Logistic map

The logistic map in this visualization

The logistic map is the map $f(x) = r x (1 - x)$ of the interval $[0, 1]$ to itself, with a parameter $0 \le r \le 4$. Starting from $x_0$ and applying $x_{n+1} = f(x_n)$ again and again gives a sequence; this page is about how its long run changes with $r$.

What the pictures show

The bifurcation diagram, the large picture: $r$ across, $x$ up. Each column takes the $r$ at its centre, starts from $x_0 = 1/\pi$, skips the first $T$ steps and records where the next $K$ land; the brighter, the more often. $T$ and $K$ are printed in the top right corner and grow as you zoom in. So each column is a long-run sample of one start, not the whole history of an orbit. “Typical” means almost every start (in the sense of length); $1/\pi$ stands in for them, without a proof that it is typical for every $r$.

The λ strip, under the diagram on the same $r$ axis: a numerical estimate of the Lyapunov exponent $\lambda(r) = \lim \frac1n \sum \ln|f'(x_i)|$ (the same typical start, 1000 steps skipped, 4000 averaged). It is the average exponential rate at which an infinitesimal difference between two starts grows: below 0 such differences shrink on average, above 0 they grow. At a superstable parameter ($\tfrac12$ lies on the attracting cycle) $\lambda = -\infty$; it is drawn as a downward arrow, labelled with the period, at the parameter found numerically.

The cobweb plot, heading the panel: the curve $y = f(x)$ (or $f^2$, $f^3$, $f^4$) and the diagonal $y = x$. Up from $x_0$ to the curve, across to the diagonal: one step. Filled dots are attracting fixed points, hollow ones repelling.

The time series, one key away: $x_n$ against $n$, and on its right the share of $10^5$ steps of this orbit in each of 40 bins; at $r = 4$ the ink line is the probability in theory.

Why it looks like this

At the fixed point $x^* = 1 - 1/r$ the slope is $f'(x^*) = 2 - r$. For $1 < r < 3$, $|f'(x^*)| < 1$ and it attracts its neighbours. As $r$ passes 3, $f'(x^*)$ passes $-1$: it loses its pull, and beside it a 2-cycle appears, $\frac{r + 1 \pm \sqrt{(r - 3)(r + 1)}}{2r}$, with multiplier $4 + 2r - r^2$. The pair is where $f^2$ newly meets the diagonal — switch the cobweb to $f^2$ to see it. At $r = 1 + \sqrt6$ the 2-cycle’s multiplier reaches $-1$ and it splits in turn.

The doublings come faster and faster and pile up at $r_\infty \approx 3.5699$. The ratio of successive spacings tends to Feigenbaum’s constant $\delta \approx 4.6692$, and not only for this map: it is the same for a whole class of one-humped maps with a quadratic maximum (found by Feigenbaum in 1978, proved later with computer assistance by Lanford).

A window inside chaos: as $r$ passes $1 + \sqrt8 \approx 3.8284$, $f^3$ touches the diagonal and a pair of 3-cycles is born, one attracting and one repelling (a saddle-node bifurcation, multiplier $+1$). Inside the window comes its own period-doubling cascade.

At $r = 4$, $f(\sin^2 \pi u) = \sin^2(2\pi u)$: putting $x = \sin^2(\pi u)$ carries the doubling map $u \mapsto 2u \bmod 1$ onto $f$ (a semiconjugacy: $u$ and $1 - u$ give the same $x$). From this one gets, exactly, the long-run density of a typical start, $\frac{1}{\pi\sqrt{x(1 - x)}}$, and $\lambda = \ln 2$.

Periodic windows occur throughout the chaotic-looking region. When an attracting cycle exists, typical starts approach it; otherwise the plotted column is a long-run sample of the stated initial value, not a classification of every possible start.

Easy to misread

• A column of the diagram is not the history of one orbit but a long-run sample after the transient is dropped; where the transient is slow (near a bifurcation) the column looks blurred.
• The period in the “long run” line is a numerical verdict: a repeat across three windows in a row, the same answer with more iterations, a multiplier smaller than 1 in size. A repeat in floating point is not proof of an attracting cycle; when the test cannot decide, the line says so.
• The sign of λ is no universal classifier: at $r = 4$ with $x_0 = 3/4$ the orbit never moves, yet λ along it is $\ln 2$ too.
• Special starts have their own fates: at $r = 4$, $x_0 = 1/2 \to 1 \to 0$ and stays at 0. All such starts together have length 0, so what is said about typical starts still holds.

History

“Logistic” comes from Verhulst’s 1838 differential equation for population growth. In 1976 May used this discrete map in Nature to show that simple models can behave in very complicated ways [1]. Ulam and von Neumann had already noted the invariant density at $r = 4$ in 1947 [2]. In 1978 Feigenbaum found the universal constants of the period-doubling cascade [3], and Singer gave the criterion for at most one attracting cycle [4].

Related concepts

Mandelbrot set · Lorenz attractor · Fourier series (page planned)

References

1. R. M. May, “Simple mathematical models with very complicated dynamics”, Nature 261 (1976) 459–467. doi:10.1038/261459a0
2. S. M. Ulam, J. von Neumann, “On combination of stochastic and deterministic processes” (abstract), Bulletin of the American Mathematical Society 53 (1947) 1120.
3. M. J. Feigenbaum, “Quantitative universality for a class of nonlinear transformations”, Journal of Statistical Physics 19 (1978) 25–52. doi:10.1007/BF01020332
4. D. Singer, “Stable orbits and bifurcation of maps of the interval”, SIAM Journal on Applied Mathematics 35 (1978) 260–267.
5. O. E. Lanford III, “A computer-assisted proof of the Feigenbaum conjectures”, Bulletin of the American Mathematical Society (N.S.) 6 (1982) 427–434.
6. M. V. Jakobson, “Absolutely continuous invariant measures for one-parameter families of one-dimensional maps”, Communications in Mathematical Physics 81 (1981) 39–88.
7. J. Graczyk, G. Świątek, “Generic hyperbolicity in the logistic family”, Annals of Mathematics 146 (1997) 1–52.
8. M. Lyubich, “Dynamics of quadratic polynomials, I–II”, Acta Mathematica 178 (1997) 185–297.

Further reading: Wikipedia: Logistic map; MacTutor History of Mathematics: Robert May; MacTutor History of Mathematics: Mitchell Feigenbaum.