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.

One of the basic astronomical pursuits throughout history has been to determine the amount and temporal nature of the flux emitted by an object as a function of wavelength. This process, termed photometry, forms one of the fundamental branches of astronomy. Photometry is important for all types of objects from planets to stars to galaxies, each with their own intricacies, procedures, and problems. At times, we may be interested in only a single measurement of the flux of some object, while at other times we could want to obtain temporal measurements on time scales from seconds or less to years or longer. Some photometric output products, such as differential photometry, require fewer additional steps, whereas to obtain the absolute flux for an object, additional CCD frames of photometric standards are needed. These standard star frames are used to correct for the Earth's atmosphere, color terms, and other possible sources of extinction that may be peculiar to a given observing site or a certain time of year (Pecker, 1970).

We start this chapter with a brief discussion of the basic methods of performing photometry when using digital data from 2-D arrays. It will be assumed here that the CCD images being operated on have already been reduced and calibrated as described in detail in the previous chapter. We will see that photometric measurements require that we accomplish only a few steps to provide output flux values. Additional steps are then required to produce light curves or absolute fluxes.

Although imaging and photometry have been and continue to be mainstays of astronomical observations, spectroscopy is indeed the premier method by which we can learn the physics that occurs within or near the object under study. Photographic plates obtained the first astronomical spectra of bright stars in the late nineteenth century, while the early twentieth century saw the hand-in-hand development of astronomical spectroscopy and atomic physics. Astronomical spectroscopy with photographic plates, or with some method of image enhancement placed in front of a photographic plate, has led to numerous discoveries and formed the basis for modern astrophysics. Astronomical spectra have also had a profound influence on the development of the fields of quantum mechanics and the physics of extreme environments. The low quantum efficiency and nonlinear response of photographic plates placed the ultimate limiting factors on their use.

During the 1970s and early 1980s, astronomy saw the introduction of numerous electronic imaging devices, most of which were applied as detectors for spectroscopic observations. Television- type devices, diode arrays, and various silicon arrays such as Reticons were called into use. They were a step up from plates in a number of respects, one of which was their ability to image not only a spectrum of an object of interest, but, simultaneously, the nearby sky background spectrum as well – a feat not always possible with photographic plates.

The current high level of understanding of CCDs in terms of their manufacture, inherent characteristics, instrumental capabilities, and data analysis techniques make these devices desirable for use in spacecraft and satellite observatories and at wavelengths other than the optical. Silicon provides at least some response to photons over the large wavelength range from about 1 to 10 000 Å. Figure 7.1 shows this response by presenting the absorption depth of silicon over an expanded wavelength range. Unless aided in some manner, the intrinsic properties of silicon over the UV and EUV spectral range (1000–3000 Å) are such that the QE of the device at these wavelengths is typically only a few percent or less. This low QE value is due to the fact that for these very short wavelengths, the absorption depth of silicon is near 30–50 Å, far less than the wavelength of the incident light itself. Thus, the majority of the light (~ 70%) is reflected with the remaining percentage passing directly through the CCD unhindered.

Observations at wavelengths shorter than about 3000 Å involve additional complexities not encountered with ground-based optical observations. Access to these short wavelengths can only be obtained via space-based telescopes or high altitude rocket and balloon flights. The latter are of short duration from only a few hours up to possibly hundreds of days and use newly developing high-altitude ultra-long duration balloon flight technologies.

Before we begin our discussion of the physical and intrinsic characteristics of charge-coupled devices (Chapter 3), we want to spend a brief moment looking into how CCDs are manufactured and some of the basic, important properties of their electrical operation.

The method of storage and information retrieval within a CCD is dependent on the containment and manipulation of electrons (negative charge) and holes (positive charge) produced within the device when exposed to light. The produced photoelectrons are stored in the depletion region of a metal insulator semiconductor (MIS) capacitor, and CCD arrays simply consist of many of these capacitors placed in close proximity. Voltages, which are static during collection, are manipulated during readout in such as way as to cause the stored charges to flow from one capacitor to another, providing the reason for the name of these devices. These charge packets, one for each pixel, are passed through readout electronics that detect and measure each charge in a serial fashion. An estimate of the numerical value of each packet is sent to the next step in this process, which takes the input analog signal and assigns a digital number to be output and stored in computer memory.

Thus, originally designed as a memory storage device, CCDs have swept the market as replacements for video tubes of all kinds owing to their many advantages in weight, power consumption, noise characteristics, linearity, spectral response, and others.

After the two-body problem, the next more complicated system consists of three bodies. Let us call these bodies the Sun, planet and asteroid. Some further assumptions are made to keep the system as simple as possible. The word restricted means here that the mass of the asteroid is so small that it does not significantly affect the motion of the primaries (the Sun and the planet). The primaries move in circular orbits, and the asteroid is assumed to move in the same plane as the primaries. The perturbations due to the third body can be neglected and the positions of the primaries can be calculated analytically for all times. The problem is now to find the trajectory of the massless body.

The assumption about the mass of the asteroid is a little problematic. If the primaries affect the motion of the asteroid, it must, of course, affect their motions according to Newton's third law. The accuracy required determines whether the third body can actually be considered massless. Discarding Newton's third law has a side effect: total energy is no longer conserved. However, the energy conservation law can be replaced by another similar law.

Coordinate frames

When studying the restricted circular three body problem, the units are usually chosen in such a way that the properties of the system depend on a single parameter.