A special case of chaos: r = 4
When r = 4, the logistic map is     ƒ(x) = 4x(1 - x),   xn+1 = 4xn( 1 - xn).
Under the variable transform:       x = sin2πθ,   (0 ≤ θ ≤ 1/2)
the map becomes:                          sin2πθn +1 = 4sin2πθn(1 - sin2πθn) = sin22πθn,

or  θn +1 = g(θn) = { figure 90 		figure 89

This is a simple tent map, where dynamics is well-known. In particular, g has periodic points of any period in the interval [0,0.5],and its dynamics on it is chaotic. If we denote T(θ) = sin2πθ, then it is easy to show that
ƒ = T o g o T -1

(topological conjugacy).

Thus ƒ is chaotic on[0,1] 		figure 91
Note: the complete definition of topological conjugacy is as follows:

Definition: Let X and Y be two subsets of the real line and let ƒ and g be functions, ƒ : X → X and g: Y → Y. Then ƒ and g are said to be topologically conjugate provided f and g are continuous, and there is a homeomorphism T: X → Y such that
T o f(x) = g o T(x),     ƒ = T -1 o g o T

holds for all x is in X 		figure 92
Note: A function T: X → Y is said to be a homeomorphism provided T is continuous, one-to-one and onto,and the inverse T -1 is also continuous.

What happens when r > 4?
figure 93
In this case, ƒ(x) > 1 for some values of x in the middle of the interval [0,1]. Suppose ƒ(x) > 1 when x is in K (refer to the graph).
Then for x is in K, ƒ(x) > 1, ƒ2(x) > 0, •••• ƒn(x) → -∞ as n → ∞. It is easy to see that all the preimages of K also escape to negative infinity.
Then what is the set

S = {x: ƒn(x) -∞} ?

It is a set similar to the Cantor set:

(1) It is totally disconnected. All points in S are separated from each other;
(2) On the other hand, it contains no "isolated points".

We call such a set a topological Cantor set. Hence, when r > 4, the invariant set of ƒ is a topological Cantor set.
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The logistic map, period-doubling to chaos