China
Africa
Explore chapters and articles related to this topic
Coda: Fractals and the Mandelbrot Set
Published in A. David Wunsch, ® Companion to Complex Variables, 2018
In this book, I have sought to create a bridge between complex variable theory and the MATLAB programming language. It seems fitting to end with a branch of mathematics, which, although dating in its origins to the early 1900s, really did not come into flower until the widespread use of high-speed digital computers in the 1980s and cannot really be appreciated without a programming language such as MATLAB®. This is the subject known as fractals, which is a branch of a larger field in mathematics called chaos theory. The public is exposed to chaos theory in newspaper articles that speak of “the butterfly effect.” The term was coined in 1969 by Edward Lorenz, an MIT Professor of Earth Sciences, who is famous for his research in weather prediction. He is usually credited with the observation that if a butterfly flaps its wings, it might trigger a storm appearing many hundreds of miles away. To a mathematician, this is saying that the solution of a differential equation, or some iterative process, is acutely sensitive to the initial conditions imposed on the equations. Lorenz was at the time working on solutions of the Navier−Stokes equations, which are nonlinear differential equations that are at the heart of the theory of the behavior of fluids and therefore of the weather.
Historical Development of HRV Analysis
Published in Herbert F. Jelinek, David J. Cornforth, Ahsan H. Khandoker, ECG Time Series Variability Analysis, 2017
Correlation dimension (CD) analysis of the heartbeat time series is based on the algorithm of Grassberger and Procaccia (1983). The CD can be thought of as a measure of the number of independent variables needed to describe the total system in phase space (Bogaert et al. 2001). It can be used to quantify the complexity of a dynamic system or a HR time series. In the presence of chaos, an attractor in phase space characterizes the dynamics of the system and its complexity can be quantified in terms of the properties of the attractor (Beckers et al. 2006). Low values of CD indicate that the complexity of the system is lost and that sympathetic and vagal stimulation are necessary to create complex dynamical systems of HR variations (Bogaert et al. 2001). Kanters et al. (1996) claimed that the CD of HRV signals is mostly due to linear correlations between the RR intervals. In general, CD values are decreased in cardiac diseases in comparison to healthy subjects. Hence, the algorithm of Grassberger and Procaccia (1983) is only applicable for the long-term time series and an increasing interest to overcome this limitation is under way.
Concept of Randomness
Published in Franklin R. Nash, Reliability Assessments, 2017
In the kinetic theory of gases, for example, chance generated determinism. There are the reverse cases in which determinism generates chance, which is referred to as deterministic chaos or simply chaos. Determinism and randomness are linked by chaos. Chaos has been defined as—“stochastic behavior occurring in a deterministic system” [15]. It has also been defined as—“persistent pseudorandom motion in a deterministic dynamical system with exponential sensitivity to initial conditions” [16]. Pseudorandom refers to the generation of a pattern that is random according to statistical tests but completely specified by deterministic equations [17]. Chaos in a mathematically deterministic system is surprising for two reasons: (a) it can occur with no random input and (b) it can occur in very simple systems.
Pattern Detection on Glioblastoma’s Waddington Landscape via Generative Adversarial Networks
Published in Cybernetics and Systems, 2022
Fractal Analysis: The fractal dimension is a non-integer dimension, FD. A fractal dimension suggests the existence of a scaling law describing the complexity and roughness (self-similarity) of the pattern (Mandelbrot 1982). A fractal dimension is also a characteristic signature of deterministic chaos in dynamical systems (Frederickson et al. 1983). Strange attractors, the causal patterns to which the trajectories of a chaotic system are bound to, occupy fractal dimensions in phase-space (Ruelle 1980). The ImageJ plugin FracLac was used to compute the fractal dimension (FD) of the GAN-generated patterns. The images of the GAN-reconstructed attractors were enhanced in saturation to 25% and converted to binarized images on the ImageJ Plugin. FracLac then performs FD calculation via the Box-counting algorithm. The description of the Box-counting algorithm is as follows: Let L be the line length, be the box size, and be the number of boxes which can divide the pattern/object into self-similar substructures. The slope of the log-log plot of N and if it exists, provides the box-count fractal dimension of the object/pattern, as given by:
Understanding climate change through Earth’s energy flows
Published in Journal of the Royal Society of New Zealand, 2020
The best example of the tremendous success of this approach is numerical weather prediction, which uses a computer-based model of the atmosphere. Beginning with crude models in the 1950s, these have developed to include 50–100 levels in the vertical and horizontal resolutions of order 20 km. They have improved enormously, as has the observing system, including data and imagery from satellites, that are combined through a data assimilation process. Together they allow the state of the atmosphere at any time to be determined to a known level of accuracy through a combination of past predictions of the model, which essentially carries forward in time all of the past observed data, with the new observations. This observed state is then used as the starting point for new predictions for up to about 2 weeks. The predictability of each analysed situation can be determined by making imperceptible perturbations in the fields, and watching how well the predictions group or spread. Hence it is common now-a-days to use an ensemble of predictions to explore these aspects. Indeed, it is well established that even for a perfect model, the deterministic predictability of the atmosphere is of order 10 days before the chaotic aspects take over (Zhang et al. 2019). Here ‘chaos’ refers to the mathematics of sensitivity to small perturbations, often called ‘the butterfly effect’ after Ed Lorenz, who first discovered and documented mathematical chaos: how ‘a butterfly flapping its wings in Brazil can produce a tornado in Texas’ (see Gleick 1987 and Lorenz 1993).
Preface
Published in International Journal of Parallel, Emergent and Distributed Systems, 2018
Chaotic systems make a vitally important part of science and engineering at the theoretical and practical levels of research. Most interesting and applicable notions are, for example, chaos control and chaos synchronisation related to secure communications and hidden attractors among others. Recently, the study of chaos is focused not only on the traditional trends but also on the understanding and analysing principles aimed to controlling and utilising chaos in real-world applications. Well-known chaotic attractors and systems can even be produced by the simple three-dimensional autonomous system of ordinary differential equations, e.g. the Lorenz system, which originates from modelling of atmospheric dynamics, or discrete chaos as the logistic equation, based on a predator–prey model showing complex dynamical behaviours. These models and others can be found in the selected papers of this issue. This issue also studies an intersection of two interesting fields of research – deterministic chaos and evolutionary computation. The aim of this issue is to show how chaos can be used as a tool in research, e.g. its use in evolutionary algorithms or identification of the geothermal processes, or as an object of research.