Chaos Theory: Periodic Windows

The bifurcation diagram is much more complex than just a simple division of regions of chaos and periodic behaviour [1]. As previously stated, in this post we will be looking at one of the most iconic features that comes from observing the bifurcation diagram; periodic windows. Amongst all the chaos, a stable period-3 orbit appears at r \approx 3.83 as shown below. These regions (periodic windows) exist for r > r_{\infty}, they are areas whereby the system isn’t chaotic.

Periodic window at approx. r=3.83

First we will explore how the period-3 orbit is created. Note, all periodic windows are created in this manner [2]. Using the logistic equation x_{n+1}=rx_{n}(1-x_{n}) then we can rewrite as x_{n+1} = f(x_n), so x_{n+2} = f(f(x_n)) = f^2(x_n), similarly x_{n+3} = f^3(x_n). This mapping is vital in understanding the birth of the period-3 window. In the figure below we plot the third iterative mapping for r = 3.84 which has 8 fixed points such that f^3(x) = x.

Third iterative map of the logistic map for r=3.84

Third iterative map of the logistic map for r=3.84

Two of the fixed points are actually solutions to f(x)=x, the other six are marked by dots coloured red and black. The red dots correspond to a stable period-3 orbit, alternatively, the black notes correspond to a unstable period-3 orbit. We can show this is true by looking at the gradient at each fixed point. If it’s negative then it’s stable, and if positive then it’s unstable. Decreasing r to r = 3.8 then the graph changes as indicated by the red line below.

Third iterative map of the logistic map for r=3.8, 3.8284, 3.84

Third iterative map for r=3.8, 3.8284, 3.84

Therefore we expect for some value between r = 3.8 and r = 3.84 the graph of f^3(x) becomes tangent to the diagonal. At this value of r then the stable and unstable fixed points unite and annihilate in a tangent bifurcation [2], indicating the birth of the period-3 window. This value of r is approximately represented by the green line above, and is said to be r = 3.8284... [3]. As previously stated we can show that each periodic window is created in this way. In addition, it can be shown that there are periodic windows in all chaotic regions and so is a infinite number of periodic windows [4].


[1] GILMORE, R. & LEFRANC, M. (2002). The Topology of Chaos. New York, USA. John Wiley & Sons.

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

[3] TAKAYASU, H. (1990). Fractals in the physical sciences. Manchester, UK. Manchester University Press.

[4] FELDMAN, D.P. (2012). Chaos and Fractals: An elementary introduction. Oxford, UK. Oxford University Press.


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