# Chaos Theory: Dependence on parameter r.

In the previous post on chaos theory we looked at the logistic map, a simple 1-D discrete map which illustrates chaotic behaviour.  This map was defined by the following equation:

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

We’re going to be looking at the maps dependence upon the parameter value r. The logistic map shows a wide range of different behaviours depending upon our choice of r. Here we’ll be using initial condition $x_0 = 0.4$. As we can see in the figure below, for $0 \leq \emph{r} \leq 1$ the orbit tends to 0, independent of the initial condition. However, when $1 < \emph{r} < 3$ the orbit will quickly converge towards a fixed point. For clarity we will connect the discrete points by line segments, note, only the blue points are significant, using initial condition $x = 0.4$ throughout. The population is decreasing and will eventually approach 0 as the number of iterations tends to infinity for $0 \leq r \leq 1$ the map tends to 0. For $1 < r < 3$ we can show mathematically that the population approaches a fixed point $\frac{r-1}{r}$.

Logistic map orbit for various r values.

Using (1) then the fixed points are when $rx(1-x) = x$. Expanding gives $rx - rx^2 = x$, we then divide through by $x$ to get $r - rx = 1$ and finally rearrange to get fixed point at $\frac{r-1}{r} = x$. For example, at $r=2.8$ the fixed point is at $0.64285...$. These results can be supported by plotting the corresponding cobweb plots (also seen below).

Logistic map: corresponding cobweb plots for various r values.

In the next post about chaos theory, we’re going to be investigating what happens when we look at higher values of r, alongside introducing ways of representing all the data onto one plot. We get some remarkable images when we do this!

As previously stated, all plots are made via MATLAB, if you wish to acquire the MATLAB codes, feel free to contact me!