Dynamical Systems And Fractals

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Lectures on Fractal Geometry and Dynamical Systems

Ya. B. Pesin,Vaughn Climenhaga
Lectures on Fractal Geometry and Dynamical Systems Book PDF

Author : Ya. B. Pesin,Vaughn Climenhaga
Publisher : American Mathematical Soc.
Page : 334 pages
File Size : 52,5 Mb
Release : 2009
Category : Mathematics
ISBN : 9780821848890

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Lectures on Fractal Geometry and Dynamical Systems by Ya. B. Pesin,Vaughn Climenhaga Pdf

Both fractal geometry and dynamical systems have a long history of development and have provided fertile ground for many great mathematicians and much deep and important mathematics. These two areas interact with each other and with the theory of chaos in a fundamental way: many dynamical systems (even some very simple ones) produce fractal sets, which are in turn a source of irregular 'chaotic' motions in the system. This book is an introduction to these two fields, with an emphasis on the relationship between them. The first half of the book introduces some of the key ideas in fractal geometry and dimension theory - Cantor sets, Hausdorff dimension, box dimension - using dynamical notions whenever possible, particularly one-dimensional Markov maps and symbolic dynamics. Various techniques for computing Hausdorff dimension are shown, leading to a discussion of Bernoulli and Markov measures and of the relationship between dimension, entropy, and Lyapunov exponents. In the second half of the book some examples of dynamical systems are considered and various phenomena of chaotic behaviour are discussed, including bifurcations, hyperbolicity, attractors, horseshoes, and intermittent and persistent chaos. These phenomena are naturally revealed in the course of our study of two real models from science - the FitzHugh - Nagumo model and the Lorenz system of differential equations. This book is accessible to undergraduate students and requires only standard knowledge in calculus, linear algebra, and differential equations. Elements of point set topology and measure theory are introduced as needed. This book is a result of the MASS course in analysis at Penn State University in the fall semester of 2008.

Dynamical Systems and Fractals

Karl-Heinz Becker,Michael Dörfler
Dynamical Systems and Fractals Book PDF

Author : Karl-Heinz Becker,Michael Dörfler
Publisher : Cambridge University Press
Page : 420 pages
File Size : 43,7 Mb
Release : 1989-10-26
Category : Computers
ISBN : 052136910X

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Dynamical Systems and Fractals by Karl-Heinz Becker,Michael Dörfler Pdf

This 1989 book is about chaos, fractals and complex dynamics.

The Beauty of Fractals

Heinz-Otto Peitgen,Peter H. Richter
The Beauty of Fractals Book PDF

Author : Heinz-Otto Peitgen,Peter H. Richter
Publisher : Springer Science & Business Media
Page : 214 pages
File Size : 45,6 Mb
Release : 2013-12-01
Category : Mathematics
ISBN : 9783642617171

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The Beauty of Fractals by Heinz-Otto Peitgen,Peter H. Richter Pdf

Now approaching its tenth year, this hugely successful book presents an unusual attempt to publicise the field of Complex Dynamics. The text was originally conceived as a supplemented catalogue to the exhibition "Frontiers of Chaos", seen in Europe and the United States, and describes the context and meaning of these fascinating images. A total of 184 illustrations - including 88 full-colour pictures of Julia sets - are suggestive of a coffee-table book. However, the invited contributions which round off the book lend the text the required formality. Benoit Mandelbrot gives a very personal account, in his idiosyncratic self-centred style, of his discovery of the fractals named after him and Adrien Douady explains the solved and unsolved problems relating to this amusingly complex set.

