Chaotic Real-Valued Sequences By Skew Tent Map

0. Note

This is the first assignment from my Masters Applied Digital Information Theory Course which has never been published anywhere and I, as the author and copyright holder, license this assignment customized CC-BY-SA where anyone can share, copy, republish, and sell on condition to state my name as the author and notify that the original and open version available here.

1. Introduction

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: