Some useful points on the Chaos Map
The values of A given below are derived from the logistic equation.
The values are different for the other equations but, remarkably, the overall pattern is identical.
A <= 1
Curve is always below the line y = x so the point is drawn to zero
A > 1 and A < 3
Point is attracted to the intersection of the first order curve and the line y = x. This is at the point y = (A - 1) / A
Up to this point, therefore, the line of stability is part of a rectangular hyperbola.
A = 3
At this point, the gradient of the curve where it crosses the y = x line becomes -1
At the same time, the gradient of the second order curve at this point becomes 1
This circumstance causes the attractor to split into two.
Around this critical point, the attractor is very weak and the point takes a long time to reach its destination.
A > 3 and A < 3.44
The attractor consists of two places where the second order curve interscts the y = x line. It does this at four points but the point x = 0 and the third point are unstable. The attractor follows the second and fourth points. At A = 3.232 the second order curve passes through the point (0.5, 0.5). This is the pivot point of this episode.
A = 3.44
The gradient of the second order curve now reaches -1 at both points simultaneously and the attractor splits into four.
A > 3.44 and A < 3.6
Further period doublings appear until the output becomes essentially chaotic at around A=3.6
A = 3.629
An episode of period 6 occurs
A = 3.676
This is where the two hyperbolas which spring from the first bifurcation intersect. All stable periods up to now (except, of course, the first) have even numbered periods. From this point on it becomes possible to find odd mumbered periods. For odd periods to occur, one of the numbers in the sequence must be 0.5. The sequence can therefore be easily displayed by starting with an initial value of 0.5
A = 3.7019
An episode of period 7 occurs
A = 3.7391
An episode of period 5 occurs.
A > 3.830 and A < 3.84
A very stable episode of period 3 occurs. At A = 3.832 the central loop of the third order curve crosses the y=x line at the point (0.5, 0.5). In general, a stable episode of period n will occur whenever the nth order curve passes through the point (0.5, 0.5)