"...a blend of erudition (fascinating and sometimes obscure historical minutiae abound), popularization (mathematical rigor is relegated to appendices) and exposition (the reader need have little knowledge of the fields involved) ...and the illustrations include many superb examples of computer graphics that are works of art in their own right." Nature

The Fractal Geometry of Nature

A new method for analysing nonlinear and non-stationary data has been developed. The key part of the method is the empirical mode decomposition method with which any complicated data set can be dec...

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The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis

Finite systems of deterministic ordinary nonlinear differential equations may be designed to represent forced dissipative hydrodynamic flow. Solutions of these equations can be identified with trajectories in phase space For those systems with bounded solutions, it is found that nonperiodic solutions are ordinarily unstable with respect to small modifications, so that slightly differing initial states can evolve into consider­ably different states. Systems with bounded solutions are shown to possess bounded numerical solutions.

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Deterministic nonperiodic flow

/pdf/determining-lyapunov-exponents-from-a-time-series-3bxxq6d0wt.pdf

Determining Lyapunov exponents from a time series

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

The available potential energy of the atmosphere may be defined as the difference between the total potential energy and the minimum total potential energy which could result from any adiabatic redistribution of mass. It vanishes if the density stratification is horizontal and statically stable everywhere, and is positive otherwise. It is measured approximately by a weighted vertical average of the horizontal variance of temperature. In magnitude it is generally about ten times the total kinetic energy, but less than one per cent of the total potential energy. Under adiabatic flow the sum of the available potential energy and the kinetic energy is conserved, but large increases in available potential energy are usually accompanied by increases in kinetic energy, and therefore involve nonadiabatic effects. Available potential energy may be partitioned into zonal and eddy energy by an analysis of variance of the temperature field. The zonal form may be converted into the eddy form by an eddy-transport of sensible heat toward colder latitudes, while each form may be converted into the corresponding form of kinetic energy. The general circulation is characterized by a conversion of zonal available potential energy, which is generated by low-latitude heating and high-latitude cooling, to eddy available potential energy, to eddy kinetic energy, to zonal kinetic energy. DOI: 10.1111/j.2153-3490.1955.tb01148.x

https://www.tandfonline.com/doi/pdf/10.3402/tellusa.v7i2.8796

Available Potential Energy and the Maintenance of the General Circulation

Glimpses of chaos a journey into chaos our chaotic weather encounters with chaos what else is chaos?

The Essence Of Chaos

Two states of the atmosphere which are observed to resemble one another are termed analogues. Eitherstate of a pair of analogues may be regarded as equal to the other state plus a small superposed "error."From the behavior of the atmosphere following each state, the growth rate of the error may be determined. Five years of twice-daily height values of the 200-, 500-, and 850-mb surfaces at a grid of 1003 pointsover the Northern Hemisphere are procured. A weighted root-mean-square height difference is used as ameasure of the difference between two states, or the error. For each pair of states occurring within onemonth of the same time of year, but in different years, the error is computed. There are numerous mediocre analogues but no truly good ones. The smallest errors have an averagedoubling time of about 8 days. Larger errors grow less rapidly. Extrapolation with the aid of a quadratichypothesis indicates that truly small errors would double in about 2.5 days. These rates may be comparedwith a ...

Edward N. Lorenz

Papers

Deterministic nonperiodic flow

Available Potential Energy and the Maintenance of the General Circulation

The Essence Of Chaos

Atmospheric Predictability as Revealed by Naturally Occurring Analogues

Predictability-A problem partly solved