The Feigenbaum constants are two mathematical constants named after the 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. ... In mathematics, particularly in dynamical systems, a bifurcation diagram shows the possible long-term values a variable of a system can obtain in function of a parameter of the system. ...

is the limiting 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. It was discovered in 1975. In number and more generally in algebra, a ratio is the linear relationship between two quantities of the same unit. ... A rendering of the Mandelbrot set In mathematics, the Mandelbrot set is defined as the connectedness locus of the family of complex quadratic polynomials. ... The logistic map is a polynomial mapping, often cited as an archetypal example of how complex, chaotic behaviour can arise from very simple non-linear dynamical equations. ... 1975 (MCMLXXV) was a common year starting on Wednesday. ...

The second Feigenbaum constant,

,

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. A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. ... In mathematics, a transcendental number is any real number that is not algebraic, that is, not the solution of a non-zero polynomial equation with integer (or, equivalently, rational) coefficients. ...

References

• Weisstein, Eric W., Feigenbaum Constant at MathWorld.