Need help with all the problems!! 1. a) Sketch phase portraits of the maps: f(x)

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Need help with all the problems!!

1. a) Sketch phase portraits of the maps: f(x) - x2, f(x) - (-2x +1) and f(x) - 2x - x3, and demonstrate stability of all fixed points. b) Find period 2 points of these maps. 2. a) Use graphical method to find fixed points and period two points of the Logistic map: X+1f(Xk)r Xk (1 - Xk) and demonstrate their stability/instability for r-1 1.5 and 3. (See 1.1 from the attached file titled: Logistic Map Period Doubling). You can use Matlab to obtain graphs of the map f(X) and second iteration of the map fx). b) Determine those values of r for which both fixed points of the logistic map are hyperbolic. c) Determine whether (5 5)/8; (5+5)/8) is period 2 orbit of the logistic map. 3. Give definition of fractal dimension 4. a) Use web site: https://users.math.vale .edulpublic htm/People/frame/Fractals/ and provide detailed description of two fractals. b) Generate a fractal by using Iterated Function System (IFS) method (see the same web site) 5. Demonstrate calculation of the fractal dimension and measure of the "middle thirds" Cantor set 6. Show that {1/7, 2/7, 4/7) is period 3 orbit of the tent map: f(x) -2x if 0 sx s % and f(x)-2(1-x) if ½ s x s 1. 7. a) Give definition of a sensitive point of a map. b) Show how symbolic sequences can be associated with the trajectories of the tent map. c) Demonstrate existence of sensitive points for the tent map.

Explanation / Answer

3) Answer

fractaldimension is a ratio providing a statistical index of complexitycomparing how detail in a pattern changes with the scale at which it is measured. It has also been characterized as a measure of the space-filling capacity of a pattern that tells how a fractal scales differently from the space it is embedded in, a fractal dimension does not have to be an integer. That is it should be an fraction.

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