David Ruelle

Bio: David Ruelle is an academic researcher from Institut des Hautes Études Scientifiques. The author has contributed to research in topics: Statistical mechanics & Dynamical systems theory. The author has an hindex of 72, co-authored 230 publications receiving 31960 citations. Previous affiliations of David Ruelle include IHS Inc. & Carnegie Mellon University.

The Ergodic Theory of Axiom A Flows.

Abstract: Let M be a compact (Riemann) manifold and (f t ): M → M a differentiable flow A closed (f t )-invariant set ∧ ⊂ M containing no fixed points is hyperbolic if the tangent bundle restricted to ∧ can be written as the Whitney sum of three (Tf t )-invariant continuous subbundles $${T_\Lambda }M = E + {E^s} + {E^u}$$ where E is the one-dimensional bundle tangent to the flow, and there are constants c λ>0 so that $$(a)||T{f^t}(v)||\underset{\raise03em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } c{e^{ - \lambda t}}||v||forv \in {E^s},t\underset{\raise03em\hbox{$\smash{\scriptscriptstyle-}$}}{ \geqslant } 0and$$ $$(b)||T{f^{ - 1}}(v)||\underset{\raise03em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } c{e^{ - \lambda t}}||v||forv \in {E^u},t\underset{\raise03em\hbox{$\smash{\scriptscriptstyle-}$}}{ \geqslant } 0$$

On the Nature of Turbulence

Abstract: A mechanism for the generation of turbulence and related phenomena in dissipative systems is proposed.

Occurrence of strange axiom a attractors near quasi periodic flows on tm, m is greater than or equal to 3

Abstract: It is shown that by a smallC 2 (resp.C ∞) perturbation of a quasiperiodic flow on the 3-torus (resp. them-torus,m>3), one can produce strange AxiomA attractors. Ancillary results and physical interpretation are also discussed.

Ergodic theory of differentiable dynamical systems

Abstract: Iff is a C1 + ɛ diffeomorphism of a compact manifold M, we prove the existence of stable manifolds, almost everywhere with respect to everyf-invariant probability measure on M. These stable manifolds are smooth but do not in general constitute a continuous family. The proof of this stable manifold theorem (and similar results) is through the study of random matrix products (multiplicative ergodic theorem) and perturbation of such products.

The Fractal Geometry of Nature

Abstract: "...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

Synchronization in chaotic systems

Abstract: 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.

Pattern formation outside of equilibrium

Abstract: 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.