Rather than starting from abstract mathematical definitions related to dynamical systems and the concepts used to analyse them, I prefer to start from the outset with a few familiar examples. The systems described below are related to the paradigms of deterministic chaos, some of which have indeed been the ones leading to the discovery, definition and understanding of chaotic behaviour. Instead of repeating here the so often quoted examples such as biological population growth, nonlinearly driven electrical oscillations, weather unpredictability, three-body Hamiltonian dynamics and chemical reaction oscillations and patterns, I shall attempt to motivate the reader by trying to find such examples among simplistic models of astrophysical systems. Obviously, the underlying mathematical structure of these will be very similar to the above mentioned paradigms. This only strengthens one of the primary lessons of nonlinear dynamics, namely that this is a generic, universal approach to natural phenomena.

Examples and analogies may sometimes be misleading and decide nothing, but they can make one feel more at home. This was, at least, the view of Sigmund Freud, the father of psychology, whose advice on matters didactic should not be dismissed. Indeed, as stressed before, these examples are the readers' old acquaintances from their astrophysics educational ‘home’. In the next chapter, where the basic notions characterising chaotic behaviour will be dealt with in detail, these examples will sometimes be used again for demonstrating abstract concepts.

Developments in the theory of nonlinear conservative dynamical systems and Hamiltonian chaos have often been motivated, as we have seen, by problems in celestial mechanics. Fluid dynamics has, likewise, motivated much of the dissipative dynamical systems and pattern theory. However, while celestial mechanics is certainly a part of astronomy (a discussion of chaotic dynamics in planetary, stellar and galactic n-body systems can be found in Chapter 9 of this book), fluid dynamics can be regarded as an essentially separate discipline of the physical sciences and applied mathematics. Its uses and applications are widespread, ranging from practical engineering problems to abstract mathematical investigations, and its importance to astrophysics stems from the fact that most of the observable cosmos is made up of hot plasma, whose physical conditions are very often such that a fluid (or sometimes magneto-fluid) dynamical description is appropriate.

As is well known, and we shall shortly discuss explicitly, fluid dynamics has as its basis the description of matter as a continuum. Various assumptions then give rise to appropriate sets of the basic equations and these are usually nonlinear PDEs. The nonlinear PDEs of fluid (or magneto-fluid) dynamics are complicated dynamical systems and they have so far defied (except for the simplest cases) a rigorous and complete mathematical understanding.

In the last two decades or so the astrophysical community – students, teachers and researchers alike – have become aware of a new kind of activity in physics. Some researchers, science historians and philosophers have gone as far as calling it a ‘new science’ or ‘new physics’, while others see it as a mere natural extension of ‘old’ classical mechanics and fluid dynamics. In any case, the subject, variously referred to as dynamical systems theory, nonlinear dynamics or simply chaos, has undergone an explosive development, causing a lot of excitement in the scientific community and even in the general public. The discoveries look fundamental and there is hope that we will quite soon gain new and basic scientific understanding of the most complex aspects of nature.

The most striking quality of this modern approach to dynamical systems theory is, in my view, its extremely diverse range of applicability. Mechanics, fluid dynamics, chemical kinetics, electronic circuits, biology and even economics, as well as astrophysics, are among the subjects in which chaotic behaviour occurs. At the heart of the theory lies the quest for the universal and the generic, from which an understanding of complicated and seemingly heterogeneous phenomena can emerge. The ideas of bifurcations, strange attractors, fractal sets and so on, seem to provide the tools for such an unexpected conceptual unification.

My own experience in discussing the subject with astrophysicists suggests that they and their students would like to know more about the new developments in nonlinear dynamics.

A dynamical system has already been loosely defined in Chapter 1 as a set of rules by the application of which the state of a physical (or other well defined) system can be found, if some initial state is known. The origin of this name is in Newtonian mechanics, where a system of particles is described by ODE in time. If the initial state is specified, the ODE can be solved (at least in principle) to give the particles' state at any other later time. It should be stressed that the above mentioned set of rules can define a meaningful dynamical system only if it implies a unique evolution. Dynamical systems are deterministic. This is obvious in mechanics, since a particle cannot be at two different positions at the same time.

