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The chaos theory is a field of study in mathematics that studies the behaviour of dynamical system. Rober L. Devaney stated a system can be chaotic if it is sensitive to initial value, topologically mixing, and have a dense periodic orbit. The theory was summarized by Edward Lorenz. Although the system was highly determined by the initial value, in the end the sequence is unpredictable. Chaos exist in many natural system such as the climate and whether. In the field of computer science, it’s applied to generate random values.
2. Tent Map
The tent map can be formulated as t = x/c for 0 ≤ x < c and t = (1-x)/(1-c) for c ≤ x ≤ 1 where:
c = critical point
x = initial value
t = tent map (next value for sequence)
For example c = 0.5 with x from 0 to 1:
Figure 1. Tent Map 1
Figure 1 is based on the equation that the value would range from 0 to 1, and when “x” reached the critical point, the value of “t” will equals to 1. It is up to us to decide the critical point, as in Figure 2 shows if “c” equals to 0.2. Another information is if “x” equals to 0 or 1 the end result will be 0, thus using this value is not recommended.
Figure 2. Tent Map 2
3. Chaotic Sequence
In some areas the tent map equation is used to generate chaotic sequence, here t(n) = x(n+1). The equation would turn into x(n+1) = (x(n))/c for 0 ≤ x(n) < c and x(n+1) = (1-x(n))/(1-c) for c ≤ x(n) ≤ 1. Why the equation is stated differently? So the value of x(n+1) will never…