Feigenbaum constants

The Feigenbaum constants are two mathematical constants named after the mathematician Mitchell Feigenbaum. Both express ratios in a bifurcation diagram.

:$delta\; =\; 4.66920160910299067185320382cdots$

OEIS|id=A006890 is the limiting ratio of each bifurcation interval to the next, or between the diameters of successive circles on the real axis of the Mandelbrot set. Feigenbaum originally related this number to the period-doubling bifurcations in the logistic map, but also showed it to hold for all one-dimensional maps with a single quadratic maximum. As a consequence of this generality, every chaotic system that corresponds to this description will bifurcate at the same rate. Feigenbaum's constant can be used to predict when chaos will arise in such systems before it ever occurs. It was discovered in 1975.

The second Feigenbaum constant OEIS|id=A006891,

:$alpha\; =\; 2.502907875095892822283902873218cdots$,

is the ratio between the width of a tine and the width of one of its two subtines (except the tine closest to the fold).

These numbers apply to a large class of dynamical systems. Both numbers are believed to be transcendental although have not been proven to be so.

References

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* [http://keithbriggs.info/documents/Keith_Briggs_PhD.pdf Keith Briggs - PhD thesis (University of Melbourne 1997) Feigenbaum scaling in discrete dynamical systems]