Chaos Theory: Dependence on parameter r (continued)

Previously we looked at how the logistic map is affected by changing the parameter r for small values. Now we’re going to look at what happens when we change it the parameter r for large values, say r = 3.4. Instead of converging to a fixed point, the orbit oscillates. For the logistic map, these orbits exist between r = 3 and r \approx 3.45 and the oscillations repeat every two iterations as shown below:

Period-2 orbit

Period-2 orbits

This is also known as a period-2 orbit. Again, for larger values of r, say r = 3.5, the orbit now repeats its oscillations every four iterations.

Period-4 orbits

Period-4 orbits

In other words, the previous orbit has doubled its period to a period-4 orbit. The splitting of the orbit is known as a bifurcation, i.e. at r = 3 the orbit bifurcates into two solutions, and for r = 3.5 the orbit bifurcates into four solutions. As we increase r even further, more bifurcations occur resulting in orbits of period 8,16,32,...,\infty. The table below shows values of r for which these bifurcations occur [1].

table

We can graph this data to get the bifurcation diagram for the logistic map, showing the possible long-term behaviour for all values of r between 0 and 4. Note, you can click on the image to see a larger version of the image.

Bifurcation for r between 0 and 4

Bifurcation for r between 0 and 4

To clarify and understand the points of bifurcations; we can also graph the bifurcation at a smaller interval for r.

Zoomed in of the above bifurcation, for values r = 2.8 to r = 4

Zoomed in of the above bifurcation. Values of r between r=2.8 and r=4.0

Clearly from both the table and graphs we can observe that the bifurcations occur faster and faster until r = 3.569946... \approx 3.57 = r_{\infty}. Here the map becomes chaotic and has infinite number of periodic orbits. Therefore, an infinite set of points or solutions. In other words, the map doesn’t converge to a fixed point or periodic orbit, instead it becomes aperiodic. This is an example into the periodic-doubling route to chaos. We can plot histograms with reference to the bifurcation diagrams. The histograms display the amount of time a typical orbit spends in some part of the attractor, for different values of r we have different attractors. Here we’ll use r = 3.2, 3.5, 3.8 and 4, with initial condition 0.4 and 10000 x values. From the bifurcation diagrams we expect that the distribution of data eventually spreads across the whole range of x values (0 to 1) as we increase the parameter r.

Logistic map histograms: Distribution of data

Logistic map histograms: Distribution of data

For r = 3.2, the data lies on two values which supports the fact that for r = 3.2 the orbit is period-2, and so the orbit oscillates between two values. Similarly for r = 3.5, the orbit is period-4 and the data lies on four values. As we approach r = 4 we begin to see the data converging to a U-shaped distribution, with the number of values intensifying at zero and one.

Sources:

[1] STROGATZ, S.H. (1994). Nonlinear dynamics and chaos. New York, USA. Westview Press, Perseus.

(I suggest you check this book out, it’s amazing and helped me a lot through my university years)!

P.S. All images generated on MATLAB, if you want my M-files then feel free to contact me!

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