In the study of dynamical systems the term Feigenbaum function has been used to describe two different functions introduced by the physicist Mitchell Feigenbaum:
In the logistic map,
we have a function , and we want to study what happens when we iterate the map many times. The map might fall into a fixed point, a fixed cycle, or chaos. When the map falls into a stable fixed cycle of length , we would find that the graph of and the graph of intersects at points, and the slope of the graph of is bounded in at those intersections.
For example, when , we have a single intersection, with slope bounded in , indicating that it is a stable single fixed point.
As increases to beyond , the intersection point splits to two, which is a period doubling. For example, when , there are three intersection points, with the middle one unstable, and the two others stable.
As approaches , another period-doubling occurs in the same way. The period-doublings occur more and more frequently, until at a certain , the period doublings become infinite, and the map becomes chaotic. This is the period-doubling route to chaos.
Looking at the images, one can notice that at the point of chaos , the curve of looks like a fractal. Furthermore, as we repeat the period-doublings, the graphs seem to resemble each other, except that they are shrunken towards the middle, and rotated by 180 degrees.
This suggests to us a scaling limit: if we repeatedly double the function, then scale it up by for a certain constant : then at the limit, we would end up with a function that satisfies . Further, as the period-doubling intervals become shorter and shorter, the ratio between two period-doubling intervals converges to a limit, the first Feigenbaum constant . The constant can be numerically found by trying many possible values. For the wrong values, the map does not converge to a limit, but when it is , it converges. This is the second Feigenbaum constant.
In the chaotic regime, , the limit of the iterates of the map, becomes chaotic dark bands interspersed with non-chaotic bright bands.
When approaches , we have another period-doubling approach to chaos, but this time with periods 3, 6, 12, ... This again has the same Feigenbaum constants . The limit of is also the same function. This is an example of universality. We can also consider period-tripling route to chaos by picking a sequence of such that is the lowest value in the period- window of the bifurcation diagram. For example, we have , with the limit . This has a different pair of Feigenbaum constants . And converges to the fixed point toAs another example, period-4-pling has a pair of Feigenbaum constants distinct from that of period-doubling, even though period-4-pling is reached by two period-doublings. In detail, define such that is the lowest value in the period- window of the bifurcation diagram. Then we have , with the limit . This has a different pair of Feigenbaum constants .
In general, each period-multiplying route to chaos has its own pair of Feigenbaum constants. In fact, there are typically more than one. For example, for period-7-pling, there are at least 9 different pairs of Feigenbaum constants.
Generally, , and the relation becomes exact as both numbers increase to infinity: .
This functional equation arises in the study of one-dimensional maps that, as a function of a parameter, go through a period-doubling cascade. Discovered by Mitchell Feigenbaum and Predrag CvitanoviÃÂ, the equation is the mathematical expression of the universality of period doubling. It specifies a function g and a parameter by the relation
with the initial conditionsFor a particular form of solution with a quadratic dependence of the solution near is one of the Feigenbaum constants.
The power series of is approximately
The Feigenbaum function can be derived by a renormalization argument.
The Feigenbaum function satisfies for any map on the real line at the onset of chaos.
The Feigenbaum scaling function provides a complete description of the attractor of the logistic map at the end of the period-doubling cascade. The attractor is a Cantor set, and just as the middle-third Cantor set, it can be covered by a finite set of segments, all bigger than a minimal size d<sub>n</sub>. For a fixed d<sub>n</sub> the set of segments forms a cover Δ<sub>n</sub> of the attractor. The ratio of segments from two consecutive covers, Δ<sub>n</sub> and Δ<sub>n+1</sub> can be arranged to approximate a function σ, the Feigenbaum scaling function.