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Author: Elhadj Zeraoulia
Publisher: World Scientific
ISBN: 9814307742
Category : Science
Languages : en
Pages : 357
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Book Description
This book is based on research on the rigorous proof of chaos and bifurcations in 2-D quadratic maps, especially the invertible case such as the Hnon map, and in 3-D ODE's, especially piecewise linear systems such as the Chua's circuit. In addition, the book covers some recent works in the field of general 2-D quadratic maps, especially their classification into equivalence classes, and finding regions for chaos, hyperchaos, and non-chaos in the space of bifurcation parameters. Following the main introduction to the rigorous tools used to prove chaos and bifurcations in the two representative systems, is the study of the invertible case of the 2-D quadratic map, where previous works are oriented toward Hnon mapping. 2-D quadratic maps are then classified into 30 maps with well-known formulas. Two proofs on the regions for chaos, hyperchaos, and non-chaos in the space of the bifurcation parameters are presented using a technique based on the second-derivative test and bounds for Lyapunov exponents. Also included is the proof of chaos in the piecewise linear Chua's system using two methods, the first of which is based on the construction of Poincar map, and the second is based on a computer-assisted proof. Finally, a rigorous analysis is provided on the bifurcational phenomena in the piecewise linear Chua's system using both an analytical 2-D mapping and a 1-D approximated Poincar mapping in addition to other analytical methods.
Author: Elhadj Zeraoulia
Publisher: World Scientific
ISBN: 9814307742
Category : Science
Languages : en
Pages : 357
Get Book Here
Book Description
This book is based on research on the rigorous proof of chaos and bifurcations in 2-D quadratic maps, especially the invertible case such as the Hnon map, and in 3-D ODE's, especially piecewise linear systems such as the Chua's circuit. In addition, the book covers some recent works in the field of general 2-D quadratic maps, especially their classification into equivalence classes, and finding regions for chaos, hyperchaos, and non-chaos in the space of bifurcation parameters. Following the main introduction to the rigorous tools used to prove chaos and bifurcations in the two representative systems, is the study of the invertible case of the 2-D quadratic map, where previous works are oriented toward Hnon mapping. 2-D quadratic maps are then classified into 30 maps with well-known formulas. Two proofs on the regions for chaos, hyperchaos, and non-chaos in the space of the bifurcation parameters are presented using a technique based on the second-derivative test and bounds for Lyapunov exponents. Also included is the proof of chaos in the piecewise linear Chua's system using two methods, the first of which is based on the construction of Poincar map, and the second is based on a computer-assisted proof. Finally, a rigorous analysis is provided on the bifurcational phenomena in the piecewise linear Chua's system using both an analytical 2-D mapping and a 1-D approximated Poincar mapping in addition to other analytical methods.
Author: Zeraoulia Elhadj
Publisher: CRC Press
ISBN: 0429647425
Category : Mathematics
Languages : en
Pages : 307
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Book Description
Chaos is the idea that a system will produce very different long-term behaviors when the initial conditions are perturbed only slightly. Chaos is used for novel, time- or energy-critical interdisciplinary applications. Examples include high-performance circuits and devices, liquid mixing, chemical reactions, biological systems, crisis management, secure information processing, and critical decision-making in politics, economics, as well as military applications, etc. This book presents the latest investigations in the theory of chaotic systems and their dynamics. The book covers some theoretical aspects of the subject arising in the study of both discrete and continuous-time chaotic dynamical systems. This book presents the state-of-the-art of the more advanced studies of chaotic dynamical systems.
