The authors explore the origin of multidimensional strange attractors and their role in describing turbulence. It includes an analytical estimation of the deminsions of strange attractors for models described by differential-difference equations and discusses the conditions in which space-homogeneous chaos is stable with respect to random perturbations in flow systems.

The present collection of reprints covers the main contributions of David Ruelle, and coauthors, to the theory of chaos and its applications. Several of the papers reproduced here are classics in the field. Others (that were published in less accessible places) may still surprise the reader.The collection contains mathematical articles relevant to chaos, specific articles on the theory, and articles on applications to hydrodynamical turbulence, chemical oscillations, etc.A sound judgement of the value of techniques and applications is crucial in the interdisciplinary field of chaos. For a critical assessment of what has been achieved in this area, the present volume is an invaluable contribution.

At Copenhagen in June 1988, the 80th Anniversary of the birth of L D Landau, the much respected Soviet physicist and author of the Course on Theoretical Physics, published by Pergamon Press, was celebrated with an International Symposium in his honour. The papers presented at that meeting are published here, providing an overview of recent progress in theoretical physics, covering super-string theories, chaos, high Tc superconductivity and biomolecules.

The present collection of reprints covers the main contributions of David Ruelle, and coauthors, to the theory of chaos and its applications. Several of the papers reproduced here are classics in the field. Others (that were published in less accessible places) may still surprise the reader. The collection contains mathematical articles relevant to chaos, specific articles on the theory, and articles on applications to hydrodynamical turbulence, chemical oscillations, etc. A sound judgement of the value of techniques and applications is crucial in the interdisciplinary field of chaos. For a critical assessment of what has been achieved in this area, the present volume is an invaluable contribution. Contents:On the Nature of TurbulenceBifurcation in the Presence of a Symmetry GroupThe Ergodic Theory of Axiom A FlowsMicroscopic Fluctuations and TurbulenceStrange AttractorsMeasures Describing a Turbulent FlowDo Turbulent Crystals Exist?Characteristic Exponents for a Viscous Fuid Subjected to Time Dependent ForcesBowen's Formula for the Hausdorff Dimension of Self-Similar SetsErgodic Theory of Chaos and Strange AttractorsLiapunov Exponents from Time SeriesFundamental Limitations for Estimating Dimensions and Lyapunov Exponents in Dynamical SystemsWhere can One Hope to Profitably Apply the Ideas of Chaos? Readership: Nonlinear scientists, researchers in fluid dynamics, mathematical physicists and mathematicians. keywords:Turbulence;Strange Attractor;Chaos;Chemical Oscillation;Ergodic Theory;Turbulent Crystal;Reaction-Diffusion;Hausdorff Dimension;Repeller;Resonance;Recurrence Plot

Since 1972 the Schools on Nonlinear Physics in Gorky have been a meeting place for Soviet Scientists working in this field. Since 1989 the proceedings appear in English. They present a good cross section of nonlinear physics in the USSR. This third volume emerged from material presented at the 1989 School. It contains sections dealing with nonlinear problems in physics and astrophysics, quantum and solid state physics, dynamical chaos and self-organization.

The book represents the first attempt to systematically deal with the use of deep neural networks to forecast chaotic time series. Differently from most of the current literature, it implements a multi-step approach, i.e., the forecast of an entire interval of future values. This is relevant for many applications, such as model predictive control, that requires predicting the values for the whole receding horizon. Going progressively from deterministic models with different degrees of complexity and chaoticity to noisy systems and then to real-world cases, the book compares the performances of various neural network architectures (feed-forward and recurrent). It also introduces an innovative and powerful approach for training recurrent structures specific for sequence-to-sequence tasks. The book also presents one of the first attempts in the context of environmental time series forecasting of applying transfer-learning techniques such as domain adaptation.