Following from the previous post, we’re going to explore the limitations of iterating the tent map for when . The graph below shows graphically what occurs given a rational number, say, . 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.

To see the behaviour of the tent map for all across the range of *x* we plot the bifurcation diagram (seen below).

Previously we stated that the behaviour of the tent map for converges to 0 and so it isn’t of any interest to us, and so we’ve plotted the bifurcation for . 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].

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