Author: Zeraoulia Elhadj
Publisher: World Scientific
ISBN: 9814464791
Category : Science
Languages : en
Pages : 357
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Book Description
This book is based on research on the rigorous proof of chaos and bifurcations in 2-D quadratic maps, especially the invertible case such as the Hénon map, and in 3-D ODE's, especially piecewise linear systems such as the Chua's circuit. In addition, the book covers some recent works in the field of general 2-D quadratic maps, especially their classification into equivalence classes, and finding regions for chaos, hyperchaos, and non-chaos in the space of bifurcation parameters.Following the main introduction to the rigorous tools used to prove chaos and bifurcations in the two representative systems, is the study of the invertible case of the 2-D quadratic map, where previous works are oriented toward Hénon mapping. 2-D quadratic maps are then classified into 30 maps with well-known formulas. Two proofs on the regions for chaos, hyperchaos, and non-chaos in the space of the bifurcation parameters are presented using a technique based on the second-derivative test and bounds for Lyapunov exponents. Also included is the proof of chaos in the piecewise linear Chua's system using two methods, the first of which is based on the construction of Poincaré map, and the second is based on a computer-assisted proof. Finally, a rigorous analysis is provided on the bifurcational phenomena in the piecewise linear Chua's system using both an analytical 2-D mapping and a 1-D approximated Poincaré mapping in addition to other analytical methods.
Author: Julien Clinton Sprott
Publisher: World Scientific
ISBN: 9814460796
Category : Science
Languages : en
Pages : 268
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Book Description
This collection of review articles is devoted to new developments in the study of chaotic dynamical systems with some open problems and challenges. The papers, written by many of the leading experts in the field, cover both the experimental and theoretical aspects of the subject. This edited volume presents a variety of fascinating topics of current interest and problems arising in the study of both discrete and continuous time chaotic dynamical systems. Exciting new techniques stemming from the area of nonlinear dynamical systems theory are currently being developed to meet these challenges. Presenting the state-of-the-art of the more advanced studies of chaotic dynamical systems, Frontiers in the Study of Chaotic Dynamical Systems with Open Problems is devoted to setting an agenda for future research in this exciting and challenging field.
Author: Zeraoulia Elhadj
Publisher: CRC Press
ISBN: 1466580720
Category : Mathematics
Languages : en
Pages : 334
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Book Description
This book is a comprehensive collection of known results about the Lozi map, a piecewise-affine version of the Henon map. Henon map is one of the most studied examples in dynamical systems and it attracts a lot of attention from researchers, however it is difficult to analyze analytically. Simpler structure of the Lozi map makes it more suitable fo
Author: Elhadj Zeraoulia
Publisher: World Scientific
ISBN: 9814374075
Category : Mathematics
Languages : en
Pages : 473
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Book Description
Robust chaos is defined by the absence of periodic windows and coexisting attractors in some neighborhoods in the parameter space of a dynamical system. This unique book explores the definition, sources, and roles of robust chaos. The book is written in a reasonably self-contained manner and aims to provide students and researchers with the necessary understanding of the subject. Most of the known results, experiments, and conjectures about chaos in general and about robust chaos in particular are collected here in a pedagogical form. Many examples of dynamical systems, ranging from purely mathematical to natural and social processes displaying robust chaos, are discussed in detail. At the end of each chapter is a set of exercises and open problems (more than 260 in the whole book) intended to reinforce the ideas and provide additional experiences for both readers and researchers in nonlinear science in general, and chaos theory in particular.
