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Full Description
Nature provides many examples of physical systems that are described by deterministic equations of motion, but that nevertheless exhibit nonpredictable behavior. The detailed description of turbulent motions remains perhaps the outstanding unsolved problem of classical physics. In recent years, however, a new theory has been formulated that succeeds in making quantitative predictions describing certain transitions to turbulence. Its significance lies in its possible application to large classes (often very dissimilar) of nonlinear systems.
Since the publication of Universality in Chaos in 1984, progress has continued to be made in our understanding of nonlinear dynamical systems and chaos. This second edition extends the collection of articles to cover recent developments in the field, including the use of statistical mechanics techniques in the study of strange sets arising in dynamics. It concentrates on the universal aspects of chaotic motions, the qualitative and quantitative predictions that apply to large classes of physical systems. Much like the previous edition, this book will be an indispensable reference for researchers and graduate students interested in chaotic dynamics in the physical, biological, and mathematical sciences as well as engineering.
Contents
Introductory articles: Strange attractors. Universal behaviour in nonlinear systems. Simple mathematical models with very complicated dynamics. Experiments: Onset of turbulence in a rotating fluid Transition to chaotic behaviour via a reproducible sequence of period-doubling bifurcations. Representation of a strange attractor fron an experimental study of chemical turbulence. One-dimensional dynamics in a multicomponent chemical reaction. Experimental evidence of subharmonic bifurcations, multistability and turbulence in a Q-switched gas laser. Evidence for universal chaotic behaviour of a driven nonlinear oscillator. Phase locking, period-doubling bifurcations, and irregular dynamics in periodically stimulated cardiac cells. Theory: Qualitative universality in one dimension. Quantitative universality for one-dimensional period-doublings. A computer-assisted proof of the Feigenbaum conjectures. The transition to aperiodic behaviour in turbulent systems. Noise: Deterministic noise. Invariant distributions and stationary correlation functions of one-dimensional discrete processes. Scaling behaviour of chaotic flows. Power spectra of strange attractors. External noise: Fluctuations and the onset of chaos. Scaling for external noise at the onset of chaos. Intermittency: Intermittent transition to turbulence in dissipative dynamical system. Period-doubling in higher dimensions: A two-dimensional mapping with a strange attractor. Period doubling bifurcations for families of maps on R. Sequences of infinite bifurcations and turbulence in a five-mode truncation of the Navier-Stokes equations. Beyond the one-dimensional theory: Scaling behaviour in a map of a circle onto itself: empirical results. Self-generated chaotic behaviour in nonlinear mechanics. Recent developments: Feigenbaum universality and the thermodynamic formalism. Fractal measures and their singularities: the characterization of strange sets. Fixed points of composition operators.
