regular motion, attractors are simple: equilibrium states and periodic oscillations, implying fixed point attractors and limit cycle attractors, respectively.

Instability and stability

Motion around an unstable state: the hyperbolic point

Let us start the analysis – contrary to the traditional approach – with the behaviour at and in the vicinity of an unstable equilibrium state.

An equilibrium state of a body at some position x* is unstable if, when released from a slightly displaced position, the body starts moving further away from x*.

A special but important class of dynamics is provided by systems in which friction is negligible, or, more generally, where dissipative effects play no role. In this case the direction of time is not specific, the process described by a differential equation is reversible: forward and backward time behaviour is similar. Think of, for example, a planet: one cannot decide whether its motion recorded on a film takes place in direct or in reversed time. In frictionless systems phase space volume is preserved, and attractors cannot exist. In such conservative systems, the manifestation of chaos is of a different nature than in dissipative cases. In this chapter we investigate persistent conservative chaos where escape is impossible, and defer the problem of transient conservative chaos to Chapter 8. We start with the area preserving baker map and the stroboscopic map of a kicked rotator. Next, the dynamics of continuous-time, non-driven frictionless systems is considered. On the basis of these examples, we summarize the general properties of conservative chaos, including one of the most important relationships, the KAM theorem. The structure of chaotic bands characteristic of conservative systems is discussed and compared with that of chaotic attractors. Finally, we present how conservative chaos of increasing strength manifests itself and we discuss the consequences.

What is a fractal?

Objects with large surfaces

It is taken for granted that the surface or volume of a traditional geometrical object, for example a sphere or a cube, is well defined. Indeed, filling the object with smaller and smaller cubes leads to better and better approximations, and the total volume of the cubes converges to that of the object in question. It is well known that the surface, S, is proportional to the second, while the volume V is proportional to the third, power of the linear size, L, of the object. Consequently, the surface-to-volume ratio, S/V, is proportional to V−⅓. (For plane figures, the ratio of the perimeter, P, to the area, A, is proportional to A−½.) The surface-to-volume ratio is therefore finite and becomes smaller as the size becomes larger. This is why surface phenomena are of little importance compared with volume phenomena for macroscopic systems of traditional geometry.

On the other hand, it is known that there exist macroscopic objects with large surface area. These are always porous, with ramified or pitted surfaces. Effective chemical catalysts, for example, must have a large surface. The need for rapid gas exchange accounts for the large surface-to-volume ratio of the respiratory organs. The surface area of the human lungs (measured at microscopic resolution), for example, is the same as that of a tennis court (approximately 100 m2), while the volume is only a few litres (10−3 m3).

The world around us is full of phenomena that seem irregular and random in both space and time. Exploring the origin of these phenomena is usually a hopeless task due to the large number of elements involved; therefore one settles for the consideration of the process as noise. A significant scientific discovery made over the past few decades has been that phenomena complicated in time can occur in simple systems, and are in fact quite common. In such chaotic cases the origin of the random-like behaviour is shown to be the strong and non-linear interaction of the few components. This is particularly surprising since these are systems whose future can be deduced from the knowledge of physical laws and the current state, in principle, with arbitrary accuracy. Our contemplation of nature should be reconsidered in view of the fact that such deterministic systems can exhibit random-like behaviour.

Under certain circumstances chaotic behaviour is of finite duration only, i.e. the complexity and unpredictability of the motion can be observed over a finite time interval. Nevertheless, there also exists in these cases a set in phase space responsible for chaos, which is, however, non-attracting. This set is again a well defined fractal, although it is more rarefied than chaotic attractors. This type of chaos is called transient chaos, and the underlying non-attracting set in invertible systems is a chaotic saddle. The concept of transient chaos is more general than that of permanent chaos studied so far, and knowledge of it is essential for a proper interpretation of several chaos-related phenomena. The basic new feature here is the finite lifetime of chaos.

In dissipative systems transient chaos appears primarily in the dynamics of approaching the attractor(s). It is therefore also called the chaotic transient. The temporal duration of the chaotic behaviour varies even within a given system, depending on the initial conditions (see Fig. 6.1 and Section 1.2.2). Despite the significant differences in the individual lifetimes, an average lifetime can be defined. To this end, it is helpful to consider several types of motion (trajectories) instead of a single one: the study of particle ensembles is even more important in transient than in permanent chaos.

Patterns such as hexagons and squares can also become unstable to phase and cross-pattern modes, while stripes in systems with additional symmetries, such as Galilean invariance, can undergo new types of instability, leading to drift, for example. This chapter looks at some of these new situations, starting with two examples of more complicated planforms – hexagons and quasipatterns – and some of their instabilities. After that we study drift instabilities where stationary or standing-wave patterns start to travel, and finally we look at the effect of Galilean invariance and conservation laws on the instabilities of stripes.

Instabilities of two-dimensional steady patterns

There are many possible extensions of the work on roll instabilities to more complicated situations. An obvious starting point is to consider what happens when the pattern that emerges at the primary pattern-forming instability is more complicated – a steady square pattern, for example, or oscillating hexagons, or maybe even a quasipattern. There is an extensive literature dealing with the phase instabilities of steady and oscillatory patterns of all sorts. We shall concentrate on two examples – steady hexagons and steady twelvefold quasipatterns – that illustrate how to extend the methods used in the previous chapter to these harder problems and lead to some interesting new results. At the end of this chapter you will find exercises on the instabilities of steady and oscillating squares as further examples.

Instabilities of hexagons

In this section we will adapt the methods used for rolls to investigate the instabilities of hexagons.

Regular patterns are found in abundance in nature, from the spots on a leopard's back to the ripples on a sandy beach or desert dune. There has a been a flurry of recent research activity seeking to explain their appearance and evolution, and the selection of one pattern over another has turned out to be an inherently nonlinear phenomenon. My aim in writing this book has been to provide an introduction to the range of methods used to analyse natural patterns, at a level suitable for final year undergraduates and beginning graduate students in UK universities.

The book brings together several different approaches used in describing pattern formation, from group theoretic methods to envelope equations and the theory of patterns in large-aspect-ratio systems. The emphasis is on using symmetries to describe universal classes of pattern rather than restricting attention to physical systems with well-known governing equations, though connections with particular systems are also explored. I have taken a wholeheartedly nonpartisan approach, unifying for perhaps the first time in a textbook a multiplicity of methods used by active researchers in the field.

It was David Crighton who originally suggested I should write this book. I had been lecturing a Cambridge Part III course on pattern formation, and David mentioned in passing that it might be a nice idea to turn my lecture notes into a book. Of course I had no idea what I was letting myself in for, but David was always persuasive and inspirational so naturally I said yes. Several years of sweat and toil later I have finally produced the book, though it bears little resemblance to my Part III course, which is probably just as well.