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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. %hile 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.

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Exponential smoothing is a formalization of the familiar learning process, which is a practical basis for statistical forecasting. Higher orders of smoothing are defined by the operator Snt(x) = αSn−1t(x) + (1 − α) Snt−1(x), where S0t(x) = xt, 0 < α < 1. If one assumes that the time series of observations {xt} is of the form xt = nt + ∑ı=Nı=0 aıtı where nt is a sample from some error population, then least squares estimates of the coefficients a, can be obtained from linear combinations of the operators S, S2, …, SN+1. Explicit forms of the forecasting equations are given for N = 0, 1, and 2. This result makes it practical to use higher order polynomials as forecasting models, since the smoothing computations are very simple, and only a minimum of historical statistics need be retained in the file from one forecast to the next.

Fractal boundaries can occur for certain situations involving chaotic Hamiltonian systems. In particular, situations are considered in which an orbit can exit from the system in one of several different ways, and the question is asked which of these ways applies for a given initial condition.As an illustration, specific examples are considered for which there are two possible ways in which a particle can exit from the system. We examine the space of initial conditions to see which of the two exit possibilities applies for each initial condition. It is found that the regions of initialcondition state space corresponding to the two exit modes are separated by a boundary that has both fractal and smooth {nonfractal) regions, for one example, and by a fractal boundary for the other example. Furthermore, it is found for the example where the boundary has fractal and smooth regions that these regions are intertwined on arbitrarily fine scale. The existence of fractal boundaries is conjectured to be a typical property of chaotic Hamiltonian dynamics with multiple exit modes. Two situations in space physics where our results may be relevant are discussed.

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Tendinopathy accounts for over 30% of primary care consultations and represents a growing healthcare challenge in an active and increasingly ageing population. Recognising critical cells involved in tendinopathy is essential in developing therapeutics to meet this challenge. Tendon cells are heterogenous and sparsely distributed in a dense collagen matrix; limiting previous methods to investigate cell characteristics ex vivo. We applied next generation CITE-sequencing; combining surface proteomics with in-depth, unbiased gene expression analysis of > 6400 single cells ex vivo from 11 chronically tendinopathic and 8 healthy human tendons. Immunohistochemistry validated the single cell findings. For the first time we show that human tendon harbours at least five distinct COL1A1/2 expressing tenocyte populations in addition to endothelial cells, T-cells, and monocytes. These consist of KRT7/SCX + cells expressing microfibril associated genes, PTX3 + cells co-expressing high levels of pro-inflammatory markers, APOD + fibro–adipogenic progenitors, TPPP3/PRG4 + chondrogenic cells, and ITGA7 + smooth muscle-mesenchymal cells. Surface proteomic analysis identified markers by which these sub-classes could be isolated and targeted in future. Chronic tendinopathy was associated with increased expression of pro-inflammatory markers PTX3 , CXCL1, CXCL6, CXCL8, and PDPN by microfibril associated tenocytes. Diseased endothelium had increased expression of chemokine and alarmin genes including IL33.