📂Dynamics

Feigenbaum Universality

Conjecture

$$ x \mapsto f_{\alpha} (x) \qquad , x \in \mathbb{R}^{1} $$ Let us consider a dynamical system represented as a map, as defined above, which exhibits a period-doubling bifurcation with $\alpha$ as the bifurcation parameter. Suppose the sequence of parameters at which the $k$th period-doubling occurs is denoted by $\left\{ \alpha_{k} \right\}_{k=1}^{\infty}$. The ratio of the lengths between them will converge to some constant $\mu_{F} \approx 4.6692 \ldots$¹: $$ \lim_{k \to \infty} {\frac{ \alpha_{k} - \alpha_{k-1} }{ \alpha_{k+1} - \alpha_{k} }} = \mu_{F} $$

Explanation

The Feigenbaum conjectures state that period-doubling occurring under the aforementioned conditions, not just for the logistic family, universally exhibits the same rate of occurrence regardless of the system. This was reported by Feigenbaum in 1978².

For instance, although the logistic map and the Hénon map are different systems, it can be observed that the periods of the period-doubling converge to the same point as shown below³.

Consequently, this conjecture was confirmed to be true, leading to the Feigenbaum universality, and $\mu_{F}$ came to be known as the Feigenbaum constant^{4 5}.

In notation, the sequence $\left\{ \alpha_{k} \right\}_{k=1}^{\infty}$ has the limit $\alpha_{\infty}$, which implies that the system is becoming chaotic.