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.

/pdf/pattern-formation-outside-of-equilibrium-1601b9znum.pdf

Pattern formation outside of equilibrium

It has often been remarked that turbulence is a subject of great scientific and technological importance, and yet one of the least understood (e.g. McComb 1990). To an outsider this may seem strange, since the basic physical laws of fluid mechanics are well established, an excellent mathematical model is available in the Navier-Stokes equations, and the results of well over a century of increasingly sophisticated experiments are at our disposal. One major difficulty, of course, is that the governing equations are nonlinear and little is known about their solutions at high Reynolds number, even in simple geometries. Even mathematical questions as basic as existence and uniqueness are unsettled in three spatial dimensions (cf Temam 1988). A second problem, more important from the physical viewpoint, is that experiments and the available mathematical evidence all indicate that turbulence involves the interaction of many degrees of freedom over broad ranges of spatial and temporal scales. One of the problems of turbulence is to derive this complex picture from the simple laws of mass and momentum balance enshrined in the NavierStokes equations. It was to this that Ruelle & Takens (1971) contributed with their suggestion that turbulence might be a manifestation in physical

/pdf/the-proper-orthogonal-decomposition-in-the-analysis-of-44vs42u998.pdf

The Proper Orthogonal Decomposition in the Analysis of Turbulent Flows

Preface Part I. Turbulence: 1. Introduction 2. Coherent structures 3. Proper orthogonal decomposition 4. Galerkin projection Part II. Dynamical Systems: 5. Qualitative theory 6. Symmetry 7. One-dimensional 'turbulence' 8. Randomly perturbed systems Part III. 9. Low-dimensional Models: 10. Behaviour of the models Part IV. Other Applications and Related Work: 11. Some other fluid problems 12. Review: prospects for rigor Bibliography.

/pdf/turbulence-coherent-structures-dynamical-systems-and-fqofnixbu9.pdf

Turbulence, Coherent Structures, Dynamical Systems and Symmetry

[1] The analysis of univariate or multivariate time series provides crucial information to describe, understand, and predict climatic variability. The discovery and implementation of a number of novel methods for extracting useful information from time series has recently revitalized this classical field of study. Considerable progress has also been made in interpreting the information so obtained in terms of dynamical systems theory. In this review we describe the connections between time series analysis and nonlinear dynamics, discuss signal-to-noise enhancement, and present some of the novel methods for spectral analysis. The various steps, as well as the advantages and disadvantages of these methods, are illustrated by their application to an important climatic time series, the Southern Oscillation Index. This index captures major features of interannual climate variability and is used extensively in its prediction. Regional and global sea surface temperature data sets are used to illustrate multivariate spectral methods. Open questions and further prospects conclude the review.

/pdf/advanced-spectral-methods-for-climatic-time-series-4hxv68i0r0.pdf

Advanced spectral methods for climatic time series

Hidden physics models: Machine learning of nonlinear partial differential equations

Contents: Introduction.- Presentation of the Approach and of the Main Results.- The Transport of Finite Dimensional Contact Elements.- Spectral Blocking Property.- Strong Squeezing Property.- Cone Invariance Properties.- Consequences Regarding the Global Attractor.- Local Exponential Decay Toward Blocked Integral Surfaces.- Exponential Decay of Volume Elements and the Dimension of the Global Attractor.- Choice of the Initial Manifold.- Construction of the Inertial Mainfold.- Lower Bound for the Exponential Rate of Convergence to the Attractor.- Asymptotic Completeness: Preparation.- Asymptotic Completeness: Proof of Theorem 12.1.- Stability with Respect to Perturbations.- Application: The Kuramoto-Sivashinsky Equation.- Application: A Nonlocal Burgers Equation.- Application: The Cahn-Hilliard Equation.- Application: A parabolic Equation in Two Space Variables.- Application: The Chaffee-Infante Reaction Diffusion Equation.- References.- Index.

Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations

Some global dynamical properties of the Kuramoto-Sivashinsky equations: Nonlinear stability and attractors

Inertial Manifolds for the Kuramoto-Sivashinsky Equation and an Estimate of their Lowest Dimension

The complex Ginzburg-Landau equation in one spatial dimension with periodic boundary conditions is studied from the viewpoint of effective low-dimensional behaviour by three distinct methods. Linear stability analysis of a class of exact solutions establishes lower bounds on the dimension of the universal, or global, attractor and the Fourier spanning dimension, defined as the number of Fourier modes required to span the universal attractor. The authors use concepts from the theory of inertial manifolds to determine rigorous upper bounds on the Fourier spanning dimension, which also establishes the finite dimensionality of the universal attractor. Upper bounds on the dimension of the attractor itself are obtained by bounding (or, for some parameter values, computing exactly) the Lyapunov dimension and invoking a recent theorem that asserts that the Lyapunov dimension, defined by the Kaplan-Yorke formula with the universal (global) Lyapunov exponents, is an upper bound on the Hausdorff dimension. This study of low dimensionality in the complex Ginzburg-Landau equation allows for an examination of the current techniques used in the rigorous investigation of finite-dimensional behaviour. Contact is made with some recent results for fluid turbulence models, and the authors discuss some unexplored directions in the area of low-dimensional behaviour in the complex Ginzburg-Landau equation.

https://iopscience.iop.org/article/10.1088/0951-7715/1/2/001/pdf

Low-dimensional behaviour in the complex Ginzburg-Landau equation

Existence, uniqueness and properties of asymptotic behavior are proved for solutions of the Milne and Kramers problems for the linearized Boltzmann equation for a gas of hard spheres. The proof uses energy-like estimates and follows the analysis of Bardos, Santos and Sentis for the equations of neutron transport.

B. Nicolaenko

Papers

Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations

Some global dynamical properties of the Kuramoto-Sivashinsky equations: Nonlinear stability and attractors

Inertial Manifolds for the Kuramoto-Sivashinsky Equation and an Estimate of their Lowest Dimension

Low-dimensional behaviour in the complex Ginzburg-Landau equation

The milne and kramers problems for the boltzmann equation of a hard sphere gas