Chaos Theory: Logistic Map Introduction

This chapter looks at dynamical systems where by time is discrete. These systems are also known as iterated maps or difference equations. In particular, we will be focusing on 1-D dynamical systems, when plotted the orbit is just a sequence of points in the real numbers (\mathbb{R}). The two systems we will be looking at (as previously stated) are the logistic map and the tent map, these maps illustrate how chaotic behaviour arises from simple systems.

The logistic map was first proposed to describe the dynamics of insect populations by biologist Robert May in 1976. Often speculated as the archetypal example of how chaotic behaviour arises from simple dynamical equations, it is a recurrence relation, written as

x_{n+1}=rx_{n}(1-x_{n}), \qquad 0 \leq x \leq 1\qquad\qquad\qquad\qquad\qquad\qquad\qquad (1)

where x_{n} represents the ratio of the existing populations to a possible maximum population at year n such that 0 \leq x_n \leq 1. For example, x_{0} represents the initial ratio at year 0. The parameter r is a positive number, representing the rate of reproduction and starvation such that 0 \leq r \leq 4.

logistic mapThe graph of (1) is a parabola with maximum value of r/4 at x=\frac{1}{2} as shown above. Increasing the value of r=0 to r=4 make the map undergo a series of period-doubling bifurcations, something we will be exploring later on. The logistic map has fixed points at x=0 and x=1, regardless of r. However, the stability of the fixed points depend upon r. Again, something we will be exploring in the future.

P.S. All graphical output produced on MATLAB.

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