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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.
Contents:Introduction and Point of ViewRecursive ConstructionDescription of Events in the Inverse GraphThe (1+1)-Dimensional Nonlinear UniverseThe Creation TableGraphical Presentation of MSS RootsGraphical Presentation of Inverse Graphs
Readership: Post-graduates from mathematics and physics backgrounds, mathematics and physics professionals with an interest in astrophysics.
Key Features:There exists a unique inverse graph. This unique graph can be constructed from a simple algorithmThe concept of a deterministic inverse graph and its relationship to complex adaptive systems is newThis monograph is of general value to readers because it illustrates nicely that research in new methodologies in Mathematics can lead to new insights in Physics