Mathematically, a dynamical system is a set of equations – algebraic, differential, integral or combination thereof, the solution of which provides the above mentioned rule of evolution with time (or some other independent variable). The equations may contain given parameters and the study of the dependence of the systems' behaviour on these parameters is often very important and interesting.

The notion of a dynamical system is used nowadays almost always as a synonym for a nonlinear system, i.e., one containing nonlinear evolution equations. In this book we shall deal mainly with such nonlinear dynamical systems; linear behaviour, when encountered, will be an exception. An overwhelming majority of systems taught and studied in the exact sciences are linear.

The first part of this book contained a rather extensive survey of the mathematical background needed for the study of various types of dynamical systems. The following, second part will be devoted to applications of some of these (often quite abstract) mathematical ideas and techniques to a selection of dynamical systems derived from astrophysical problems. The connection between mathematics and astronomy is a well known fact in the history of science. Sir James Jeans, the great British astronomer and mathematician, illustrated this by writing in 1930 that ‘… the Great Architect of the Universe now begins to appear as a pure mathematician.’

The mathematical theory of non-integrable Hamiltonian systems indeed followed directly from the gravitational n-body problem – a model of the most ancient and basic astronomical dynamical system. Dissipative chaos was first explicitly demonstrated in numerical calculations of simplistic models of thermal convection, a fluid-dynamical phenomenon having an obvious relevance to astrophysics. It is thus only natural to look for additional applications of dynamical system theory in astrophysics. Astrophysics is a relatively young science and many of its achievements have been made possible by modern advances in technology. The space programmes of the competing superpowers gave rise to enormous progress in electronics, enabling the development of modern observational instruments and powerful digital computers. Astronomers are today able to collect a wealth of data in virtually all the bands of the electromagnetic spectrum and store and analyse them effectively.

The classical astronomical sources of radiation, stars and star-like objects, are spatially unresolvable. This fact does not exclude, however, the possibility of timevariability, and indeed a variety of point sources have been found to possess such intrinsic variability, that is, one that remains in the light curve after atmospheric and other local effects are properly eliminated. Different classes of objects exhibit a wide range of variability timescales, often depending also on the spectral range.

When an astronomical source emits a time-variable signal, the natural first step in the data analysis is to search for periodicity. The identification of well-defined periods provides extremely valuable information, which can be used in understanding the relevant physical processes and therefore in constructing viable physical models of the astronomical source. This is obvious if we consider as an example the simplest periodic physical system of them all, the harmonic oscillator. Its period immediately reveals the ratio of the inertia to the restoring force and since every sufficiently small oscillation is to a good approximation harmonic (i.e., linear), the number of physical systems modelled with the help of this paradigm and its generalisations (multidimensional linear systems) has been very large. In astronomy the most prominent examples of this kind are pulsating stars. The famous period–luminosity relations of the classical Cepheids and other pulsating variables have not only been instrumental in the development of stellar pulsation theory, they have also played an important role in establishing the cosmic distance scale.

The great majority of astronomical systems are at enormous distances from Earth, but the extent of some of them is large enough as to be spatially resolvable in observations. Surveys in the optical as well as in other spectral ranges have revealed an inherently complex spatial distribution of matter (stars and gaseous nebulae) in the Galaxy and beyond. In this chapter we shall discuss some possible applications of nonlinear dynamics and pattern theory to the study of two (a priori unrelated) topics in this context: the complexity of the interstellar medium (ISM), and the properties of the large scale distribution of matter in the universe.

Interstellar clouds are invariably spatially complex and they pose a rather nontrivial challenge to the theorist, whose primary goal is the understanding of the physical processes shaping them. It is plausible that fluid turbulence plays an important role in this respect. This enigmatic problem is still largely unsolved and we shall not discuss here its possible applications to the ISM complexity (some selected aspects of turbulence will be addressed, however, in the next chapter). After briefly reviewing some observational features of the interstellar medium, which are probably relevant to our discussion, we shall mention some theoretical approaches, among them a pattern and complexity forming toy-model and a couple of global models, motivated by mathematical methods of statistical mechanics.

Hamiltonian systems constitute a very important subclass of dynamical systems. The basic model systems of classical mechanics – point masses moving in external potentials (modelling, e.g., planets in the Solar System, oscillating bodies attached to springs etc.) – are among the simplest Hamiltonian systems. The Lagrangian and Hamiltonian formalism of Newtonian dynamics provides the mathematical framework for the study of such (literally) dynamical systems. As these were the first dynamical systems to be studied extensively, the subject owes its name to them.

