QUADRATIC MAP

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