Screen LibraryThe (1+1)-nonlinear universe of the parabolic map and combinatorics /
This monograph develops chaos theory from properties of the graphs inverse to the parabolic map of the interval [0, 2], where the height at the midpoint x = 1 may be viewed as a time-like parameter, which together with the x-coordinate, provide the two parameters that uniquely characterize the parab...
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| Published: |
World Scientific Pub. Co.,
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| Publisher Address: | Singapore ; Hackensack, N.J. : |
| Publication Dates: | 2015. |
| Literature type: | eBook |
| Language: | English |
| Subjects: | |
| Online Access: |
http://www.worldscientific.com/worldscibooks/10.1142/9370#t=toc |
| Summary: |
This monograph develops chaos theory from properties of the graphs inverse to the parabolic map of the interval [0, 2], where the height at the midpoint x = 1 may be viewed as a time-like parameter, which together with the x-coordinate, provide the two parameters that uniquely characterize the parabola, and which are used throughout the monograph. There is only one basic mathematical operation used: function composition. The functions studied are the n-fold composition of the basic parabola with itself. However, it is the properties of the graph inverse to this n-fold composition that are the objects whose properties are developed. The reflection symmetry of the basic parabola through the vertical line x = 1 gives rise to two symmetry classes of inverse graphs: the inverse graphs and their conjugates. Quite remarkably, it turns out that there exists, among all the inverse graphs and their conjugates, a completely deterministic class of inverse graphs and their conjugates. Deterministic in the sense that this class is uniquely determined for all values of the time-like parameter and the x-coordinate, the entire theory, of course, being highly nonlinear - it is polynomial in the time-like parameter and in the x-coordinate. The deterministic property and its implementation are key to the argument that the system is a complex adaptive system in the sense that a few axioms lead to structures of unexpected richness. This monograph is about working out the many details that advance the notion that deterministic chaos theory, as realized by a complex adaptive system, is indeed a new body of mathematics that enriches our understanding of the world around us. But now the imagination is also opened to the possibility that the real universe is a complex adaptive system. |
| Carrier Form: | 1 online resource (xii,179pages) : illustrations |
| Bibliography: | Includes bibliographical references (pages 171-174) and index. |
| ISBN: | 9789814632423 |
| Index Number: | QA427 |
| CLC: | O157.1 |
| Contents: | 1. Introduction and point of view. 1.1. Function composition and graphs. 1.2. Inverse graphs created at [symbol] = 1. 1.3. Preview of the full [symbol]-evolution. 1.4. The baseline. 1.5. Vocabulary, symbol definitions, and explanations -- 2. Recursive construction. 2.1. Construction of the baseline B[symbol]. 2.2. Reducible and irreducible sequences -- 3. Description of events in the inverse graph. 3.1. Domains of definition of branches and curves. 3.2. Concatenation, harmonics, and antiharmonics. 3.3. Fixed points as dynamical objects. 3.4. The fabric of bifurcation events. 3.5. The anatomy of period-doubling bifurcations. 3.6. Signature properties of fixed points. 3.7. Young Hook tableaux and Gelfand-Tsetlin patterns -- 4. The (1+1)-dimensional nonlinear universe. 4.1. The parabolic map. 4.2. Complex adaptive systems -- 5. The creation table. 5.1. The creation intervals. 5.2. Properties of the creation table -- 6. Graphical presentation of MSS roots -- 7. Graphical Presentation of inverse graphs. |