Chaos Theory: Universality – Feigenbaum constant and U-sequence

In this post we’ll be investigating some amazing results gained from non-linear dynamics. First we’ll be looking at the Feigenbaum constant, a quantitative universality followed by the U-sequence, a qualitative universality [1],[2].

       Feigenbaum constant

We can make some extraordinary observations from the bifurcation diagram. The ratio between each bifurcation converges to Feigenbaum’s constant, a universal constant for functions that have a periodic-doubling route to chaos, a fact discovered by Mitchell Feigenbaum in 1975. This constant is approximately 4.669 [2]. We can interpret from this constant that as we approach chaos each periodic region is smaller than the previous region by a factor of 4.669 (the Feigenbaum constant).

Bifurcation diagram showing source of the Feigenbaum constant

Diagram showing source of the Feigenbaum constant

The extraordinary thing about this constant is that its the same for all uni-modal functions, i.e. functions having a quadratic maximum and approach chaos via period doubling. If we call r_n the value of r for which the 2^n period becomes unstable then \delta = \lim_{n\to\infty}\frac{r_{n+1}-r_n}{r_{n+2}-r_{n+1}}=4.669.... The plot above shows the source of the Feigenbaum constant, the red lines indicate where the logistic map bifurcates to a period 2, 4 and 8 orbit respectively.

      U-sequence

The universal sequence, or U-sequence is commonly demonstrated on iterated maps whereby the mathematical model is a uni-modal mapping such as the logistic map [1,p.100]. In 1973, Metropolis, Stein and Stein investigated periods in uni-modal maps, proving that stable periodic solutions appear in an order independent of the map being iterated. In other words the periodic attractors always occur in the same sequence [2]. Up to period 6 the U-sequence is 1,2,4,6,5,3,6,5,6,4,6,5,6 [2]. The values represent the periodic-orbit of the mapping, so the first numbers show that the system goes from a period 1 to a period 2, to a period 4 and so on (see the bifurcation diagram for the logistic map in previous posts to view locations of these periodic orbits). We have highlighted the position where the period-3 window appears, a conspicuous occurrence and feature of the bifurcation diagram; something we will be exploring in the next post. To summarise, these phenomenal results imply that the form of a function isn’t relevant, however, the overall shape is.

Sources:

[1] ZELINKA, I. (2010). Evolutionary Algorithms and Chaotic Systems (eBook). Berlin, Germany. Springer.

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

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5 thoughts on “Chaos Theory: Universality – Feigenbaum constant and U-sequence

  1. How is the curve which is showing the source of Feigenbaum constant plotted?
    The recurrence relation is x(n+1)=ax(n)(1-x(n)) where 1<a<=4 and 0<x(0)<1
    where the quantity inside the bracket denotes the subscript.

    • Basically plotted bifurcation diagram for logistic map. Focusing on the interval 2.8 < r < 4 since it's more interesting and appropriate to use since we're interested at where period doubling occurs.

      Generated the plot with some Matlab code (which I could give you if you wish), the annotations help show how the fiegenbaum constant is calculated.

  2. I get back from work about 7pm gmt, I’ll give you around that time 🙂 think eventually if I get some time I’ll update blog some more and make all my m-files accessible.

    All the best 🙂

  3. Pingback: Chaos Theory: Hénon map (Part 2) | the world is mysterious

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