Dynamical systems and ergodic theory Definition

Period-doubling cascade

Builds on Bifurcation in a family of maps Period-doubling bifurcation Periodic solution of a difference equation

Definition

Let $I \subseteq \R$ be an open interval and let $f_\mu \colon X \to X$ , $\mu \in I$ , be a family of maps of a set $X \subseteq \R$ with $(\mu, x) \mapsto f_\mu(x)$ smooth. A period-doubling cascade of the family is a strictly increasing sequence of parameters $\mu_1 < \mu_2 < \mu_3 < \cdots$ in $I$ with a limit $\mu_\infty \in I$ , such that for each $k \ge 1$ :

for $\mu$ slightly less than $\mu_k$ , the family has an attracting periodic solution of prime period $2^{k-1}$ ;
at $\mu = \mu_k$ this cycle undergoes a period-doubling bifurcation: it loses stability, and an attracting cycle of prime period $2^k$ is born from it.

Remarks

Example: the logistic family $f_r(x) = r x (1 - x)$ on $[0, 1]$ has such a cascade with $r_1 = 3$ and $r_2 = 1 + \sqrt{6} \approx 3.449$ . At $r_1 = 3$ the fixed point $1 - 1/r$ has multiplier $f_r'(1 - 1/r) = 2 - r = -1$ , and a $2$ -cycle appears. The limit is $r_\infty \approx 3.5699$ .

Non-example: the family $f_\mu(x) = \mu x$ on $\R$ has no period-doubling cascade. Its only possible attracting cycle is the fixed point $0$ for $\abs{\mu} < 1$ , and it never has an attracting cycle of period $2$ , so clause 2 fails already for $k = 1$ .

For typical smooth unimodal families with a quadratic maximum, such as the logistic family, the ratios $(\mu_k - \mu_{k-1})/(\mu_{k+1} - \mu_k)$ tend to the Feigenbaum constant $\delta \approx 4.6692$ .

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Bifurcation in a family of maps concept
Period-doubling bifurcation concept
Periodic solution of a difference equation concept
Period-doubling cascade definition

Tag	`21FMF8VKXY`
URI	`m2://21FMF8VKXY`
Kind	definition
Writing	W0 stub
MSC	37E05, 39A28, 39A33
Defines	Period-doubling cascade
Uses	Bifurcation in a family of maps, Period-doubling bifurcation, Periodic solution of a difference equation

Period-doubling cascade

Definition

Remarks

Reading path

Further reading

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