Dynamics with Chaos and Fractals

Marat Akhmet,Mehmet Onur Fen,Ejaily Milad Alejaily
Dynamics with Chaos and Fractals Book PDF

Author : Marat Akhmet,Mehmet Onur Fen,Ejaily Milad Alejaily
Publisher : Springer Nature
Page : 226 pages
File Size : 40,9 Mb
Release : 2020-01-01
Category : Mathematics
ISBN : 9783030358549

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Dynamics with Chaos and Fractals by Marat Akhmet,Mehmet Onur Fen,Ejaily Milad Alejaily Pdf

The book is concerned with the concepts of chaos and fractals, which are within the scopes of dynamical systems, geometry, measure theory, topology, and numerical analysis during the last several decades. It is revealed that a special kind of Poisson stable point, which we call an unpredictable point, gives rise to the existence of chaos in the quasi-minimal set. This is the first time in the literature that the description of chaos is initiated from a single motion. Chaos is now placed on the line of oscillations, and therefore, it is a subject of study in the framework of the theories of dynamical systems and differential equations, as in this book. The techniques introduced in the book make it possible to develop continuous and discrete dynamics which admit fractals as points of trajectories as well as orbits themselves. To provide strong arguments for the genericity of chaos in the real and abstract universe, the concept of abstract similarity is suggested.

Chaotic Dynamics and Fractals

Michael F. Barnsley,Stephen G. Demko
Chaotic Dynamics and Fractals Book PDF

Author : Michael F. Barnsley,Stephen G. Demko
Publisher : Academic Press
Page : 304 pages
File Size : 47,5 Mb
Release : 2014-05-10
Category : Mathematics
ISBN : 9781483269085

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Chaotic Dynamics and Fractals by Michael F. Barnsley,Stephen G. Demko Pdf

Chaotic Dynamics and Fractals covers the proceedings of the 1985 Conference on Chaotic Dynamics, held at the Georgia Institute of Technology. This conference deals with the research area of chaos, dynamical systems, and fractal geometry. This text is organized into three parts encompassing 16 chapters. The first part describes the nature of chaos and fractals, the geometric tool for some strange attractors, and other complicated sets of data associated with chaotic systems. This part also considers the Henon-Hiles Hamiltonian with complex time, a Henon family of maps from C2 into itself, and the idea of turbulent maps in the course of presenting results on iteration of continuous maps from the unit interval to itself. The second part discusses complex analytic dynamics and associated fractal geometry, specifically the bursts into chaos, algorithms for obtaining geometrical and combinatorial information, and the parameter space for iterated cubic polynomials. This part also examines the differentiation of Julia sets with respects to a parameter in the associated rational map, permitting the formulation of Taylor series expansion for the sets. The third part highlights the applications of chaotic dynamics and fractals. This book will prove useful to mathematicians, physicists, and other scientists working in, or introducing themselves to, the field.

Invitation to Dynamical Systems

Edward R. Scheinerman
Invitation to Dynamical Systems Book PDF

Author : Edward R. Scheinerman
Publisher : Courier Corporation
Page : 408 pages
File Size : 42,7 Mb
Release : 2013-05-13
Category : Mathematics
ISBN : 9780486275321

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Invitation to Dynamical Systems by Edward R. Scheinerman Pdf

This text is designed for those who wish to study mathematics beyond linear algebra but are unready for abstract material. Rather than a theorem-proof-corollary exposition, it stresses geometry, intuition, and dynamical systems. 1996 edition.

Fractals and Chaos

A.J. Crilly,Rae Earnshaw,Huw Jones
Fractals and Chaos Book PDF

Author : A.J. Crilly,Rae Earnshaw,Huw Jones
Publisher : Springer Science & Business Media
Page : 278 pages
File Size : 43,5 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461230342

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Fractals and Chaos by A.J. Crilly,Rae Earnshaw,Huw Jones Pdf

This volume is based upon the presentations made at an international conference in London on the subject of 'Fractals and Chaos'. The objective of the conference was to bring together some of the leading practitioners and exponents in the overlapping fields of fractal geometry and chaos theory, with a view to exploring some of the relationships between the two domains. Based on this initial conference and subsequent exchanges between the editors and the authors, revised and updated papers were produced. These papers are contained in the present volume. We thank all those who contributed to this effort by way of planning and organisation, and also all those who helped in the production of this volume. In particular, we wish to express our appreciation to Gerhard Rossbach, Computer Science Editor, Craig Van Dyck, Production Director, and Nancy A. Rogers, who did the typesetting. A. J. Crilly R. A. Earnshaw H. Jones 1 March 1990 Introduction Fractals and Chaos The word 'fractal' was coined by Benoit Mandelbrot in the late 1970s, but objects now defined as fractal in form have been known to artists and mathematicians for centuries. Mandelbrot's definition-"a set whose Hausdorff dimension is not an integer" -is clear in mathematical terms. In addition, related concepts are those of self-similarity and sub-divisibility. A fractal object is self-similar in that subsections of the object are similar in some sense to the whole object.