The problems of classical mechanics are by no means the only examples of Hamiltonian systems. Hamiltonian maps and differential systems also arise in the study of such diverse physical systems as propagating waves, fluids and plasmas, electric circuits etc. In addition, the Hamiltonian formulation is central to such important branches of physics as statistical mechanics, field theory and, most notably, quantum mechanics. It is thus important to understand the fundamental properties of classical Hamiltonian systems and the possibility of chaotic behaviour in them.

A basic attribute of Hamiltonian systems is that they are conservative. As we have already defined, a dynamical system is conservative if it preserves phase volume. Volume preservation in Hamiltonian systems endows them with special properties. One such important feature is that they do not have attractors. As we have seen in Chapter 4, the existence of strange attractors in dissipative systems is a primary characteristic of chaos.

In this part of the book we provide the basic mathematical background for dynamical systems and chaos theory. Some of the ideas introduced here will be applied to various astrophysical systems in the second part of the book. Our discussion here, while not particularly rigorous, will, however, be rather theoretical and abstract. I believe that a reasonable precision in building the basis for further understanding and research is mandatory. Throughout the discussion we continually give specific examples and often return to them in other places in the book. These examples, including some systems that are important by themselves, illustrate the various abstract concepts.

I have made an effort to interest readers, whose background is astronomy and astrophysics, by starting with astrophysical examples. After all, dynamical system theory and chaos have their origins in the studies of the three-body problem and celestial mechanics by Poincaré at the end of the nineteenth century. Fluid turbulence, an important unsolved scientific problem, is now being approached using methods from chaos and dynamical system theory. It has also had many important applications in astrophysics. Readers who are interested more in applications and less in theory and mathematical structure are particularly encouraged to become acquainted with the main concepts and results of this part of the book. Technical details may be skipped, certainly during first reading. When dealing with the second part (applications), the interested reader may return to the relevant material in the first part and study it more deeply.

Poincaré's important (and surprising) discovery that classical Hamiltonian systems, the paradigm of determinism, may be non-integrable and exhibit a seemingly erratic behaviour, had its roots in one of the most fundamental astronomical problems. In his book, entitled New Methods in Celestial Mechanics, Poincaré laid the basis for a geometrical approach to differential equations and made a number of significant mathematical discoveries. One of the consequences of these findings, phrased in a language that is relevant to us here, is that the gravitational n-body problem is generally non-integrable already for n = 3. We shall start this chapter with a short historical note on this subject.

Beyond their historical significance, Poincaré's findings have obvious relevance to astrophysics, as some of the most basic astronomical systems are naturally modelled by a number of mutually attracting point masses. For example, the Sun–Earth–Moon system can be viewed as a three-body problem, and its investigation was actually the initial motivation for Poincaré's work. The dynamics of the constituents of the Solar System (another obvious example of the n-body problem), and the question of the stability of this system as a whole, have always naturally enjoyed widespread attention. The masses of the objects in this system are very unequal and it is only natural to neglect the very minor planets, the planet moons and other small bodies if one is interested in the dynamics of the major planets. One is thus left with 3 < n < 6, say.

Spatially extended physical systems are naturally described by partial differential equations (PDE), which reflect the spatial structure of the system in addition to its time evolution. Classical examples in this context are fluid systems, described by the Euler or Navier–Stokes equations. Extended systems having several constituents that can react with each other and diffuse in space, are another well known example. These systems (like, e.g., mixtures of appropriate chemical species) are described by PDEs of the reaction–diffusion type. Many systems are only properly described when several types of process (e.g., fluid–dynamical and reactive–diffusive) are taken into account. A fluid system with heating and cooling processes as well as heat conduction operating within it is an example of such a system.

When we are dealing with only temporal evolution of a physical system, the dynamical system describing it consists of ODE or discrete maps. We have seen in Chapter 3 that near the threshold of an instability it is often possible to reduce the dimension of such systems and obtain a small set of amplitude equations, which faithfully describe the essential features of the system near marginality. A spatiotemporal PDE (the independent variables include coordinates and time) can be regarded as the infinite limit (n → ∞), of a system of n-coupled ODEs.