A primer for researchers and graduate students; introduces and applies chaos techniques to specific astrophysical systems.

The per iod of an oscillator tells us much about its structure. J. J. Thomson's deduction that a particle with the e/rn of an electron was in the atom is perhaps the most stunning instance. For us, the deduction of the mean density of a star from its oscillation period is another important example. What then can we deduce about an oscillator that is not periodic? If there are several frequencies or if the behavior is chaotic, may we not hope to learn even more delicate vital statistics about its workings? The recent progress in the theory of dynamical systems, particularly in the elucidat ion of the nature of chaos, makes it seem reasonable to ask this now. This is an account of some of the happenings of a workshop at which this question was raised and discussed. ~iTe were inc0rested in seeing ways in which the present understanding of chaos might guide astrophysical modelling and the interpretation of observations. But we did not try to conceal that we were also interested in chaos itself, and that made for a pleasant rapport between the chaoticists and astrophysicists at the meeting. We have several introductory papers on chaos in these proceedings, particularly on the analysis of data from systems that may be suspected of chaotic behavior. The papers of Geisel, Grassberger and Guckenheimer introduce the ways of characterizing chaos and Perdang illustrates how some of these ideas may be put into practice in explicit cases.

Nonlinear dynamics and chaos pervade dynamical problems on all astrophysical scales, ranging from the sun and solar system to galaxies and cosmology. This volume, the 13th in a series devoted to problems in nonlinear astronomy and physics, presents the work of 18 senior scientists from around the world as well as that of several postdoctoral associates to honour their mentor and colleague George Contopoulos, a seminal figure in this area of astrophysical research. Some of the topics considered are plasma physics, accelerator dynamics, several formal problems in nonlinear dynamics and several applied to astronomical problems on cosmology, accretion phenomena, and the structure and evolution of galaxies.

This book is one of the first to provide a general overview of order and chaos in dynamical astronomy. The progress of the theory of chaos has a profound impact on galactic dynamics. It has even invaded celestial mechanics, since chaos was found in the solar system which in the past was considered as a prototype of order. The book provides a unifying approach to these topics from an author who has spent more than 50 years of research in the field. The first part treats order and chaos in general. The other two parts deal with order and chaos in galaxies and with other applications in dynamical astronomy, ranging from celestial mechanics to general relativity and cosmology.

The conference 'Chaos in Astronomy' was held in Athens on 17-20 Sept. 2007. This book contains edited refereed contributions. It offers an overview to students and newcomers entering various fields of dynamical astronomy.

'he year was 1889. The French physicist-mathematician Henry T Poincare could not believe his eyes. He had worked for months on one of the most famous problems in science-the problem of three bodies moving around one another under mutual gravita tional attraction-and what he was seeing dismayed and trou bled him. Since Newton's time it had been assumed that the problem was solvable. All that was needed was a little ingenuity and considerable perseverance, but Poincare saw that this was not the case. Strange, unexplainable things happened when he delved into the problem; it was not solvable after all. Poincare was shocked and dismayed by the result-so disheartened he left the problem and went on to other things. What Poincare was seeing was the first glimpse of a phe nomenon we now call chaos. With his discovery the area lay dormant for almost 90 years. Not a single book was written about the phenomenon, and only a trickle of papers appeared. Then, about 1980 a resurgence of interest began, and thousands of papers appeared along with dozens of books. The new science of chaos was born and has attracted as much attention in recent years as breakthroughs in superconductivity and superstring theory.

On August 2000 in the Lomonosov Moscow State University the first scientific conference dedicated to chaos in the real astronomical systems was held. The most prominent astrophysisists - specialist in the field of stochastic dynamics - attended the conference. A broad scope of the problems related to the observed manifes tations of chaotic motions in galactic and stellar objects, with the involvement of basic theory and numerical modeling, were addressed. The idea (not so obvious, as we believe, to many astrophysicists) was to show that, while great progress in the field of stochastic mechanics was accomplished, the science of chaos in actually observed systems is only just being born. Basically, the situation described prompted the organizers to hold the meeting in order to discuss chaotic processes in real systems. It seemed worthwhile to begin these introductory remarks with a brief descrip tion of some events that preceeded the conference. Since actually existing systems are the subject of the natural sciences, and in the latter experiments play the key role, we shall begin our account with the experimental results.

The per iod of an oscillator tells us much about its structure. J. J. Thomson's deduction that a particle with the e/rn of an electron was in the atom is perhaps the most stunning instance. For us, the deduction of the mean density of a star from its oscillation period is another important example. What then can we deduce about an oscillator that is not periodic? If there are several frequencies or if the behavior is chaotic, may we not hope to learn even more delicate vital statistics about its workings? The recent progress in the theory of dynamical systems, particularly in the elucidat ion of the nature of chaos, makes it seem reasonable to ask this now. This is an account of some of the happenings of a workshop at which this question was raised and discussed. ~iTe were inc0rested in seeing ways in which the present understanding of chaos might guide astrophysical modelling and the interpretation of observations. But we did not try to conceal that we were also interested in chaos itself, and that made for a pleasant rapport between the chaoticists and astrophysicists at the meeting. We have several introductory papers on chaos in these proceedings, particularly on the analysis of data from systems that may be suspected of chaotic behavior. The papers of Geisel, Grassberger and Guckenheimer introduce the ways of characterizing chaos and Perdang illustrates how some of these ideas may be put into practice in explicit cases.

This unique volume is a collection of papers on various problems in astrophysics and cosmology ? from planetary motion to the arrow of time ? that are closely linked by the common spirit, technique and methodology of chaos.

This overview of classical celestial mechanics focuses the interplay with dynamical systems. Paradigmatic models introduce key concepts – order, chaos, invariant curves and cantori – followed by the investigation of dynamical systems with numerical methods.