In the study of dynamical systems the term Feigenbaum function has been used to describe two different functions introduced by the physicist Mitchell Feigenbaum: [1]
In the logistic map,
(1) |
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 :
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 . [2] And converges to the fixed point to
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. [2]
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 Cvitanović, [3] the equation is the mathematical expression of the universality of period doubling. It specifies a function g and a parameter α by the relation
with the initial conditions
The power series of is approximately [4]
The Feigenbaum function can be derived by a renormalization argument. [5]
The Feigenbaum function satisfies [6]
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 dn. For a fixed dn the set of segments forms a cover Δn of the attractor. The ratio of segments from two consecutive covers, Δn and Δn+1 can be arranged to approximate a function σ, the Feigenbaum scaling function.
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cite book}}
: CS1 maint: location missing publisher (
link)
In the study of dynamical systems the term Feigenbaum function has been used to describe two different functions introduced by the physicist Mitchell Feigenbaum: [1]
In the logistic map,
(1) |
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 :
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 . [2] And converges to the fixed point to
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. [2]
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 Cvitanović, [3] the equation is the mathematical expression of the universality of period doubling. It specifies a function g and a parameter α by the relation
with the initial conditions
The power series of is approximately [4]
The Feigenbaum function can be derived by a renormalization argument. [5]
The Feigenbaum function satisfies [6]
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 dn. For a fixed dn the set of segments forms a cover Δn of the attractor. The ratio of segments from two consecutive covers, Δn and Δn+1 can be arranged to approximate a function σ, the Feigenbaum scaling function.
{{
cite book}}
: CS1 maint: location missing publisher (
link)