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

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Table Of Integrals Series And Products

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)

Since the subject of traffic dynamics has captured the interest of physicists, many surprising effects have been revealed and explained. Some of the questions now understood are the following: Why are vehicles sometimes stopped by ``phantom traffic jams'' even though drivers all like to drive fast? What are the mechanisms behind stop-and-go traffic? Why are there several different kinds of congestion, and how are they related? Why do most traffic jams occur considerably before the road capacity is reached? Can a temporary reduction in the volume of traffic cause a lasting traffic jam? Under which conditions can speed limits speed up traffic? Why do pedestrians moving in opposite directions normally organize into lanes, while similar systems ``freeze by heating''? All of these questions have been answered by applying and extending methods from statistical physics and nonlinear dynamics to self-driven many-particle systems. This article considers the empirical data and then reviews the main approaches to modeling pedestrian and vehicle traffic. These include microscopic (particle-based), mesoscopic (gas-kinetic), and macroscopic (fluid-dynamic) models. Attention is also paid to the formulation of a micro-macro link, to aspects of universality, and to other unifying concepts, such as a general modeling framework for self-driven many-particle systems, including spin systems. While the primary focus is upon vehicle and pedestrian traffic, applications to biological or socio-economic systems such as bacterial colonies, flocks of birds, panics, and stock market dynamics are touched upon as well.

Traffic and related self-driven many-particle systems

to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.

Linear and nonlinear waves, by G. B. Whitham. Pp.636. £50. 1999. ISBN 0 471 35942 4 (Wiley).

We present molecular dynamics simulations of granular material submitted to vibrations in a two-dimensional system. We find various types of convection cells, due either to the existence of walls or to spatial modulations in the amplitude of the vibration. The direction of the motion relative to the walls depends on shear friction. We measure the strength of the convection velocity and find a characteristic resonance frequency as in experiments. We propose an explanation for the mechanisms that are at the origin of the different motions.

Convection cells in vibrating granular media

The parameter space of the H\'enon map is reported to contain a regular structure-parallel-to-structure sequence of shrimp-shaped robust isoperiodic domains. They appear densely concentrated on a neighborhood along a main \ensuremath{\alpha} direction, extending across both orientation-preserving and -reversing domains. There is also a secondary \ensuremath{\beta} direction, roughly perpendicular to a very dense ``foliation of legs'' emanating from the isoperiodic domains. Familiar bifurcation phenomena observed in unimodal maps correspond to particular cuts along the \ensuremath{\beta} direction. The \ensuremath{\alpha} direction is rich in new phenomena. The topology along \ensuremath{\alpha} is conjectured to be typical of bimodal maps.

Structure of the parameter space of the Hénon map

This paper describes how certain shrimp-like clusters of stability organize themselves in the parameter space of dynamical systems. Clusters are composed of an infinite affine-similar repetition of a basic elementary cell containing two primary noble points, a head and a tail, defining an axis of approximate symmetry. Knowledge of the axis and the skewness of the k-periodic main cell of a k × 2" cluster is enough to define the orientation of the whole cluster in space. Peculiarly simple directions along which shrimp-like clusters align are formed by the locus of doubly degenerate saddle zero multipliers corresponding to the main shrimp head. In addition, we report a family of models having the boundaries of all isoperiodic domains of stability totally degenerate and describe different aspects of their mathematical arrangement and some of their consequences for example, that shrimps are diffeomorphic copies of shrimps.

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Dissecting shrimps: results for some one-dimensional physical models

It is well known, that the dynamics of small particles moving in a viscous fluid is strongly influenced by the long-range hydrodynamical interaction between them Motion at high viscosity is usually treated by means of the Stokes equations, which are linear and instantaneous Nevertheless, the hydrodynamical interaction mediated by the liquid is nonlinear; therefore the dynamics of more than two particles can be rather complex Here we present a high resolution numerical analysis of the classical three-particle Stokeslet problem in a vertical plane We show that a chaotic saddle in the phase space is responsible for the extreme sensitivity to initial configurations, which has been mentioned several times in the literature without an explanation A detailed analysis of the transiently chaotic dynamics and the underlying fractal patterns is given @S1063-651X~97!10408-1#

/pdf/chaotic-particle-dynamics-in-viscous-flows-the-three-1iju0oajyz.pdf

Chaotic particle dynamics in viscous flows: the three-particle stokeslet problem

We report the discovery of a remarkable "periodicity hub" inside the chaotic phase of an electronic circuit containing two diodes as a nonlinear resistance. The hub is a focal point from where an infinite hierarchy of nested spirals emanates. By suitably tuning two reactances simultaneously, both current and voltage may have their periodicity increased continuously without bound and without ever crossing the surrounding chaotic phase. Familiar period-adding current and voltage cascades are shown to be just restricted one-parameter slices of an exceptionally intricate and very regular onionlike parameter surface centered at the focal hub which organizes all the dynamics.

/pdf/periodicity-hub-and-nested-spirals-in-the-phase-diagram-of-a-2mvw8sho33.pdf

Jason A. C. Gallas

Papers

Convection cells in vibrating granular media

Structure of the parameter space of the Hénon map

Dissecting shrimps: results for some one-dimensional physical models

Chaotic particle dynamics in viscous flows: the three-particle stokeslet problem

Periodicity hub and nested spirals in the phase diagram of a simple resistive circuit.