An Introduction to Dynamical Systems and Chaos

G.C. Layek
An Introduction to Dynamical Systems and Chaos Book PDF

Author : G.C. Layek
Publisher : Springer
Page : 622 pages
File Size : 40,8 Mb
Release : 2015-12-01
Category : Mathematics
ISBN : 9788132225560

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An Introduction to Dynamical Systems and Chaos by G.C. Layek Pdf

The book discusses continuous and discrete systems in systematic and sequential approaches for all aspects of nonlinear dynamics. The unique feature of the book is its mathematical theories on flow bifurcations, oscillatory solutions, symmetry analysis of nonlinear systems and chaos theory. The logically structured content and sequential orientation provide readers with a global overview of the topic. A systematic mathematical approach has been adopted, and a number of examples worked out in detail and exercises have been included. Chapters 1–8 are devoted to continuous systems, beginning with one-dimensional flows. Symmetry is an inherent character of nonlinear systems, and the Lie invariance principle and its algorithm for finding symmetries of a system are discussed in Chap. 8. Chapters 9–13 focus on discrete systems, chaos and fractals. Conjugacy relationship among maps and its properties are described with proofs. Chaos theory and its connection with fractals, Hamiltonian flows and symmetries of nonlinear systems are among the main focuses of this book. Over the past few decades, there has been an unprecedented interest and advances in nonlinear systems, chaos theory and fractals, which is reflected in undergraduate and postgraduate curricula around the world. The book is useful for courses in dynamical systems and chaos, nonlinear dynamics, etc., for advanced undergraduate and postgraduate students in mathematics, physics and engineering.

Chaos

Kathleen Alligood,Tim Sauer,J.A. Yorke
Chaos Book PDF

Author : Kathleen Alligood,Tim Sauer,J.A. Yorke
Publisher : Springer
Page : 620 pages
File Size : 41,8 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783642592812

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Chaos by Kathleen Alligood,Tim Sauer,J.A. Yorke Pdf

BACKGROUND Sir Isaac Newton hrought to the world the idea of modeling the motion of physical systems with equations. It was necessary to invent calculus along the way, since fundamental equations of motion involve velocities and accelerations, of position. His greatest single success was his discovery that which are derivatives the motion of the planets and moons of the solar system resulted from a single fundamental source: the gravitational attraction of the hodies. He demonstrated that the ohserved motion of the planets could he explained hy assuming that there is a gravitational attraction he tween any two ohjects, a force that is proportional to the product of masses and inversely proportional to the square of the distance between them. The circular, elliptical, and parabolic orhits of astronomy were v INTRODUCTION no longer fundamental determinants of motion, but were approximations of laws specified with differential equations. His methods are now used in modeling motion and change in all areas of science. Subsequent generations of scientists extended the method of using differ ential equations to describe how physical systems evolve. But the method had a limitation. While the differential equations were sufficient to determine the behavior-in the sense that solutions of the equations did exist-it was frequently difficult to figure out what that behavior would be. It was often impossible to write down solutions in relatively simple algebraic expressions using a finite number of terms. Series solutions involving infinite sums often would not converge beyond some finite time.

