Period Doubling and the Route to Chaos

Definition A bifurcation is a parameter value at which the long-term behavior of a family of rules changes its kind. Take a family of rules $f_r$ on an interval whose slopes $f_r'$ change continuously. A cycle of length $k$ is a list of $k$ different states $x_1, \ldots, x_k$ that the rule visits in turn, sending each one to the next and $x_k$ back to $x_1$. Its multiplier is the product $f_r'(x_1) \cdots f_r'(x_k)$ of the slopes along the cycle. The cycle is stable if the multiplier lies strictly between $-1$ and $1$, and unstable if it is larger than $1$ in size. A period-doubling bifurcation happens at $r_0$ when, as $r$ increases through $r_0$, the multiplier of a stable cycle of length $k$ decreases through $-1$, so that the cycle becomes unstable, while a new cycle of length $2k$ branches off from it. In the logistic map $f_r(x) = r\,x(1 - x)$ the new cycle is stable just after $r_0$, and its points lie in pairs on either side of the points of the old cycle. The logistic map goes through a whole cascade of these bifurcations, called the period-doubling route to chaos. At $r = 3$ the fixed point $1 - 1/r$, whose multiplier is $2 - r$, gives way to a two-cycle. At $r = 1 + \sqrt{6} \approx 3.4495$ the two-cycle, whose multiplier is $4 + 2r - r^2$, gives way to a four-cycle. Doublings to $8$, $16$ and $32$ follow at about $3.5441$, $3.5644$ and $3.5688$, and the doubling points accumulate at $r_\infty \approx 3.569946$. The ratios of successive gaps between doubling points tend to Feigenbaum’s constant $\delta \approx 4.6692$. Mitchell Feigenbaum found in the 1970s that the same limit appears for a broad class of smooth one-hump maps whose top is shaped like a parabola, such as $r \sin(\pi x)$, whenever they go through such a cascade, which is why the constant is called universal.

Definition A bifurcation diagram of a family of rules $f_r$ plots long-term behavior against the parameter. For each value of $r$ on a fine grid, start an orbit at a chosen $x_0$, discard a fixed number of early steps as the transient, and plot the next values of the orbit as dots above $r$, with $r$ on the horizontal axis and the state on the vertical axis. Each vertical slice is a separate experiment with its own fixed $r$, so the diagram is not the path of one orbit moving from left to right. A slice with one sharp dot suggests a stable fixed point, a slice with $k$ sharp dots suggests a cycle of length $k$, and a slice with a cloud shows an orbit that did not settle during the plotted steps. The picture depends on the start, on the numbers of discarded and plotted steps and on the precision of the arithmetic, so it is evidence about long-term behavior and not a proof. For the logistic map the choice of start rarely matters, because it is known that whenever the map has a stable cycle, almost every start approaches it, meaning all starts except a set of total length zero.

Definition A system is deterministic when its current state fixes every later state exactly, with no random step. Deterministic chaos is behavior of such a system that never settles into a regular pattern and depends extremely sensitively on the starting state. Several precise definitions are in use. A widely used one, due to Robert Devaney, calls a rule $f$ that sends an interval $S$ into itself chaotic on $S$ when three conditions hold. The first is sensitive dependence on initial conditions. There is a distance $d > 0$ such that, as close as we like to every starting value in $S$, there is another starting value in $S$ whose orbit is more than $d$ away from the first orbit at some later step. The second is transitivity. For any two intervals inside $S$, however short, some orbit that starts in the first later visits the second. The third is that cycles are dense, meaning that every interval inside $S$, however short, contains a start whose orbit is a cycle. The logistic map with $r = 4$ is chaotic on $[0, 1]$ in this sense. A plotted cloud, or two computed orbits that drift apart, illustrates chaos but does not prove it, because a computer uses finitely many digits and finitely many steps.

Definition A periodic window of the logistic map is an interval of parameter values beyond $r_\infty$ in which a stable cycle appears and attracts the orbits, so that the bifurcation diagram shows a few sharp branches in place of the cloud. A window begins where a stable cycle is born, together with an unstable cycle of the same length, and the stable cycle then goes through its own period-doubling cascade before chaos returns. The most visible one is the period-three window, which begins at $r = 1 + \sqrt{8} \approx 3.8284$. Its three-cycle doubles to a six-cycle near $r = 3.8415$, then to twelve and so on, with the doublings accumulating near $3.8494$. After that the orbits are chaotic again, first within three separate bands, which merge into a single cloud near $r = 3.857$. Windows are found throughout the chaotic range. It has been proved that every interval of parameters between $r_\infty$ and $4$, however short, contains parameters with a stable cycle, and yet the parameters at which typical orbits never settle into a cycle take up a positive total length. Increasing $r$ beyond $r_\infty$ therefore does not mean steadily increasing disorder.