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.

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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.

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

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.

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