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 .
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.
 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!