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

/pdf/testing-for-nonlinearity-in-time-series-the-method-of-3obg021lq4.pdf

Testing for nonlinearity in time series: the method of surrogate data

We describe a statistical approach for identifying nonlinearity in time series; in particular, we want to avoid claims of chaos when simpler models (such as linearly correlated noise) can explain the data. The method requires a careful statement of the null hypothesis which characterizes a candidate linear process, the generation of an ensemble of surrogate'' data sets which are similar to the original time series but consistent with the null hypothesis, and the computation of a discriminating statistic for the original and for each of the surrogate data sets. The idea is to test the original time series against the null hypothesis by checking whether the discriminating statistic computed for the original time series differs significantly from the statistics computed for each of the surrogate sets. We present algorithms for generating surrogate data under various null hypotheses, and we show the results of numerical experiments on artificial data using correlation dimension, Lyapunov exponent, and forecasting error as discriminating statistics. Finally, we consider a number of experimental time series -- including sunspots, electroencephalogram (EEG) signals, and fluid convection -- and evaluate the statistical significance of the evidence for nonlinear structure in each case. 56 refs., 8 figs.

/pdf/testing-for-nonlinearity-in-time-series-the-method-of-3dx4o87iwx.pdf

Testing for nonlinearity in time series: The method of surrogate data

Recurrence plots for the analysis of complex systems

A practical method for calculating largest Lyapunov exponents from small data sets

We analyze in detail an algorithm for computing Liapunov exponents from an experimental time series. As an application, a hydrodynamic experiment is investigated.

Liapunov exponents from time series.

One-dimensional profiles f(x) can be characterized by a Minkowski-Bouligand dimension D and by a scale-dependent generalized roughness W(f, epsilon ). This roughness can be defined as the dispersion around a chosen fit to f(x) in an epsilon -scale. It is shown that D=limepsilon to 0(2-In W(f, epsilon )/In epsilon ) holds for profiles nowhere differentiable. This establishes a close connection between the roughness and the fractal dimension and proves that D=2-H for self-affine profiles (H is the roughness or Hurst exponent). Two numerical algorithms based on the roughness, one around the local average (f(x))epsilon (usual roughness) and the other around the local RMS straight line (a generalized roughness), are discussed. The estimates of D for standard self-affine profiles are reliable and robust, especially for the last method.

https://iopscience.iop.org/article/10.1088/0305-4470/27/24/018/pdf

On the fractal dimension of self-affine profiles

The breathing circle is a two-dimensional generalization of the Fermi accelerator. It is shown that the billiard map associated to this model has invariant curves in phase space, implying that any particle will have bounded gain of energy.

http://iopscience.iop.org/article/10.1088/0951-7715/12/5/310/pdf

Bounded gain of energy on the breathing circle billiard

In this paper we use discrete dynamical systems formalism to carefully investigate the effect of a time-dependent perturbation on a classical mechanical model. We present a study of a one-dimensional Fermi accelerator model and its parametric dependence on the amplitude of the movement of the wall. We focus on the low-energy region, in which the system may present chaotic behavior illustrated by orbits with stochastic behavior and sensitive dependence on initial conditions. Through numerical analysis, this behavior is quantitatively characterized by a positive Lyapunov exponent and by the position of the boundary of the observed ergodic component. Our results will be analyzed by comparing this model to a simplified version and to the Standard Map. These results indicate that the dynamics of the model is, in a certain sense, very weakly dependent of the parameter.

On the dynamical properties of a Fermi accelerator model

This is an attempt to study mathematically billiards with moving boundaries. We assume that the boundary remains closed, regular and strictly convex, deforming periodically in time, in the normal direction. We describe the associated billiard diffeomorphism and the corresponding invariant measure. We discuss the stability of 2-periodic orbits and investigate the boundedness of the velocity in some precise examples. Finally, we present the Hamiltonian formalism and the symplectic structure, considering that a moving billiard is a billiard with rigid boundary on an augmented configuration space, with a singular metric.

Sylvie Oliffson Kamphorst

Papers

Liapunov exponents from time series.

On the fractal dimension of self-affine profiles

Bounded gain of energy on the breathing circle billiard

On the dynamical properties of a Fermi accelerator model

Time-dependent billiards