Xn+1 = Xn*Xn + C, with X0 = 0

Xn+1 = Xn*Xn + C, with X0 = 0

X0 = 0

X1 = C

X2 = C*C + C

X3 = (C*C + C)*(C*C + C) + C

and so on...

Graphically:

Green parabola corresponds to C = -2

Black parabola corresponds to C = -1

Red parabola corresponds to C = 0

Blue parabola corresponds to C = 0.25

Fixed point:

Xn+1 = Xn

Xn^2 + C = Xn

Xn^2 - Xn + C = 0

Is a quadratic equation ( ax^2 + bx + c = 0) with a = 1, b = -1, c = C

Solutions: Xn = -1/2 +/- (1/2)(1-4C)^1/2

has real solutions if: 1-4C>0

or: C<1/4

Lets try a few cases:

 C= -2 Xn+1= Xn*Xn+(-2) n Xn 0 0 1 -2 2 2 3 2 4 2 5 2 6 2 7 2 8 2 9 2 10 2

 C= -1 Xn+1= Xn*Xn+(-1) n Xn 0 0 1 -1 2 0 3 -1 4 0 5 -1 6 0 7 -1 8 0 9 -1 10 0

 C= 0 Xn+1= Xn*Xn+0 n Xn 0 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 10 0

 C= 0.25 Xn+1= Xn*Xn+(0.25) n Xn 0 0 1 0.25 2 0.3125 3 0.34765625 4 0.370864868164062 5 0.387540750438347 6 0.400187833250317 7 0.410150301881584 8 0.418223270133554 9 0.424910703681204 10 0.430549106102856 É. tends to 0.5

Other cases:

 C= 1 Xn+1= Xn*Xn+(1) n Xn 0 0 1 1 2 2 3 5 4 26 5 677 6 458330 7 210066388901 8 4.42E+22 9 1.94E+45 10 3.79E+90

BIFURCATION DIAGRAM

C=-2............................................................................................................................................................................C=1/4

-2

Relationship between Quadratic and Logistic Map:

Xn+1 = Xn*Xn + C

Xn+1 = a*Xn*(1-Xn)

represent the same map if we identify:

Xn = (a/2) - a*Xn

C= (1-(a-1)^2)/4

Notice that:

a=0 --> C=1/4,

a=4 --> C=-2,

Xn = 0 --> Xn = a/2

Xn = 1 --> Xn = -a/2