The Logistic Map is hiding in the Mandelbrot Set

The logistic map is a simple second-order function on the unit interval:

$\displaystyle x_{n+1} = r x_n (1-x_n) \,,$

where ${x_n}$ is the variable value at stage ${n}$ and ${r}$ is the “growth rate”. For ${1 \le r \le 4}$ , the map sends the unit interval [0,1] into itself. It is a simple nonlinear difference equation, whose solutions exhibit both regular and erratic behaviour, and it is often used to demonstrate some important characteristics of chaotic motion. It describes behaviour found in a wide range of disciplines: physics, engineering, economics and biology.

Logistic Map hiding in the Mandelbrot Set

The discrete logistic map was popularised by ecologist Robert May in a 1976 paper in Nature entitled “Simple mathematical models with very complicated dynamics”. He used the form of the map shown above. However, a simple transformation converts the map to a form equivalent to (a real variable form of) the map used by Benoit Mandelbrot to generate the enormously complex fractal set that is named after him. This is the quadratic map

$\displaystyle z_{n+1} = z_n^2 + c \,.$

Transforming the variable and parameter of the logistic map ${x_{n+1} = r x_n (1-x_n)}$ to new variables,

$\displaystyle z = r\left({\frac {1}{2}}-x\right)\,,\qquad c = \frac {r}{2}\left(1-\frac {r}{2}\right) \,,$

we get a correspondence between the parameter space of the logistic map and that of the Mandelbrot set.

The connection between the logistic map and the (real variable) Mandelbrot map is shown in the Figure below. As ${r}$ ranges between 1 and 4, ${c}$ ranges between ${\frac{1}{4}}$ and ${-2}$ . The cusp of the cardiod is at ${r = 1}$ and ${c = \frac{1}{4}}$ . The tip of the antenna at the left extremity of the Mandelbrot set corresponds to ${r=4}$ for the logistic map and ${c=-2}$ for the Mandelbrot map.

The logistic map lurking within the Mandelbrot set. Figure adapted from Peitgen and Richter (1986).

The Road to Chaos

The following list illustrates the wide variety of behaviours of the solutions and how they depend critically on the value of the parameter ${r}$ or ${c}$ and, in the chaotic regime, on the initial conditions.

For ${1 < r < 2}$ , or ${\frac{1}{4} > c > 0}$ , the solution tends rapidly to ${\frac{r-1}{r}}$ for all initial conditions.
For ${2 < r < 3}$ , or ${0 > c > -\frac{3}{4}}$ , the solution tends eventually to ${\frac{r-1}{r}}$ , after oscillating about that value for some time.
For ${3 < r < 1+\sqrt{6}}$ , or ${ -\frac{3}{4} > c > -\frac{5}{4}}$ the solution approaches a period 2 cycle, oscillating betweeen two values.
For increasing ${r}$ , or decreasing (real) ${c}$ , solutions with successive doublings of period, 4, 8, 16, … are found.
At ${r = 1+\sqrt{8}}$ , or ${c = -\frac{7}{4}}$ a stable period 3 cycle emerges.

Many of the bifurcation points of the logistic map are at irrational values of ${r}$ , for example, the transition from period 2 solutions to period 4 solutions is at ${r = 1 + \sqrt{6}}$ and the onset of period 3 solutions is at ${r = 1 + \sqrt{8}}$ . The corresponding values of ${c}$ are low-order rational numbers, ${-\frac{5}{4}}$ and ${-\frac{7}{4}}$ .

There is a much richer spectrum of solution types than listed above. Most books on chaos and dynamics give a more exhaustive description of the solutions; see, for example Ott (1993).

Sources

${\bullet}$ Ott, Edward, 1993: Chaos in Dynamical Systems. Cambridge University Press, 385pp. ISBN: 9-780-5214-3799-8.

${\bullet}$ May, R., 1976: Simple mathematical models with very complicated dynamics. Nature, 261, 459–467 (1976). \url{https://doi.org/10.1038/261459a0}

${\bullet}$ Peitgen, Heinz-Otto and Peter Richter, 1986: The Beauty of Fractals. Springer-Verlag, 199pp. ISBN: 0-387-15851-0.

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