"...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

/pdf/determining-lyapunov-exponents-from-a-time-series-3bxxq6d0wt.pdf

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

Publisher Summary
This chapter focuses on differentiable dynamical systems. Physics gives many examples of differentiable dynamical systems in infinite-dimensional Banach spaces. In fact, Banach spaces, also defined as finite-or infinite-dimensional, provide the natural framework for the study of differentiable maps. The time evolution of natural systems is often given by a differential equation. A hyperbolic fixed point or periodic orbit is also called a sink if it is attracting and a source if it is repelling. Otherwise, it is a saddle or of saddle type. The Grobman–Hartman theorem, for maps of flow, is satisfactory in providing a local topological model for dynamics. The study of fixed points and periodic orbits becomes more delicate when they are not hyperbolic. One is, however, forced to consider non-hyperbolic behavior in the theory of bifurcations.

Differentiable Dynamical Systems

There seems to be no general theoretical argument to support the idea that thermodynamic functions are piecewise analytic. We suggest that nonanalyticity may be associated with Gibbs states which are quasiperiodic under space translations, or have a more general nonperiodic (“turbulent”) behavior.

Must thermodynamic functions be piecewise analytic

In earlier papers, we have studied the turbulent flow exponents $$\zeta _p$$
 , where $$\langle |\Delta \mathbf{v}|^p\rangle \sim \ell ^{\zeta _p}$$
 and $$\Delta \mathbf{v}$$
 is the contribution to the fluid velocity at small scale $$\ell $$
 . Using ideas of non-equilibrium statistical mechanics we have found $$\begin{aligned} \zeta _p={p\over 3}-{1\over \ln \kappa }\ln \Gamma \left( {p\over 3}+1\right) \end{aligned}$$
 where $$1/\ln \kappa $$
 is experimentally $$\approx \,0.32\,\pm \,0.01$$
 . The purpose of the present note is to propose a somewhat more physical derivation of the formula for $$\zeta _p$$
 . We also present an estimate $$\approx \,100$$
 for the Reynolds number at the onset of turbulence.

A Theory of Hydrodynamic Turbulence Based on Non-equilibrium Statistical Mechanics

Field theory and statistical mechanics

We consider a quantum spin system consisting of a finite subsystem connected to infinite reservoirs at different temperatures. In this setup we define nonequilibrium steady states and prove that the rate of entropy production in such states is nonnegative.

David Ruelle

Papers

Differentiable Dynamical Systems

Must thermodynamic functions be piecewise analytic

A Theory of Hydrodynamic Turbulence Based on Non-equilibrium Statistical Mechanics

Field theory and statistical mechanics

Entropy production in quantum spin systems