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).

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 the value of *r* for which the period becomes unstable then . 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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*Related*

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.

I would be grateful if you could give the code.

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 🙂