"...a blend of erudition (fascinating and sometimes obscure historical minutiae abound), popularization (mathematical rigor is relegated to appendices) and exposition (the reader need have little knowledge of the fields involved) ...and the illustrations include many superb examples of computer graphics that are works of art in their own right." Nature

The Fractal Geometry of Nature

Detecting strange attractors in turbulence

Certain subsystems of nonlinear, chaotic systems can be made to synchronize by linking them with common signals. The criterion for this is the sign of the sub-Lyapunov exponents. We apply these ideas to a real set of synchronizing chaotic circuits.

Synchronization in chaotic systems

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Determining Lyapunov exponents from a time series

A comprehensive review of spatiotemporal pattern formation in systems driven away from equilibrium is presented, with emphasis on comparisons between theory and quantitative experiments. Examples include patterns in hydrodynamic systems such as thermal convection in pure fluids and binary mixtures, Taylor-Couette flow, parametric-wave instabilities, as well as patterns in solidification fronts, nonlinear optics, oscillatory chemical reactions and excitable biological media. The theoretical starting point is usually a set of deterministic equations of motion, typically in the form of nonlinear partial differential equations. These are sometimes supplemented by stochastic terms representing thermal or instrumental noise, but for macroscopic systems and carefully designed experiments the stochastic forces are often negligible. An aim of theory is to describe solutions of the deterministic equations that are likely to be reached starting from typical initial conditions and to persist at long times. A unified description is developed, based on the linear instabilities of a homogeneous state, which leads naturally to a classification of patterns in terms of the characteristic wave vector q0 and frequency ω0 of the instability. Type Is systems (ω0=0, q0≠0) are stationary in time and periodic in space; type IIIo systems (ω0≠0, q0=0) are periodic in time and uniform in space; and type Io systems (ω0≠0, q0≠0) are periodic in both space and time. Near a continuous (or supercritical) instability, the dynamics may be accurately described via "amplitude equations," whose form is universal for each type of instability. The specifics of each system enter only through the nonuniversal coefficients. Far from the instability threshold a different universal description known as the "phase equation" may be derived, but it is restricted to slow distortions of an ideal pattern. For many systems appropriate starting equations are either not known or too complicated to analyze conveniently. It is thus useful to introduce phenomenological order-parameter models, which lead to the correct amplitude equations near threshold, and which may be solved analytically or numerically in the nonlinear regime away from the instability. The above theoretical methods are useful in analyzing "real pattern effects" such as the influence of external boundaries, or the formation and dynamics of defects in ideal structures. An important element in nonequilibrium systems is the appearance of deterministic chaos. A greal deal is known about systems with a small number of degrees of freedom displaying "temporal chaos," where the structure of the phase space can be analyzed in detail. For spatially extended systems with many degrees of freedom, on the other hand, one is dealing with spatiotemporal chaos and appropriate methods of analysis need to be developed. In addition to the general features of nonequilibrium pattern formation discussed above, detailed reviews of theoretical and experimental work on many specific systems are presented. These include Rayleigh-Benard convection in a pure fluid, convection in binary-fluid mixtures, electrohydrodynamic convection in nematic liquid crystals, Taylor-Couette flow between rotating cylinders, parametric surface waves, patterns in certain open flow systems, oscillatory chemical reactions, static and dynamic patterns in biological media, crystallization fronts, and patterns in nonlinear optics. A concluding section summarizes what has and has not been accomplished, and attempts to assess the prospects for the future.

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Pattern formation outside of equilibrium

Preface vii Chapter 1: Scientific Thinking 1 Chapter 2: What Is Mathematics? 5 Chapter 3: The Erlangen Program 11 Chapter 4: Mathematics and Ideologies 17 Chapter 5: The Unity of Mathematics 23 Chapter 6: A Glimpse into Algebraic Geometry and Arithmetic 29 Chapter 7: A Trip to Nancy with Alexander Grothendieck 34 Chapter 8: Structures 41 Chapter 9: The Computer and the Brain 46 Chapter 10: Mathematical Texts 52 Chapter 11: Honors 57 Chapter 12: Infinity: The Smoke Screen of the Gods 63 Chapter 13: Foundations 68 Chapter 14: Structures and Concept Creation 73 Chapter 15: Turing's Apple 78 Chapter 16: Mathematical Invention: Psychology and Aesthetics 85 Chapter 17: The Circle Theorem and an Infinite-Dimensional Labyrinth 91 Chapter 18: Mistake! 97 Chapter 19: The Smile of Mona Lisa 103 Chapter 20: Tinkering and the Construction of Mathematical Theories 108 Chapter 21: The Strategy of Mathematical Invention 113 Chapter 22: Mathematical Physics and Emergent Behavior 119 Chapter 23: The Beauty of Mathematics 127 Notes 131 Index 157

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The mathematician's brain

http://www.ihes.fr/~ruelle/PUBLICATIONS/%5B27%5D.pdf

Integral representation of states on a C∗-algebra

What Are the Measures Describing Turbulence

A number of results are derived which are pertinent to the description of physical systems by states on C*-algebras invariant under a symmetry group. In particular an integral decomposition of a state into states of lower symmetry is obtained which is relevant to the study of spontaneously broken symmetries which occur in equilibrium statistical mechanics as existence of crystals, ferromagnetic states, etc. A characterization is given of strongly clustering euclidean invariant states, and it is shown that they cannot be decomposed into states of lower symmetry.

Extremal invariant states

Previous results on reduced density matrices of quantum gases are used to obtain theorems on the existence and analyticity of Green's functions.

David Ruelle

Papers

The mathematician's brain

Integral representation of states on a C∗-algebra

What Are the Measures Describing Turbulence

Extremal invariant states

Analyticity of Green's Functions of Dilute Quantum Gases