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.
"…essential reading for the expert, while the mathematically sophisticated outsider should enjoy dipping into it."
Introduction. Part 1: Introductory articles: Strange attractors (D Ruelle). Universal behaviour in nonlinear systems (M J Feigenbaum). Simple mathematical models with very complicated dynamics (R M May). Roads to turbulence in dissipative dynamical systems (J P Eckmann). Part 2: Experiments: Fluid mechanics. A Rayleigh Benard experiment: helium in a small box (A Libchaber and J Maurer). Period doubling cascade in mercury, a quantitative measurement (A Libchader et al). Onset of turbulence in a rotating fluid (J P Gollub and H L Swinney). Transition to chaotic behaviour via a reproducible sequence of period-doubling bifurcations (M Giglio et al). Intermittency in Rayleigh-Benard convection (P Berge et al). Chemical systems: Representation of a strange attractor fron an experimental study of chemical turbulence (J C Roux et al). Chaos in the Belousov-Zhabotinskii reaction (J L Hudson and J C Mankin) One-dimensional dynamics in a multicomponent chemical reaction (R H Simoyi et al). Intermittent behaviour in the Belousov-Zhabotinsky reaction (Y Pomeau et al). Optical experiments: Experimental evidence of subharmonic bifurcations, multistability and turbulence in a Q-switched gas laser (F T Arecchi et al). Electronic experiments: Evidence for universal chaotic behaviour of a driven nonlinear oscillator (J Testa et al). Biological experiments: Phase locking, period-doubling bifurcations, and irregular dynamics in periodically stimulated cardiac cells (M R Guevara et al). Part 3: Theory: Qualitative universality in one dimension. On finite limit sets for transformations on the unit interval (M Metropolis et al). Quantitative universality for one-dimensional period-doublings. The universal metric properties of nonlinear transformations (M J Feigenbaum) A computer-assisted proof of the Feigenbaum conjectures (O E Landford III). Subharmonic spectrum. The transition to aperiodic behaviour in turbulent systems (M J Feigenbaum). Universality and the power spectrum at the onset of chaos (M Nauenberg and J Rudnick). Part 4: Noise: Deterministic noise. Invariant distributions and stationary correlation functions of one-dimensional discrete processes (S Grossmann and S Thomae). Noisy periodicity and reverse bifurcation (E N Lorenz). Scaling behaviour of chaotic flows (B A Huberman and J Rudnick). Universal power spectra for the reverse bifurcation sequence (A Wolf and J Swift). Power spectra of strange attractor (B A Huberman and A B Zisook). Spectral broadening of period-doubling bifurcation sequences (J D Farmer). External noise: Fluctuations and the onset of chaos (J P Crutchfield and B A Huberman). Scaling theory for noisy period-doubling transitions to chaos (B Shraiman et al). Scaling for external noise at the onset of chaos (J Crutchfield et al). Part 5: Intermittency: Intermittent transition to turbulence in dissipative dynamical systsm (Y Pomeaur and P Manneville). Intermittency in the presence of noise: a renormalization group formulation (J Hirsch et al). Part 6: Period-doubling in higher dimensions: A two-dimensional mapping with a strange attractor (M Henon). Universal effects of dissipation in two-dimensional mappings (A B Zisook). Period doubling bifurcations for families of maps on R (P Colletee et al). Deterministic nonperiodic flow (E N Lorenz). Sequences of infinite bifurcations and turbulence in a five-mode truncation of the Navier-Stokes equations (V Franceschini and C Tebaldi). Power spectral analysis of a dynamical system (J Crutchfield et al). Part 7: Beyond the one-dimensional theory. Scaling behaviour in a map of a circle onto itself: empirical results (S J Shenker). Period doubling as a universal route to stocasticity (R S MacKay). Self-generated chaotic behaviour in nonlinear mechanics (R H G Helleman). Part 8: Recent developments. Feigenbaum universality and the thermodynamic formalism (E B Vui et al.) Mode-locking and the transition to chaos in dissipative systems (P Bak et al). Fractal measures and their singularities: the characterization of strange sets (T C Halsey et al). Presentation functions, fixed points, and a theory of scaling function dynamics (M J Feigenbaum). Fixed points of composition operators II (H Epstein). Bounded structure of infinitely renormalizable mappings (D Sullivan). References. Indexes.