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 . Instead of converging to a fixed point, the orbit oscillates. For the logistic map, these orbits exist between and and the oscillations repeat every two iterations as shown below:

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

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 the orbit bifurcates into two solutions, and for the orbit bifurcates into four solutions. As we increase *r* even further, more bifurcations occur resulting in orbits of period . The table below shows values of *r* for which these bifurcations occur [1].

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.

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

Clearly from both the table and graphs we can observe that the bifurcations occur faster and faster until . 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 and , 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*.

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

**Sources:
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[1] STROGATZ, S.H. (1994). *Nonlinear dynamics and chao*s. 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!