The fifth International School ~ Mathematical Physics was held at the Ettore Majorana Centro della Culture Scientifica, Erice, Sicily, 2 to 14 July 1983. The present volume collects lecture notes on the session which was devoted to'Regular and Chaotic Motions in Dynamlcal Systems. The School was a NATO Advanced Study Institute sponsored by the Italian Ministry of Public Education, the Italian Ministry of Scientific and Technological Research and the Regional Sicilian Government. Many of the fundamental problems of this subject go back to Poincare and have been recognized in recent years as being of basic importance in a variety of physical contexts: stability of orbits in accelerators, and in plasma and galactic dynamics, occurrence of chaotic motions in the excitations of solids, etc. This period of intense interest on the part of physicists followed nearly a half a century of neglect in which research in the subject was almost entirely carried out by mathematicians. It is an in dication of the difficulty of some of the problems involved that even after a century we do not have anything like a satisfactory solution.

This book treats nonlinear dynamics in both Hamiltonian and dissipative systems. The emphasis is on the mechanics for generating chaotic motion, methods of calculating the transitions from regular to chaotic motion, and the dynamical and statistical properties of the dynamics when it is chaotic. The new edition brings the subject matter in a rapidly expanding field up to date, and has greatly expanded the treatment of dissipative dynamics to include most important subjects.

Discoveries of chaotic, unpredictable behaviour in physical deterministic systems has brought about new analytic and experimental techniques in dynamics. The modern study of the new phenomena requires the analyst to become familiar with experiments (at least with numerical ones), since chaotic solutions cannot be written down, and it requires the experimenter to master the new concepts of the theory of nonlinear dynamical systems. This book is unique in that it presents both viewpoints: the viewpoint of the analyst and of the experimenter. In the first part F. Moon outlines the new experimental techniques which have emerged from the study of chaotic vibrations. These include Poincaré sections, fractial dimensions and Lapunov exponents. In the text by W. Szemplinska-Stupnicka the relation between the new chaotic phenomena and classical perturbation techniques is explored for the first time. In the third part G. Iooss presents methods of analysis for the calculations of bifurcations in nonlinear systems based on modern geometric mathematical concepts.

Usually, the first edition of a book still contains a multiplicity of typographic, con ceptional, and computational errors even if one believes the opposite at the time of publication. As this book did not represent a counterexample to this rule, the current second edition offers a chance to remove at least the known shortcomings. The book has been partly re-organized. The previously rather long Chapter 4 has been split into two separate chapters dealing with discrete-time and continuous time approaches to nonlinear economic dynamics. The short summary of basic properties of linear dynamical systems has been banned to an appendix because the line of thought in the chapter seems to have been unnecessarily interrupted by these technical details and because the book concentrates on nonlinear systems. This appendix, which mainly deals with special formal properties of dynamical sys tems, also contains some new material on invariant subspaces and center-manifold reductions. A brief introduction into the theory of lags and operators is followed by a few remarks on the relation between the 'true' properties of dynamical systems and their behavior observable in numerical experiments. Additional changes in the main part of the book include a re-consideration of Popper's determinism vs. inde terminism discussion in the light of chaotic properties of deterministic, nonlinear systems in Chapter 1. An investigation of a simultaneous price-quantity adjustment process, a more detailed inquiry into the uniqueness property of limit cycles, and a short presentation of relaxation oscillations are included in Chapter 2.

The Solar System is a complex and fascinating dynamical system. This is the first textbook to describe comprehensively the dynamical features of the Solar System and to provide students with all the mathematical tools and physical models they need to understand how it works. It is a benchmark publication in the field of planetary dynamics and destined to become a classic. Clearly written and well illustrated, Solar System Dynamics shows how a basic knowledge of the two- and three-body problems and perturbation theory can be combined to understand features as diverse as the tidal heating of Jupiter's moon Io, the origin of the Kirkwood gaps in the asteroid belt, and the radial structure of Saturn's rings. Problems at the end of each chapter and a free Internet Mathematica® software package are provided. Solar System Dynamics provides an authoritative textbook for courses on planetary dynamics and celestial mechanics. It also equips students with the mathematical tools to tackle broader courses on dynamics, dynamical systems, applications of chaos theory and non-linear dynamics.

This book is devoted to the phenomenon of quasi-periodic motion in dynamical systems. Such a motion in the phase space densely fills up an invariant torus. This phenomenon is most familiar from Hamiltonian dynamics. Hamiltonian systems are well known for their use in modelling the dynamics related to frictionless mechanics, including the planetary and lunar motions. In this context the general picture appears to be as follows. On the one hand, Hamiltonian systems occur that are in complete order: these are the integrable systems where all motion is confined to invariant tori. On the other hand, systems exist that are entirely chaotic on each energy level. In between we know systems that, being sufficiently small perturbations of integrable ones, exhibit coexistence of order (invariant tori carrying quasi-periodic dynamics) and chaos (the so called stochastic layers). The Kolmogorov-Arnol'd-Moser (KAM) theory on quasi-periodic motions tells us that the occurrence of such motions is open within the class of all Hamiltonian systems: in other words, it is a phenomenon persistent under small Hamiltonian perturbations. Moreover, generally, for any such system the union of quasi-periodic tori in the phase space is a nowhere dense set of positive Lebesgue measure, a so called Cantor family. This fact implies that open classes of Hamiltonian systems exist that are not ergodic. The main aim of the book is to study the changes in this picture when other classes of systems - or contexts - are considered.

The book is concerned with the concepts of chaos and fractals, which are within the scopes of dynamical systems, geometry, measure theory, topology, and numerical analysis during the last several decades. It is revealed that a special kind of Poisson stable point, which we call an unpredictable point, gives rise to the existence of chaos in the quasi-minimal set. This is the first time in the literature that the description of chaos is initiated from a single motion. Chaos is now placed on the line of oscillations, and therefore, it is a subject of study in the framework of the theories of dynamical systems and differential equations, as in this book. The techniques introduced in the book make it possible to develop continuous and discrete dynamics which admit fractals as points of trajectories as well as orbits themselves. To provide strong arguments for the genericity of chaos in the real and abstract universe, the concept of abstract similarity is suggested.