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

The description of coherent features of fluid flow is essential to our understanding of fluid-dynamical and transport processes. A method is introduced that is able to extract dynamic information from flow fields that are either generated by a (direct) numerical simulation or visualized/measured in a physical experiment. The extracted dynamic modes, which can be interpreted as a generalization of global stability modes, can be used to describe the underlying physical mechanisms captured in the data sequence or to project large-scale problems onto a dynamical system of significantly fewer degrees of freedom. The concentration on subdomains of the flow field where relevant dynamics is expected allows the dissection of a complex flow into regions of localized instability phenomena and further illustrates the flexibility of the method, as does the description of the dynamics within a spatial framework. Demonstrations of the method are presented consisting of a plane channel flow, flow over a two-dimensional cavity, wake flow behind a flexible membrane and a jet passing between two cylinders.

https://hal-polytechnique.archives-ouvertes.fr/hal-01020654/document

Dynamic mode decomposition of numerical and experimental data

Bose-Einstein condensation (BEC) has been observed in a dilute gas of sodium atoms. A Bose-Einstein condensate consists of a macroscopic population of the ground state of the system, and is a coherent state of matter. In an ideal gas, this phase transition is purely quantum-statistical. The study of BEC in weakly interacting systems which can be controlled and observed with precision holds the promise of revealing new macroscopic quantum phenomena that can be understood from first principles.

Bose-Einstein condensation in a gas of sodium atoms

In principle, the exponential of a matrix could be computed in many ways. Methods involving approximation theory, differential equations, the matrix eigenvalues, and the matrix characteristic polyn...

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Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later ⁄

Numerical Initial Value Problems in Ordinary Differential Equations

The flat interface between two fluids in a vertically vibrating vessel may be parametrically excited, leading to the generation of standing waves. The equations constituting the stability problem for the interface of two viscous fluids subjected to sinusoidal forcing are derived and a Floquet analysis is presented. The hydrodynamic system in the presence of viscosity cannot be reduced to a system of Mathieu equations with linear damping. For a given driving frequency, the instability occurs only for certain combinations of the wavelength and driving amplitude, leading to tongue-like stability zones. The viscosity has a qualitative effect on the wavelength at onset: at small viscosities, the wavelcngth decreases with increasing viscosity, while it increases for higher viscosities. The stability threshold is in good agreement with experimental results. Based on the analysis, a method for the measurement of the interfacial tension, and the sum of densities and dynamic viscosities of two phases of a fluid near the liquid-vapour critical point is proposed.

Parametric instability of the interface between two fluids

Free Oscillator - Damped Oscillator Forced Oscillator - Parametric Oscillator The Fourier Transform Poincare Sections Three Examples of Dynamical Systems To Chaos: Temporal Chaos in Dissipative System Strange Attractors Quasiperiodicity The Subharmonic Cascade Intermittency Debate Appendixes Index.

Order within chaos : towards a deterministic approach to turbulence

Two-dimensional reaction-diffusion equations with simple reaction kinetics are used to study the dynamics of spiral waves in excitable media. Detailed numerical results are presented for the transition from simple (periodic) rotation to compound (quasiperiodic) rotation of spiral waves. It is shown that this transition occurs via a supercritical Hopf bifurcation and that there is no frequency locking within the quasiperiodic regime.

Spiral-wave dynamics in a simple model of excitable media: The transition from simple to compound rotation.

Spherical Couette flow is studied with a view to elucidating the transitions between various axisymmetric steady‐state flow configurations. A stable, equatorially asymmetric state discovered by Buhler [Acta Mech. 81, 3 (1990)] consists of two Taylor vortices, one slightly larger than the other and straddling the equator. By adapting a pseudospectral time‐stepping formulation to enable stable and unstable steady states to be computed (by Newton’s method) and linear stability analysis to be conducted (by Arnoldi’s method), the bifurcation‐theoretic genesis of the asymmetric state is analyzed. It is found that the asymmetric branch originates from a pitchfork bifurcation; its stabilization, however, occurs via a subsequent subcritical Hopf bifurcation.

Asymmetry and Hopf bifurcation in spherical Couette flow

Turbulent-laminar patterns near transition are simulated in plane Couette flow using an extension of the minimal-flow-unit methodology. Computational domains are of minimal size in two directions but large in the third. The long direction can be tilted at any prescribed angle to the streamwise direction. Three types of patterned states are found and studied: periodic, localized, and intermittent. These correspond closely to observations in large-aspect-ratio experiments.

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Laurette S. Tuckerman

Papers

Parametric instability of the interface between two fluids

Order within chaos : towards a deterministic approach to turbulence

Spiral-wave dynamics in a simple model of excitable media: The transition from simple to compound rotation.

Asymmetry and Hopf bifurcation in spherical Couette flow

Computational study of turbulent laminar patterns in couette flow.