One-Dimensional Maps.- Two-Dimensional Maps.- Chaos.- Fractals.- Chaos in Two-Dimensional Maps.- Chaotic Attractors.- Differential Equations.- Periodic Orbits and Limit Sets.- Chaos in Differential Equations.- Stable Manifolds and Crises.- Bifurcations.- Cascades.- State Reconstruction from Data.

Chaos: An Introduction to Dynamical Systems

We postulated that aging is associated with disruption in the fractallike long-range correlations that characterize healthy sinus rhythm cardiac interval dynamics. Ten young (21-34 yr) and 10 elderly (68-81 yr) rigorously screened healthy subjects underwent 120 min of continuous supine resting electrocardiographic recording. We analyzed the interbeat interval time series using standard time and frequency domain statistics and using a fractal measure, detrended fluctuation analysis, to quantify long-range correlation properties. In healthy young subjects, interbeat intervals demonstrated fractal scaling, with scaling exponents (alpha) from the fluctuation analysis close to a value of 1.0. In the group of healthy elderly subjects, the interbeat interval time series had two scaling regions. Over the short range, interbeat interval fluctuations resembled a random walk process (Brownian noise, alpha = 1.5), whereas over the longer range they resembled white noise (alpha = 0.5). Short (alpha s)- and long-range (alpha 1) scaling exponents were significantly different in the elderly subjects compared with young (alpha s = 1.12 +/- 0.19 vs. 0.90 +/- 0.14, respectively, P = 0.009; alpha 1 = 0.75 +/- 0.17 vs. 0.99 +/- 0.10, respectively, P = 0.002). The crossover behavior from one scaling region to another could be modeled as a first-order autoregressive process, which closely fit the data from four elderly subjects. This implies that a single characteristic time scale may be dominating heartbeat control in these subjects. The age-related loss of fractal organization in heartbeat dynamics may reflect the degradation of integrated physiological regulatory systems and may impair an individual's ability to adapt to stress.

/pdf/age-related-alterations-in-the-fractal-scaling-of-cardiac-m56nmf40sr.pdf

Age-related alterations in the fractal scaling of cardiac interbeat interval dynamics

Advanced Local Fractional Calculus and Its Applications

Introduction. Essentials of Complexity Theory for Organization Studies. Complexity Perspectives on Organizational Processes. Implications of Complexity Theory for Management Processes. Philosophical Issues in Applying Complexity Theory to Organizations.

/pdf/complex-systems-and-evolutionary-perspectives-on-3r95ld7fms.pdf

Complex systems and evolutionary perspectives on organisations: the application of complexity theory to organisations

Multilevel theory and research have advanced organizational science but are limited because the research focus is incomplete. Most quantitative research examines top-down, contextual, cross-level relationships. Emergent phenomena that manifest from the bottom up from the psychological characteristics, processes, and interactions among individuals—although examined qualitatively—have been largely neglected in quantitative research. Emergence is theoretically assumed, examined indirectly, and treated as an inference regarding the construct validity of higher level measures. As a result, quantitative researchers are investigating only one fundamental process of multilevel theory and organizational systems. This article advances more direct, dynamic, and temporally sensitive quantitative research methods designed to unpack emergence as a process. We argue that direct quantitative approaches, largely represented by computational modeling or agent-based simulation, have much to offer with respect to illuminatin...

Advancing Multilevel Research Design Capturing the Dynamics of Emergence

Chaos Under Control introduces the rich concepts driving the new science of complexity. Here you will discover the principles behind some of the most talked-about research findings of our time and see some astonishing examples and applications. And with the book's supplementary software, you can turn a Macintosh computer into a window of discovery, to watch fractal and chaotic phenomena come to life. "...outstanding book ...Their wide-ranging discussion of complexity is a mixture of fascinating description, unexpected examples, fancy graphics and equations." Times Higher

Chaos Under Control: The Art and Science of Complexity

Part I. Introductory Essays: 1. Some reasons for the effectiveness of fractal geometry in mathematics education Benoit B. Mandelbrot and Michael Frame 2. Unsolved problems and still emerging concepts Benoit B. Mandelbrot 3. Fractals, graphics and mathematics education Benoit B. Mandelbrot 4. Mathematics and society in the twentieth century Benoit B. Mandelbrot Part II. Classroom Experiences: 5. Teaching fractals and dynamical systems at the Hotchkiss school Melkana Brakalova and David Coughlin 6. Reflection on Wada basins: some fractals with a twist Dane Camp 7. Learning and teaching about fractals Donald M. Davis 8. The fractal geometry of the Mandelbrot set Robert L. Devaney 9. Fractals - energizing the mathematics classroom Viki Fegers and Mary Beth Johnson 10. Other chaos games Sandy Fillebrown 11. Creating and teaching undergraduate courses in fractal geometry: a personal experience Michel Lapidus 12. Exploring Fractal dimensions by experiment Ron Lewis 13. Fractal themes on all levels Kenneth G. Monks 14. Art and fractals: artistic explorations of natural self-similarity Brianna Murati and Michael Frame 15. Order and chaos, art and magic: a first college course in quantitative reasoning based on fractals and chaos David Peak and Michael Frame 16. A software driven undergraduate fractals course Douglas C. Ravenel Part III. A Final Word: 17. The fractal ring from art to art through mathematics, finance and the sciences Benoit B. Mandelbrot Part IV. Appendices: 18. Panorama of fractals and their uses. An alphabetic workbook-index Michael Frame and Benoit B. Mandelbrot 19. Reports of some field experiences.

Fractals, Graphics, and Mathematics Education

Mathematics is about theorems. The job of a mathematician is to prove theorems and to explain the proofs to others. This was more or less the vision of the profession I got as an undergraduate and graduate student in the 1970s. Yet when I worked to understand a proof, and especially when I tried to produce my own proofs, I always kept examples close at hand. Slowly I began to realize that there is an experimental aspect of mathematics: conjectures are tested against examples. The more tests a conjecture passes, the more likely it is to survive the transition from conjecture to theorem. Moreover, studying an especially well-constructed example can reveal hints of a proof of a theorem. This was not a profound discovery of a secret hidden from the uninitiated, a rite of passage through which every young mathematician must go. Indeed, many of my teachers emphasized again and again the importance of examples. But in the 1970s, in most fields there was little difference in the methodologies of constructing examples and proofs. To be sure, there were exceptions. Some finite-group theorists, for instance, embraced the computer early. I remember days as a graduate student, listening to Mark Benard enthusiastically describe his computer analyses. At the time, this seemed an anomaly: most people constructed examples and theorems by pure cogitation. An immense chasm appeared to separate mathematics from, say, experimental physics. The landscape has changed over the last two decades. Now computer experiments are done in all fields of mathematics-certainly not by every practitioner, but so far as I know, every field has its experimentalists. If someone has a counterexample, I'd be happy to hear it. This addition of computer experiments to the standard tools of mathematics has not been painless. Not many years ago, a colleague objected, rather heatedly it seemed to me, to my statement that computer experiments would influence all fields of mathematical research: he proclaimed that he would never use a computer for anything other than calculating

The Computational Beauty of Nature: Computer Explorations of Fractals, Chaos, Complex Systems, and Adaptation.By Gary William Flake

A generalized mandelbrot set and the role of critical points

An infinite circle inversion limit set fractal

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