#OK to post homework
#Nicholas DiMarzio, 10/11/21, Assignment 11
#
#Problem 1
SFPe(k*x*(1 - x), x);
                   [        [k - 1        ]]
                   [[0, k], [-----, -k + 2]]
                   [        [  k          ]]

#The difference equation x[n]=k*n[n-1]*(1-x[n-1]) Always has 2 fixed points 0 and (k-1)/k
#x=0 is stable iff 0=k<1

#Problem 2

{op(Orb(3.1*x*(1 - x), x, 0.5, 1000, 1020))};
                  {0.5580141245, 0.7645665203}

{op(Orb(3.2*x*(1 - x), x, 0.5, 1000, 1020))};
                  {0.5130445091, 0.7994554906}

{op(Orb(3.3*x*(1 - x), x, 0.5, 1000, 1020))};
                  {0.4794270198, 0.8236032832}

{op(Orb(3.4*x*(1 - x), x, 0.5, 1000, 1020))};
                  {0.4519632478, 0.8421543994}

{op(Orb(3.5*x*(1 - x), x, 0.5, 1000, 1020))};
    {0.3828196827, 0.5008842111, 0.8269407062, 0.8749972637}

{op(Orb(3.51*x*(1 - x), x, 0.5, 1000, 1020))};
    {0.3777221554, 0.5067130559, 0.8250189317, 0.8773418215}

#Bifurcation is in Parameter space

#We can estimate that at k=3.5 the second birfurcation point happens

#Problem 3
#
f := b*x/a*b*x;
                                 2  2
                                b  x 
                           f := -----
                                  a  

SFPe(f, x);
                       [        [a    ]]
                       [[0, 0], [--, 2]]
                       [        [ 2   ]]
                       [        [b    ]]


g := x*exp(r*(1 - x/k));
                               /  /    x\\
                     g := x exp|r |1 - -||
                               \  \    k//

SFPe(g, x);
             [[0, exp(r)], [k, exp(0) - r exp(0)]]


h := b*x*(a*x + 1);
                       h := b x (a x + 1)

SFPe(h, x);
                  [        [  b - 1        ]]
                  [[0, b], [- -----, -b + 2]]
                  [        [   a b         ]]