An exploration of chaos: An introduction for natural scientists and engineers: By John Argyris, Gunter Faust and Maria Haase. North-Holland, Amsterdam. (1994). 751 pages. $193.00

Abstract

Preface. 1. Descriptive Conspectus of the Text. 2. Preliminaries. Causality - determinism. Dynamical systems - examples. Phase space. First integrals and manifolds. Qualitative and quantitative approach. 3. Mathematical Introduction to Dynamical Systems. Linear autonomous systems. Non-linear systems and stability. Invariant manifolds. Discretisation in time. Poincar6 map. Fixed points and cycles of discrete systems. An example of discrete dynamics - the logistic map. 4. Dynamical Systems without Dissipation. Hamilton equations for conservative systems. Canonical transformations, integrability. f-dimensional tori and trajectories. An outline of the KAM theory. Unstable tori, chaotic regions. A numerical example: the H6non map. 5. Dynamical Systems with Dissipation. Volume contraction - a basic characteristic of dissipative systems. Strange attractor: Lorenz attractor. Power spectrum and autocorrelation. Lyapunov exponents. Dimensions. Kolmogorov-Sinai entropy. 6. Local Bifurcation Theory. Motivation. Centre manifold. Normal forms. Normal forms of bifurcations for one-parametric flows. Stability of bifurcations subject to perturbations. Bifurcations of the fixed points of one-parametric maps. Renormalisation and self-similarity with the example of the logistic map. A descriptive introduction to synergetics. 7. Convection Flows: Benard Problem. Basic hydrodynamic equations. Boussinesq-Oberbeck approximation. Lorenz model. Evolution of the Lorenz system. 8. Routes to Turbulence. Landau scenario. Ruelle-Takens scenario. Universal characteristics of the transition from quasi-periodicity to chaos. The Feigenbaum route to chaos via period doublings. Quasi-periodic transition for a fixed winding number. The route to chaos via intermittency. Routes out of chaos, control of chaos. 9. Computer Experiments. Introduction to bone remodelling processes. H6non map. The Lorenz system revisited. Van der Pol equation. Doffing equation. Julia sets and their ordering principle. Morphology of the Amol'd tongues. Oscillatory kinetics of chemical reactions on solid surfaces. An apercu of chaotic behaviour in our solar system. Colour Plates. Bibliography. Index.