#OK to post homework #Nicholas DiMarzio, 10/04/21, Assignment 9 # Problem 1 # f := 2*x*(1 - x); f := 2 x (1 - x) Orb(f, x, 0.5, 1000, 1020); [0.5000000000, 0.5000000000, 0.5000000000, 0.5000000000, 0.5000000000, 0.5000000000, 0.5000000000, 0.5000000000, 0.5000000000, 0.5000000000, 0.5000000000, 0.5000000000, 0.5000000000, 0.5000000000, 0.5000000000, 0.5000000000, 0.5000000000, 0.5000000000, 0.5000000000, 0.5000000000, 0.5000000000, 0.5000000000] SFP(f, x); [0.5000000000] g := 2.5*x*(1 - x); g := 2.5 x (1 - x) Orb(g, x, 0.5, 1000, 1020); [0.6000000000, 0.6000000000, 0.6000000000, 0.6000000000, 0.6000000000, 0.6000000000, 0.6000000000, 0.6000000000, 0.6000000000, 0.6000000000, 0.6000000000, 0.6000000000, 0.6000000000, 0.6000000000, 0.6000000000, 0.6000000000, 0.6000000000, 0.6000000000, 0.6000000000, 0.6000000000, 0.6000000000, 0.6000000000] SFP(g, x); [0.6000000000] h := 3.1*x*(1 - x); h := 3.1 x (1 - x) Orb(h, x, 0.5, 1000, 1020); [0.5580141245, 0.7645665203, 0.5580141245, 0.7645665203, 0.5580141245, 0.7645665203, 0.5580141245, 0.7645665203, 0.5580141245, 0.7645665203, 0.5580141245, 0.7645665203, 0.5580141245, 0.7645665203, 0.5580141245, 0.7645665203, 0.5580141245, 0.7645665203, 0.5580141245, 0.7645665203, 0.5580141245, 0.7645665203] SFP(h, x); [] j := (4 + x)/(3 + x); 4 + x j := ----- 3 + x Orb(j, x, 0.5, 1000, 1020); [1.236067977, 1.236067978, 1.236067977, 1.236067978, 1.236067977, 1.236067978, 1.236067977, 1.236067978, 1.236067977, 1.236067978, 1.236067977, 1.236067978, 1.236067977, 1.236067978, 1.236067977, 1.236067978, 1.236067977, 1.236067978, 1.236067977, 1.236067978, 1.236067977, 1.236067978 ] SFP(j, x); [1.236067977] k := (3 + x)/(4 + x); 3 + x k := ----- 4 + x Orb(k, x, 0.5, 1000, 1020); [0.7912878475, 0.7912878475, 0.7912878475, 0.7912878475, 0.7912878475, 0.7912878475, 0.7912878475, 0.7912878475, 0.7912878475, 0.7912878475, 0.7912878475, 0.7912878475, 0.7912878475, 0.7912878475, 0.7912878475, 0.7912878475, 0.7912878475, 0.7912878475, 0.7912878475, 0.7912878475, 0.7912878475, 0.7912878475] SFP(k, x); [0.791287848] l := (x^2 + x + 3)/(2*x^2 + x + 4); 2 x + x + 3 l := ------------ 2 2 x + x + 4 Orb(l, x, 0.5, 1000, 1020); [0.7351392591, 0.7351392591, 0.7351392591, 0.7351392591, 0.7351392591, 0.7351392591, 0.7351392591, 0.7351392591, 0.7351392591, 0.7351392591, 0.7351392591, 0.7351392591, 0.7351392591, 0.7351392591, 0.7351392591, 0.7351392591, 0.7351392591, 0.7351392591, 0.7351392591, 0.7351392591, 0.7351392591, 0.7351392591] SFP(l, x); [0.7351392587] # #Problem 2 # h := solve((x + a)/(x + b)); h := {a = -x, b = b, x = x} diff(h, a, b, x); {0 = 0} d := (x + 1)/(x + 2); x + 1 d := ----- x + 2 Orb(d, x, 0.5, 1000, 1020); [0.6180339888, 0.6180339888, 0.6180339888, 0.6180339888, 0.6180339888, 0.6180339888, 0.6180339888, 0.6180339888, 0.6180339888, 0.6180339888, 0.6180339888, 0.6180339888, 0.6180339888, 0.6180339888, 0.6180339888, 0.6180339888, 0.6180339888, 0.6180339888, 0.6180339888, 0.6180339888, 0.6180339888, 0.6180339888] FP(d, x); [-1.618033988, 0.6180339880] SFP(d, x); [0.6180339880] f := (x + 2)/(3 + x); x + 2 f := ----- 3 + x Orb(f, x, 0.5, 1000, 1020); [0.7320508076, 0.7320508076, 0.7320508076, 0.7320508076, 0.7320508076, 0.7320508076, 0.7320508076, 0.7320508076, 0.7320508076, 0.7320508076, 0.7320508076, 0.7320508076, 0.7320508076, 0.7320508076, 0.7320508076, 0.7320508076, 0.7320508076, 0.7320508076, 0.7320508076, 0.7320508076, 0.7320508076, 0.7320508076] FP(f, x); [-2.732050808, 0.732050808] SFP(f, x); [0.732050808] g := (x + 12)/(x + 17); x + 12 g := ------ x + 17 Orb(g, x, 0.5, 1000, 1020); [0.7177978871, 0.7177978871, 0.7177978871, 0.7177978871, 0.7177978871, 0.7177978871, 0.7177978871, 0.7177978871, 0.7177978871, 0.7177978871, 0.7177978871, 0.7177978871, 0.7177978871, 0.7177978871, 0.7177978871, 0.7177978871, 0.7177978871, 0.7177978871, 0.7177978871, 0.7177978871, 0.7177978871, 0.7177978871] FP(g, x); [-16.71779789, 0.717797888] SFP(g, x); [0.717797888] # #Problem 3 # #We solve for x #The equation is in equilibrium at x=0 and x=1 f := x*(1 - x); f := x (1 - x) g := 2*x*(1 - x); g := 2 x (1 - x) h := 3*x*(1 - x); h := 3 x (1 - x) j := 4*x*(1 - x); j := 4 x (1 - x) SFP(f, x); [] SFP(g, x); [0.5000000000] SFP(h, x); [] SFP(j, x); [] #We see that 0 is never stable but 0.5 is stable for k=2 #We must take the derivative of our equation first and then set it equal to 0 #For k=2, g'(x)=2-4x #g(1)=2-4(1)=-2 #Therefore k=2 is the first Bifurcation value # #problem4 # f1 := Comp(f, x); / 2 \ f1 := -x (-1 + x) \x - x + 1/ SFP(f1, x); [] g1 := Comp(g, x); / 2 \ g1 := -4 x (-1 + x) \2 x - 2 x + 1/ SFP(g1, x); [0.5000000000] h1 := Comp(h, x); / 2 \ h1 := -9 x (-1 + x) \3 x - 3 x + 1/ SFP(h1, x); [] j1 := Comp(j, x); / 2 \ j1 := -16 x (-1 + x) \4 x - 4 x + 1/ SFP(j1, x); []