Fractal Control And Its Applications

The book focuses on fractal control and applications in various fields. Fractal phenomena occur in nonlinear models, and since the behaviors depicted by fractals need to be controlled in practical applications, an understanding of fractal control is necessary. This book introduces readers to Julia set fractals and Mandelbrot set fractals in a range of models, such as physical systems, biological systems and SIRS models, and discusses controllers designed to control these fractals. Further, it demonstrates how the fractal dimension can be calculated in order to describe the complexity of various systems.Offering a comprehensive and systematic overview of the practical issues in fractal control, this book is a valuable resource for readers interested in practical solutions in fractal control. It will also appeal to researchers, engineers, and graduate students in fields of fractal control and applications, as well as chaos control and applications.

This book focuses on the control of fractal behaviors in nonlinear dynamics systems, addressing both the principles and purposes of control. For fractals in different systems, it presents revealing studies on the theory and applications of control, reflecting a spectrum of different control methods used with engineering technology. As such, it will benefit researchers, engineers, and graduate students in fields of fractals, chaos, engineering, etc.

Applications of Fractals and Chaos presents new developments in this rapidlydeveloping subject area. The presentation is more than merely theoretical, it specifically presents particular applications in a wide range of applications areas. Under the oceans, we consider the ways in which sponges and corals grow; we look, too, at the stability of ships on their surfaces. Land itself is modelled and applications to art, medicineand camouflage are presented. Readers should find general interest in the range of areas considered and should also be able to discover methods of value for their own specific areas of interest from studying the structure of related activities.

Fractal analysis has entered a new era. The applications to different areas of knowledge have been surprising. Let us begin with the fractional calculus-fractal geometry relationship, which allows for modeling with extreme precision of phenomena such as diffusion in porous media with fractional partial differential equations in fractal objects. Where the order of the equation is the same as the fractal dimension, this allows us to make calculations with enormous precision in diffusion phenomena-particularly in the oil industry, for new spillage prevention. Main applications to industry, design of fractal antennas to receive all frequencies and that is used in all cell phones, spacecraft, radars, image processing, measure, porosity, turbulence, scattering theory. Benoit Mandelbrot, creator of fractal geometry, would have been surprised by the use of fractal analysis presented in this book: "Part I: Petroleum Industry and Numerical Analysis"; "Part II: Fractal Antennas, Spacecraft, Radars, Image Processing, and Measure"; and "Part III: Scattering Theory, Porosity, and Turbulence." It's impossible to picture today's research without fractal analysis.

Fractals: A User's Guide for the Natural Sciences explains Mandelbrot's fractal geometry and describes some of its applications in the natural world. Written to enable students and researchers to master the methods of this timely subject, the book steers a middle course between the formality of many papers in mathematics and the informality of picture-orientated books on fractals. It is both a logically developed text and an essential `fractals for users' handbook.

Many are familiar with the beauty and ubiquity of fractal forms within nature. Unlike the study of smooth forms such as spheres, fractal geometry describes more familiar shapes and patterns, such as the complex contours of coastlines, the outlines of clouds, and the branching of trees. In this Very Short Introduction, Kenneth Falconer looks at the roots of the 'fractal revolution' that occurred in mathematics in the 20th century, presents the 'new geometry' of fractals, explains the basic concepts, and explores the wide range of applications in science, and in aspects of economics. This is essential introductory reading for students of mathematics and science, and those interested in popular science and mathematics. ABOUT THE SERIES: The Very Short Introductions series from Oxford University Press contains hundreds of titles in almost every subject area. These pocket-sized books are the perfect way to get ahead in a new subject quickly. Our expert authors combine facts, analysis, perspective, new ideas, and enthusiasm to make interesting and challenging topics highly readable.

Most books on fractals focus on deterministic fractals as the impact of incorporating randomness and time is almost absent. Further, most review fractals without explaining what scaling and self-similarity means. This book introduces the idea of scaling, self-similarity, scale-invariance and their role in the dimensional analysis. For the first time, fractals emphasizing mostly on stochastic fractal, and multifractals which evolves with time instead of scale-free self-similarity, are discussed. Moreover, it looks at power laws and dynamic scaling laws in some detail and provides an overview of modern statistical tools for calculating fractal dimension and multifractal spectrum.

The application of fractals in the engineering sciences is evolving swiftly and the editors have turned to Springer for the third time to bring you the latest research emerging from the rapid growth in techniques available for the employment of the ideas of fractals and complexity to a variety of disciplines in and associated with the engineering field. The strong potential of this research can be seen in real industrial situations with recent progress being made in areas such as chemical engineering, internet traffic, physics and finance. Image processing continues to be a major field of application for fractal analysis and is well-represented here. It is important to note that the applications models are presented with a firm basis in theoretical argument, the qualitative observation of fractal phenomena no longer being sufficient. Consisting of papers written by a world-wide pool of experts, the multidisciplinary approach of this third volume will be of particular interest to industrial researchers and practitioners as well as to academics from many backgrounds. Fractals in Engineering: New Trends in Theory and Applications continues the publication of engineering-related research in fractal techniques begun in Fractals in Engineering and Fractals: Theory and Applications in Engineering (Springer London 1997 and 1999).

I know that most men, including those at ease with the problems of the greatest complexity, can seldom accept even the simplest and most obvious truth if it be such as would oblige them to admit the falsity of conclusions which they have delighted in explaining to colleagues, which they have proudly taught to others, and which they have woven, thread by thread, into the fabric of their lives. Joseph Ford quoting Tolstoy (Gleick, 1987) We are used to thinking that natural objects have a certain form and that this form is determined by a characteristic scale. If we magnify the object beyond this scale, no new features are revealed. To correctly measure the properties of the object, such as length, area, or volume, we measure it at a resolution finer than the characteristic scale of the object. We expect that the value we measure has a unique value for the object. This simple idea is the basis of the calculus, Euclidean geometry, and the theory of measurement. However, Mandelbrot (1977, 1983) brought to the world's attention that many natural objects simply do not have this preconceived form. Many of the structures in space and processes in time of living things have a very different form. Living things have structures in space and fluctuations in time that cannot be characterized by one spatial or temporal scale. They extend over many spatial or temporal scales.