Feigenbaum constant

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The Feigenbaum delta constant has the value

$$\delta =4.669201609102990671853203820466\mathrm{\dots }$$

It governs the structure and behavior of many types of dynamical systems.It was discovered in the 1970s byhttp://www-groups.dcs.st-and.ac.uk/ history/Mathematicians/Feigenbaum.htmlMitchell Feigenbaum,while studying the logistic map

$$y^{\prime }=\mu \cdot y(1-y),$$

which produces the Feigenbaum tree:

Generated by GNU Octave and GNUPlot.

If the bifurcations in this tree (first few shown as dotted blue lines)are at points $b_1,b_2,b_3,\mathrm{\dots }$, then

$$\underset{n\to \mathrm{\infty }}{lim}\frac{b_n-b_{n-1}}{b_{n+1}-b_n}=\delta .$$

That is, the ratio of the intervals between the bifurcation pointsapproaches Feigenbaum’s constant.

However, this is only the beginning.Feigenbaum discovered that this constantarose in any dynamical systemthat approaches chaotic behavior via period-doubling bifurcation,and has a single quadratic maximum.So in some sense, Feigenbaum’s constantis a universal constant of chaos theory.

Feigenbaum’s constant appears in problems of fluid-flow turbulence,electronic oscillators, chemical reactions, and even the Mandelbrot set(the “budding” of the Mandelbrot set along the negative real axisoccurs at intervals determined by Feigenbaum’s constant).