FeigenbaumConstants - crowlogic/arb4j GitHub Wiki

The Feigenbaum constants are linked with bifurcation theory, the mathematical study of changes in the qualitative or topological structure of a given family. In particular, these constants are associated with period-doubling bifurcations in maps and differential equations.

Here's a formal, abstract mathematical description:

1. Consider a one-parameter family of unimodal maps $f_\mu: I \rightarrow I$.

A unimodal map is a continuous real function with exactly one local maximum. $I$ denotes an interval, and $\mu$ is a real parameter.

2. Assume that the family $\lbrace f_\mu \rbrace$ undergoes a cascade of period-doubling bifurcations at parameter values $\mu_n$, for $n = 1, 2, 3, ...$, which means for each $\mu_n$, the map $f_{\mu_n}$ has a periodic point of period $2^n$.

3. Then, the ratios of successive differences of these parameter values converge to the first Feigenbaum constant (δ)

$$ \lim_{n \rightarrow \infty} \frac{\mu_{n+1} - \mu_n}{\mu_n - \mu_{n-1}} = \delta \approx 4.669201609102990671853 $$

4. Consider the n-periodic orbits $\lbrace x_{i,n} \rbrace_{i=0}^{2^n - 1}$ at the bifurcation points, where $x_{i,n}$ denotes the i-th point in the orbit under $f_{\mu_n}$. Define $a_n = \max |f_{\mu_n}'(x_{i,n})|$ to be the maximum derivative along the period $2^n$ cycle.

5. Then the ratios of the differences of these derivatives at periodic points for successive bifurcations converge to the second Feigenbaum constant (α):

$$ \lim_{n \rightarrow \infty} \frac{a_{n+1}}{a_n} = \alpha \approx 2.502907875095892822283 $$

These results are remarkable because they are "universal", in the sense that they apply to a broad class of nonlinear systems, not just to specific equations. The constants δ and α have been found to be the same for a wide variety of mathematical functions, suggesting an underlying pattern or structure in the apparently chaotic behavior of these systems.