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

1980 Preface * 1999 Preface * 1999 Acknowledgements * Introduction * 1 Circular Logic * 2 Phase Singularities (Screwy Results of Circular Logic) * 3 The Rules of the Ring * 4 Ring Populations * 5 Getting Off the Ring * 6 Attracting Cycles and Isochrons * 7 Measuring the Trajectories of a Circadian Clock * 8 Populations of Attractor Cycle Oscillators * 9 Excitable Kinetics and Excitable Media * 10 The Varieties of Phaseless Experience: In Which the Geometrical Orderliness of Rhythmic Organization Breaks Down in Diverse Ways * 11 The Firefly Machine 12 Energy Metabolism in Cells * 13 The Malonic Acid Reagent ('Sodium Geometrate') * 14 Electrical Rhythmicity and Excitability in Cell Membranes * 15 The Aggregation of Slime Mold Amoebae * 16 Numerical Organizing Centers * 17 Electrical Singular Filaments in the Heart Wall * 18 Pattern Formation in the Fungi * 19 Circadian Rhythms in General * 20 The Circadian Clocks of Insect Eclosion * 21 The Flower of Kalanchoe * 22 The Cell Mitotic Cycle * 23 The Female Cycle * References * Index of Names * Index of Subjects

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/596B270197FE27DE5FABB598B6210622/S0025557200150892a.pdf/div-class-title-span-class-italic-the-geometry-of-biological-time-span-by-a-t-winfree-pp-544-dm68-corrected-second-printing-1990-isbn-3-540-52528-9-springer-div.pdf

The geometry of biological time , by A. T. Winfree. Pp 544. DM68. Corrected Second Printing 1990. ISBN 3-540-52528-9 (Springer)

Low Molecular Mass Gelators of Organic Liquids and the Properties of Their Gels

The hippocampal formation can encode relative spatial location, without reference to external cues, by the integration of linear and angular self-motion (path integration). Theoretical studies, in conjunction with recent empirical discoveries, suggest that the medial entorhinal cortex (MEC) might perform some of the essential underlying computations by means of a unique, periodic synaptic matrix that could be self-organized in early development through a simple, symmetry-breaking operation. The scale at which space is represented increases systematically along the dorsoventral axis in both the hippocampus and the MEC, apparently because of systematic variation in the gain of a movement-speed signal. Convergence of spatially periodic input at multiple scales, from so-called grid cells in the entorhinal cortex, might result in non-periodic spatial firing patterns (place fields) in the hippocampus.

Path integration and the neural basis of the 'cognitive map'

Unexpected sudden catastrophic shifts may occur in ecosystems, with concomitant losses or gains of ecological and economic resources. Such shifts have been theoretically attributed to positive feedback and bistability of ecosystem states. However, verifications and predictive power with respect to catastrophic responses to a changing environment are lacking for spatially extensive ecosystems. This situation impedes management and recovery strategies for such ecosystems. Here, we review recent studies on various ecosystems that link self-organized patchiness to catastrophic shifts between ecosystem states.

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Self-Organized Patchiness and Catastrophic Shifts in Ecosystems

We report the experimental observation of a sustained standing nonequilibrium chemical pattern in a single-phase open reactor. Considering the properties of the pattern (symmetry breaking, intrinsic wavelength), it is interpreted as the first unambiguous experimental evidence of a Turing structure.

Experimental evidence of a sustained standing Turing-type nonequilibrium chemical pattern.

Transitions from bistability to limit cycle oscillations. Theoretical analysis and experimental evidence in an open chemical system

Turing-type chemical patterns in the chlorite-iodide-malonic acid reaction

Experimental studies of the formation of Turing patterns in the (chlorine dioxide, iodine, malonic acid) reaction are performed in a spatial open gel disk reactor where all the input species are fed onto one side by a continuous stirred tank reactor. This setup is shown to fit the pool-chemical approximation used in most theoretical approaches. Nonequilibrium phase diagrams are established as a function of concentrations in the input flows. In agreement with theoretical predictions, the location of the transition from uniform steady states to Turing patterns is found to be almost independent of the concentrations of the complexing agent which controls the effective diffusion of activatory species. Extensive analytical and numerical calculations in two and three dimensions are performed on the basis of the Lengyel−Rabai−Epstein kinetic model and its two-variable reduction. This particular experimental configuration is shown to minimize the problems encountered with more commonly used versions of spatial op...

Experimental Studies and Quantitative Modeling of Turing Patterns in the (Chlorine Dioxide, Iodine, Malonic Acid) Reaction

We report the experimental observation of an endogeneous antisymmetric wave source in a quasi-one-dimensional chemical system. Substantiated by numerical simulations, a theoretical interpretation relying on the interaction between Turing and Hopf modes is proposed.

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P. De Kepper

Papers

Experimental evidence of a sustained standing Turing-type nonequilibrium chemical pattern.

Transitions from bistability to limit cycle oscillations. Theoretical analysis and experimental evidence in an open chemical system

Turing-type chemical patterns in the chlorite-iodide-malonic acid reaction

Experimental Studies and Quantitative Modeling of Turing Patterns in the (Chlorine Dioxide, Iodine, Malonic Acid) Reaction

One-dimensional "spirals": Novel asynchronous chemical wave sources.