a) Sketch phase portraits of the maps: f(x)-x, f(x) = (2x + 1) and f(x) = 2x-x?
ID: 3283230 • Letter: A
Question
a) Sketch phase portraits of the maps: f(x)-x, f(x) = (2x + 1) and f(x) = 2x-x? and demonstrate stability of all fixed points. b) Find period 2 points of these maps. 1. 2. a) Use graphical method to find fixed points and period two points of the Logistic map: Xk+ 1-1%)-r Xk ( 1-%) 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 Px) b) Determine those values of r for which both fixed points of the logistic map are hyperbolic c) Determine whether((5 v5y8; (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 yale edu/public 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 0sxs and f(x)-2(1-x) if ½ xs 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) Ans
Fractal dimensionn is a ratio providing a statistical index of complexity comparing how detail in a pattern changes with the scale at which it is measured.
Example : Koch snowflake
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