Chaotic Phenomenal and One-dimensional Logistic Map

Abstract

Introducing the basic properties of logistic map. The paper includes the proofing of the stability of fixed points in different domain of  and determining the property of period doubling of Logistic map, describing Chaos phenomenon in logistic map by showing the stability of fixed points in different intervals of , researching 3-period limit cycle in the Logistic Map bifurcation diagram, showing a special characteristic of the logistic map on the region on the bifurcation diagram that exist 3-period limit cycles, collecting the different value of  at one, two, four, eight, sixteen and thirty-two pitchfork points from the Logistic Map Bifurcation diagram to calculate Feigenbaum constant and proving the Feigenbaum can also be used to determine the pitchfork bifurcation speed of logistic map when r is between zero to negative two. Showing the basic properties of the Mandelbrot Set. Showing the result of computer simulation in real part to the Mandelbrot Set to determine the relationship between the Logistic Map bifurcation diagram and Mandelbrot Set.