http://www.maths.qmul.ac.uk/%7Elatora/report_06.pdf

Complex networks: Structure and dynamics

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

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)

Synchronization phenomena in large populations of interacting elements are the subject of intense research efforts in physical, biological, chemical, and social systems. A successful approach to the problem of synchronization consists of modeling each member of the population as a phase oscillator. In this review, synchronization is analyzed in one of the most representative models of coupled phase oscillators, the Kuramoto model. A rigorous mathematical treatment, specific numerical methods, and many variations and extensions of the original model that have appeared in the last few years are presented. Relevant applications of the model in different contexts are also included.

/pdf/the-kuramoto-model-a-simple-paradigm-for-synchronization-2akwf0t24m.pdf

The Kuramoto model: A simple paradigm for synchronization phenomena

/pdf/from-kuramoto-to-crawford-exploring-the-onset-of-3wpvjokr2x.pdf

From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators

https://www.math.fsu.edu/~okhanmoh/media/Cox,%20Matthews,%20JCP,%202002,%20Exponential%20Time%20Differencing%20for%20Stiff%20Systems.pdf

Exponential Time Differencing for Stiff Systems

We analyze a model of globally coupled nonlinear oscillators with randomly distributed frequencies. Twenty-five years ago it was conjectured that, for coupling strengths below a certain threshold, this system would always relax to an incoherent state. We prove this conjecture for the system linearized about the incoherent state, for frequency distributions with compact support. The relaxation is exponentially fast at intermediate times but slower than exponential at long times. The decay mechanism is remarkably similar to Landau damping in plasmas, even though the model was originally inspired by biological rhythms.

Coupled nonlinear oscillators below the synchronization threshold: Relaxation by generalized Landau damping.

Dynamics of a large system of coupled nonlinear oscillators

We analyze a large dynamical system of limit-cycle oscillators with mean-field coupling and randomly distributed natural frequencies. Depending on the choice of coupling strength and the spread of natural frequencies, the system exhibits frequency locking, amplitude death, and incoherence, as well as novel unsteady behavior characterized by periodic, quasiperiodic, or chaotic evolution of the system's order parameter. The phase boundaries between several of these states are obtained analytically.

Phase diagram for the collective behavior of limit-cycle oscillators.

Pattern formation in systems with a conserved quantity is considered by studying the appropriate amplitude equations. The conservation law leads to a
 large-scale neutral mode that must be included in the asymptotic analysis for pattern formation near onset. Near a stationary bifurcation, the usual
 Ginzburg--Landau equation for the amplitude of the pattern is then coupled to an equation for the large-scale mode. These amplitude equations show
 that for certain parameters all roll-type solutions are unstable. This new instability differs from the Eckhaus instability in that it is amplitude-driven and is
 supercritical. Beyond the stability boundary, there exist stable stationary solutions in the form of strongly modulated patterns. The envelope of these
 modulations is calculated in terms of Jacobi elliptic functions and, away from the onset of modulation, is closely approximated by a sech profile.
 Numerical simulations indicate that as the modulation becomes more pronounced, the envelope broadens. A number of applications are considered,
 including convection with fixed-flux boundaries and convection in a magnetic field, resulting in new instabilities for these systems.

/pdf/pattern-formation-with-a-conservation-law-1myawt8rlv.pdf

Paul C. Matthews

Papers

Exponential Time Differencing for Stiff Systems

Coupled nonlinear oscillators below the synchronization threshold: Relaxation by generalized Landau damping.

Dynamics of a large system of coupled nonlinear oscillators

Phase diagram for the collective behavior of limit-cycle oscillators.

Pattern formation with a conservation law