Author: Robert Gilmore
Publisher: John Wiley & Sons
ISBN: 352763942X
Category : Mathematics
Languages : en
Pages : 618
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Book Description
A highly valued resource for those who wish to move from the introductory and preliminary understandings and the measurement of chaotic behavior to a more sophisticated and precise understanding of chaotic systems. The authors provide a deep understanding of the structure of strange attractors, how they are classified, and how the information required to identify and classify a strange attractor can be extracted from experimental data. In its first edition, the Topology of Chaos has been a valuable resource for physicist and mathematicians interested in the topological analysis of dynamical systems. Since its publication in 2002, important theoretical and experimental advances have put the topological analysis program on a firmer basis. This second edition includes relevant results and connects the material to other recent developments. Following significant improvements will be included: * A gentler introduction to the topological analysis of chaotic systems for the non expert which introduces the problems and questions that one commonly encounters when observing a chaotic dynamics and which are well addressed by a topological approach: existence of unstable periodic orbits, bifurcation sequences, multistability etc. * A new chapter is devoted to bounding tori which are essential for achieving generality as well as for understanding the influence of boundary conditions. * The new edition also reflects the progress which had been made towards extending topological analysis to higher-dimensional systems by proposing a new formalism where evolving triangulations replace braids. * There has also been much progress in the understanding of what is a good representation of a chaotic system, and therefore a new chapter is devoted to embeddings. * The chapter on topological analysis program will be expanded to cover traditional measures of chaos. This will help to connect those readers who are familiar with those measures and tests to the more sophisticated methodologies discussed in detail in this book. * The addition of the Appendix with both frequently asked and open questions with answers gathers the most essential points readers should keep in mind and guides to corresponding sections in the book. This will be of great help to those who want to selectively dive into the book and its treatments rather than reading it cover to cover. What makes this book special is its attempt to classify real physical systems (e.g. lasers) using topological techniques applied to real date (e.g. time series). Hence it has become the experimenter?s guidebook to reliable and sophisticated studies of experimental data for comparison with candidate relevant theoretical models, inevitable to physicists, mathematicians, and engineers studying low-dimensional chaotic systems.
Author: Ramon Alonso-Sanz
Publisher: World Scientific
ISBN: 9814343633
Category : Science
Languages : en
Pages : 478
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Book Description
Memory is a universal function of organized matter. What is the mathematics of memory? How does memory affect the space-time behaviour of spatially extended systems? Does memory increase complexity? This book provides answers to these questions. It focuses on the study of spatially extended systems, i.e., cellular automata and other related discrete complex systems. Thus, arrays of locally connected finite state machines, or cells, update their states simultaneously, in discrete time, by the same transition rule. The classical dynamics in these systems is Markovian: only the actual configuration is taken into account to generate the next one. Generalizing the conventional view on spatially extended discrete dynamical systems evolution by allowing cells (or nodes) to be featured by some trait state computed as a function of its own previous state-values, the transition maps of the classical systems are kept unaltered, so that the effect of memory can be easily traced. The book demonstrates that discrete dynamical systems with memory are not only priceless tools for modeling natural phenomena but unique mathematical and aesthetic objects.
Author: Ellis D. Cooper
Publisher: World Scientific
ISBN: 9814289701
Category : Mathematics
Languages : en
Pages : 390
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Book Description
This unprecedented book offers all the details of the mathematical mechanics underlying state-of-the-art modeling of skeletal muscle contraction. The aim is to provide an integrated vision of mathematics, physics, chemistry and biology for this one understanding. The method is to take advantage of modern mathematical technology — Eilenberg-Mac Lane category theory, Robinson infinitesimal calculus and Kolmogorov probability theory — to examine a succession of distinguishable universes of particles, and continuous, thermodynamic, chemical, and molecular bodies, all with a focus on proofs by algebraic calculation without set theory. Also provided are metaphors and analogies, and careful distinction between representational pictures, mental model drawings, and mathematical diagrams.High school mathematics teachers, undergraduate and graduate college students, and researchers in mathematics, physics, chemistry, and biology may use this integrated publication to broaden their perspective on science, and to experience the precision that mathematical mechanics brings to understanding the muscular mechanism of nearly all animal behavior.
Author: Leon O. Chua
Publisher: World Scientific
ISBN: 9814317306
Category : Science
Languages : en
Pages : 405
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Book Description
Annotation This text introduces cellular automata from a rigorous nonlinear dynamics perspective. It supplies the missing link between nonlinear differential and difference equations to discrete symbolic analysis. It provides an analysis, and classification of the empirical results presented in Wolfram's 'New Kind of Science'.