Chaos, Fractals, and Noise

Andrzej Lasota,Michael C. Mackey
Chaos Fractals and Noise Book PDF

Author : Andrzej Lasota,Michael C. Mackey
Publisher : Springer Science & Business Media
Page : 481 pages
File Size : 46,8 Mb
Release : 2013-11-27
Category : Mathematics
ISBN : 9781461242864

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Chaos, Fractals, and Noise by Andrzej Lasota,Michael C. Mackey Pdf

The first edition of this book was originally published in 1985 under the ti tle "Probabilistic Properties of Deterministic Systems. " In the intervening years, interest in so-called "chaotic" systems has continued unabated but with a more thoughtful and sober eye toward applications, as befits a ma turing field. This interest in the serious usage of the concepts and techniques of nonlinear dynamics by applied scientists has probably been spurred more by the availability of inexpensive computers than by any other factor. Thus, computer experiments have been prominent, suggesting the wealth of phe nomena that may be resident in nonlinear systems. In particular, they allow one to observe the interdependence between the deterministic and probabilistic properties of these systems such as the existence of invariant measures and densities, statistical stability and periodicity, the influence of stochastic perturbations, the formation of attractors, and many others. The aim of the book, and especially of this second edition, is to present recent theoretical methods which allow one to study these effects. We have taken the opportunity in this second edition to not only correct the errors of the first edition, but also to add substantially new material in five sections and a new chapter.

Chaos, Fractals, and Dynamics

P. Fischer
Chaos Fractals and Dynamics Book PDF

Author : P. Fischer
Publisher : CRC Press
Page : 282 pages
File Size : 44,9 Mb
Release : 2020-11-26
Category : Mathematics
ISBN : 9781000154221

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Chaos, Fractals, and Dynamics by P. Fischer Pdf

This book contains eighteen papers, all more-or-less linked to the theory of dynamical systems together with related studies of chaos and fractals. It shows many fractal configurations that were generated by computer calculations of underlying two-dimensional maps.

Fractal Control Theory

Shu-Tang Liu,Pei Wang
Fractal Control Theory Book PDF

Author : Shu-Tang Liu,Pei Wang
Publisher : Springer
Page : 293 pages
File Size : 46,8 Mb
Release : 2018-04-21
Category : Technology & Engineering
ISBN : 9789811070501

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Fractal Control Theory by Shu-Tang Liu,Pei Wang Pdf

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.

Discrete Dynamical Systems, Chaos Theory and Fractals

Linda Sundbye
Discrete Dynamical Systems Chaos Theory and Fractals Book PDF

Author : Linda Sundbye
Publisher : Createspace Independent Publishing Platform
Page : 228 pages
File Size : 50,6 Mb
Release : 2018-10-05
Category : Electronic
ISBN : 172716153X

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Discrete Dynamical Systems, Chaos Theory and Fractals by Linda Sundbye Pdf

An introductory undergraduate level text on chaos theory, nonlinear dynamics and fractal geometry.

Chaos, Dynamics, and Fractals

Joseph L. McCauley
Chaos Dynamics and Fractals Book PDF

Author : Joseph L. McCauley
Publisher : Cambridge University Press
Page : 352 pages
File Size : 44,9 Mb
Release : 1994-05-26
Category : Science
ISBN : 9781107393271

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Chaos, Dynamics, and Fractals by Joseph L. McCauley Pdf

This book develops deterministic chaos and fractals from the standpoint of iterated maps, but the emphasis makes it very different from all other books in the field. It provides the reader with an introduction to more recent developments, such as weak universality, multifractals, and shadowing, as well as to older subjects like universal critical exponents, devil's staircases and the Farey tree. The author uses a fully discrete method, a 'theoretical computer arithmetic', because finite (but not fixed) precision cannot be avoided in computation or experiment. This leads to a more general formulation in terms of symbolic dynamics and to the idea of weak universality. The connection is made with Turing's ideas of computable numbers and it is explained why the continuum approach leads to predictions that are not necessarily realized in computation or in nature, whereas the discrete approach yields all possible histograms that can be observed or computed.

Chaotic Dynamics

Geoffrey R. Goodson
Chaotic Dynamics Book PDF

Author : Geoffrey R. Goodson
Publisher : Cambridge University Press
Page : 419 pages
File Size : 45,7 Mb
Release : 2017
Category : Mathematics
ISBN : 9781107112674

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Chaotic Dynamics by Geoffrey R. Goodson Pdf

This rigorous undergraduate introduction to dynamical systems is an accessible guide for mathematics students advancing from calculus.