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