# Choas Theory: Tent Map (Part 3)

Following from the previous post, we’re going to explore the limitations of iterating the tent map for when $\mu = 2$. The graph below shows graphically what occurs given a rational number, say, $x_0 = 0.4$. For clarity, we’ve zoomed in on the point in which the system jumps straight to 0. This is the point in which the binary notation for x is all 0’s.

Limitations when using $\mu = 2$

To see the behaviour of the tent map for all $\mu$ across the range $[0,1]$ of x we plot the bifurcation diagram (seen below).

Bifurcation diagram for the Tent map.

Previously we stated that the behaviour of the tent map for $\mu < 1$ converges to 0 and so it isn’t of any interest to us, and so we’ve plotted the bifurcation for $1 \leq \mu \leq 2$. Unlike the Logistic map, the tent map doesn’t follow the period-doubling route to chaos. In fact it can be shown that there are no period-doublings [1].

Tent map histograms: distribution of data

Similarly to the logistic map we can represent the distribution of x values and the range on a histogram. Note we cannot use $\mu = 2$ due to numerical limitations when computing the results as previously discussed. For $\mu \leq 1$ all the points converge to a fixed point and so nearly all the points are distributed to one value. For $\mu > 1$ the points converge to a number of fixed points as shown by the histogram for $\mu = 1.2$. As we can see from the graph above, the distribution for the tent map tends towards a uniform distribution as $\mu$ tends to 2.

Sources:

[1] LAM, L. (1998). Non-Linear Physics for Beginners: Fractals, Chaos, Pattern Formation, Solutions, Cellular Automata and Complex Systems. London, UK. World Scientific Publishing.