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

We explore several basic aspects of chaos, chaotic systems, and non-linear dynamics through three different setups. The logistic map, a canonical one-dimensional system exhibiting surprisingly complex and aperiodic behavior, is modeled as a function of its chaotic parameter. We consider maps of its phase space, and the progression through period-doubling bifurcations to the onset of chaos. The Feigenbaum ratio of successive bifurcation periods is estimated at 4.674, in good agreement with the accepted value. The Liapunov exponent, governing the exponential growth of small perturbations in chaotic systems, is calculated and its fractal structure compared to the corresponding bifurcation diagram for the logistic map. Using a non-linear p-n junction circuit we analyze the return maps and power spectrums of the resulting time series at various types of system behavior. Similarly, an electronic analog to a ball bouncing on a vertically driven table provides insight into real-world applications of chaotic motion. For both systems we calculate the fractal information dimension and compare with theoretical behavior for dissipative versus Hamiltonian systems. Subject headings: non-linear dynamics; non-linear dynamical systems; fractal dimension; chaos; strange attractors; logistic map

Nonlinear dynamics and Chaos

Recurrence plots for the analysis of complex systems

Chaotic time series data are observed routinely in experiments on physical systems and in observations in the field. The authors review developments in the extraction of information of physical importance from such measurements. They discuss methods for (1) separating the signal of physical interest from contamination ("noise reduction"), (2) constructing an appropriate state space or phase space for the data in which the full structure of the strange attractor associated with the chaotic observations is unfolded, (3) evaluating invariant properties of the dynamics such as dimensions, Lyapunov exponents, and topological characteristics, and (4) model making, local and global, for prediction and other goals. They briefly touch on the effects of linearly filtering data before analyzing it as a chaotic time series. Controlling chaotic physical systems and using them to synchronize and possibly communicate between source and receiver is considered. Finally, chaos in space-time systems, that is, the dynamics of fields, is briefly considered. While much is now known about the analysis of observed temporal chaos, spatio-temporal chaotic systems pose new challenges. The emphasis throughout the review is on the tools one now has for the realistic study of measured data in laboratory and field settings. It is the goal of this review to bring these tools into general use among physicists who study classical and semiclassical systems. Much of the progress in studying chaotic systems has rested on computational tools with some underlying rigorous mathematics. Heuristic and intuitive analysis tools guided by this mathematics and realizable on existing computers constitute the core of this review.

The analysis of observed chaotic data in physical systems

: This thesis applies neural network feature selection techniques to multivariate time series data to improve prediction of a target time series. Two approaches to feature selection are used. First, a subset enumeration method is used to determine which financial indicators are most useful for aiding in prediction of the S&P 500 futures daily price. The candidate indicators evaluated include RSI, Stochastics and several moving averages. Results indicate that the Stochastics and RSI indicators result in better prediction results than the moving averages. The second approach to feature selection is calculation of individual saliency metrics. A new decision boundary-based individual saliency metric, and a classifier independent saliency metric are developed and tested. Ruck's saliency metric, the decision boundary based saliency metric, and the classifier independent saliency metric are compared for a data set consisting of the RSI and Stochastics indicators as well as delayed closing price values. The decision based metric and the Ruck metric results are similar, but the classifier independent metric agrees with neither of the other metrics. The nine most salient features, determined by the decision boundary based metric, are used to train a neural network and the results are presented and compared to other published results. (AN)

/pdf/nonlinear-time-series-analysis-1j3a35n4y3.pdf

Nonlinear Time Series Analysis.

This essential handbook provides the theoretical and experimental tools necessary to begin researching the nonlinear behavior of mechanical, electrical, optical, and other systems. The book describes several nonlinear systems which are realized by desktop experiments, such as an apparatus showing chaotic string vibrations, an LRC circuit displaying strange scrolling patterns, and a bouncing ball machine illustrating the period doubling route to chaos. Fractal measures, periodic orbit extraction, and symbolic analysis are applied to unravel the chaotic motions of these systems. The simplicity of the examples makes this an excellent book for undergraduate and graduate-level physics and mathematics courses, new courses in dynamical systems, and experimental laboratories.

An experimental approach to nonlinear dynamics and chaos

Accurate methods are needed to measure changing soil water content from meter to kilometer scales. Laboratory results demonstrate the feasibility of the heat pulse method implemented with fiber optic temperature sensing to obtain accurate distributed measurements of soil water content. A fiber optic cable with an electrically conductive armoring was buried in variably saturated sand and heated via electrical resistance to create thermal pulses monitored by observing the distributed Raman backscatter. A new and simple interpretation of heat data that takes advantage of the characteristics of fiber optic temperature measurements is presented. The accuracy of the soil water content measurements varied approximately linearly with water content. At volumetric moisture content of 0.05 m3/m3 the standard deviation of the readings was 0.001 m3/m3, and at 0.41 m3/m3 volumetric moisture content the standard deviation was 0.046 m3/m3. This uncertainty could be further reduced by averaging several heat pulse interrogations and through use of a higher?performance fiber optic sensing system.

/pdf/feasibility-of-soil-moisture-monitoring-with-heated-fiber-4hpn8jet9q.pdf

Feasibility of soil moisture monitoring with heated fiber optics

A well-defined order-disorder transition occurs in the capillary waves on a fluid layer driven by vertical oscillation. The transition is characterized by a sharp decline in both the translational correlation length and the long-range orientational order of the pattern, and an onset in the characteristic frequency ${f}^{*}$ of chaotic fluctuations varying approximately as ${(A\ensuremath{-}{A}_{d})}^{\frac{1}{2}}$, for driving amplitudes above a threshold ${A}_{d}$. The transition is geometry dependent even when the container size is 50-100 times the wavelength.

/pdf/order-disorder-transition-in-capillary-ripples-2kkz3hoguw.pdf

Order-disorder transition in capillary ripples.

We show how to characterize a strange attractor by a set of integers. These are extracted from the chaotic time-series data by first reconstructing the low-period orbits and then determining the template, or knot holder, which supports all periodic orbits embedded in the strange attractor, and the strange attractor itself. The template is identified by a set of integers which therefore characterize the strange attractor. This identification is explicitly demonstrated for the Pirogon using a relatively small data set (5000 points}.

https://www.physics.drexel.edu/~bob/Papers/RRR90b.pdf

Classification of strange attractors by integers.

An undergraduate experiment is described that illustrates the period doubling route to chaos in a simple dissipative mechanical system, a bouncing ball subject to repeated impacts with a vibrating table.

Nicholas B. Tufillaro

Papers

An experimental approach to nonlinear dynamics and chaos

Feasibility of soil moisture monitoring with heated fiber optics

Order-disorder transition in capillary ripples.

Classification of strange attractors by integers.

Chaotic dynamics of a bouncing ball