There are two mathematical constants called Feigenbaum constants, named after mathematician Mitchell Feigenbaum. Both express ratios in a bifurcation diagram. A mathematical constant is a quantity, usually a real number or a complex number, that arises naturally in mathematics and does not change. ... Mitchell Jay Feigenbaum (born December 19, 1944; Philadelphia, USA) is a mathematical physicist whose pioneering studies in chaos theory led to the discovery of the Feigenbaum constant. ... A bifurcation diagram shows the possible values a function can have as a function of a parameter of that function. ...

is the ratio between successive bifurcation intervals, or between the diameters of successive circles on the 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 displaying a single hump. 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. In algebra, a ratio is the relationship between two quantities. ... A rendering of the Mandelbrot set: black points represent the stable points under the iterative map In mathematics, the Mandelbrot set is a fractal that is defined as the set of points c in the complex plane for which the iteratively defined sequence: does not tend to infinity. ... The logistic map is a polynomial mapping, often cited as an archetypical example of how complex, chaotic behaviour can arise from very simple non-linear dynamical equations. ...

The second Feigenbaum constant,

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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. In engineering and mathematics, a dynamical system is a deterministic process in which a functions value changes over time according to a rule that is defined in terms of the functions current value. ... In mathematics, a transcendental number is any irrational number that is not an algebraic number, i. ...