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The (1+1)-nonlinear universe of the parabolic map and combinatorics

Author: James D Louck; M L Stein
Publisher: New Jersey : World Scientific, [2015]
Edition/Format:   Print book : EnglishView all editions and formats
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  Read more...
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Details

Document Type: Book
All Authors / Contributors: James D Louck; M L Stein
ISBN: 9789814632416 9814632414
OCLC Number: 894032954
Description: xii, 179 pages : illustrations ; 26 cm
Contents: Introduction and point of view --
Recursive construction --
Description of events in the inverse graph --
The (1+1)-dimensional nonlinear universe --
The creation table --
Graphical presentation of MSS roots --
Graphical presentation of inverse graphs.
Other Titles: Nonlinear universe of the parabolic map and combinatorics
Responsibility: James D. Louck, Myron L. Stein.

Abstract:

"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"--The publisher.

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