Logistic Equation

The logistic equation (sometimes called the Verhulst Model since it was first published in 1845 by the Belgian P.-F. Verhulst) is defined by

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

where $r$ (sometimes also denoted $\mu$) is a Positive constant (the ``biotic potential''). We will start $x_0$ in the interval $[0,1]$. In order to keep points in the interval, we must find appropriate conditions on $r$. The maximum value $x_{n+1}$ can take is found from

$${dx_{n+1}\over dx_n} = r(1-2x_n) = 0,$$ (2)

so the largest value of $x_{n+1}$ occurs for $x_n={1/2}$. Plugging this in, $\max(x_{n+1}) = r/4$. Therefore, to keep the Map in the desired region, we must have $r\in (0,4]$. The Jacobian is

$$J = \left\vert{dx_{n+1}\over dx_n}\right\vert = \vert r (1-2x_n)\vert,$$ (3)

and the Map is stable at a point $x_0$ if $J(x_0) < 1$. Now we wish to find the Fixed Points of the Map, which occur when $x_{n+1} = x_n$. Drop the $n$ subscript on $x_n$

$$f(x)= r x(1-x)=x$$ (4)

$$x[1-r (1-x)] = x(1-r+r x) = r x\left[{x-\left({1-r^{-1}}\right)}\right]= 0,$$ (5)

so the Fixed Points are $x^{(1)}_1 = 0$ and $x^{(1)}_2 = 1-r^{-1}$. An interesting thing happens if a value of $r$ greater than 3 is chosen. The map becomes unstable and we get a Pitchfork Bifurcation with two stable orbits of period two corresponding to the two stable Fixed Points of $f^2(x)$. The fixed points of order two must satisfy $x_{n+2}=x_n$, so

$$x_{n+2} = r x_{n+1}(1-x_{n+1})$$

$$= r [r x_n(1-x_n)][1-r x_n(1-x_n)]$$

$$= r^2 x_n(1-x_n)(1-r x_n+r {x_n}^2) = x_n.$$ (6)

Now, drop the $n$ subscripts and rewrite

$$x\{r^2[1-x(1+r)+2r x^2-r x^3]-1\} = 0$$ (7)

$$x[-r^3 x^3+2r^3 x^2-r^2(1+r)x+(r^2-1)]=0$$ (8)

$$-r^3 x[x-(1-r^{-1})] [x^2-(1+r^{-1})x+r^{-1}(1+r^{-1})]=0.$$ (9)

Notice that we have found the first-order Fixed Points as well, since two iterations of a first-order Fixed Point produce a trivial second-order Fixed Point. The true 2-Cycles are given by solutions to the quadratic part

$$x^{(2)}_\pm = {\textstyle{1\over 2}}[(1+r^{-1})\pm\sqrt{(1+r^{-1})^2-4r^{-1}(1+r^{-1})}\,]$$

$$= {\textstyle{1\over 2}}[(1+r^{-1})\pm\sqrt{1+2r^{-1}+r^{-2}-4r^{-1}-4r^{-2}}\,]$$

$$= {\textstyle{1\over 2}}[(1+r^{-1})\pm\sqrt{1-2r^{-1}-3r^{-2}}\,]$$

$$= {\textstyle{1\over 2}}[(1+r^{-1})\pm r^{-1}\sqrt{(r-3)(r+1)}\,].$$ (10)

These solutions are only Real for $r\geq 3$, so this is where the 2-Cycle begins. Now look for the onset of the 4-Cycle. To eliminate the 2- and 1-Cycles, consider

$${f^4(x)-x\over f^2(x)-x}=0.$$ (11)

This gives

$1+{r^2}+(-{r^2}-{r^3}-{r^4}-{r^5})x$
$+(2{r^3}+{r^4}+4{r^5}+{r^6}+2{r^7}){x^2}$
$+(-{r^3}-5{r^5}-4{r^6}-5{r^7}-4{r^8}-{r^9}){x^3}$
$+(2{r^5}+6{r^6}+4{r^7}+14{r^8}+5{r^9}+3{r^{10}}){x^4}$
$+(-4{r^6}-{r^7}-18{r^8}-12{r^9}-12{r^{10}}-3{r^{11}}){x^5}$
$+({r^6}+10{r^8}+17{r^9}+18{r^{10}}+15{r^{11}}+{r^{12}}){x^6}$
$+(-2{r^8}-14{r^9}-12{r^{10}}-30{r^{11}}-6{r^{12}}){x^7}$
$+(6{r^9}+3{r^{10}}+30{r^{11}}+15{r^{12}}){x^8}$
$+(-{r^9}-15{r^{11}}-20{r^{12}}){x^9}+(3{r^{11}}+15{r^{12}}){x^{10}}$
	(12)

The Roots of this equation are all Imaginary for $r<1+\sqrt{6}$, but two of them convert to Real roots at this value (although this is difficult to show except by plugging in). The 4-Cycle therefore starts at $1+\sqrt{6}=3.449489\ldots$. The Bifurcations come faster and faster (8, 16, 32, ...), then suddenly break off. Beyond a certain point known as the Accumulation Point, periodicity gives way to Chaos.

A table of the Cycle type and value of $r_n$ at which the cycle $2^n$ appears is given below.

$n$	cycle ($2^n$)	$r_n$
1	2	3
2	4	3.449490
3	8	3.544090
4	16	3.564407
5	32	3.568750
6	64	3.56969
7	128	3.56989
8	256	3.569934
9	512	3.569943
10	1024	3.5699451
11	2048	3.569945557
$\infty$	acc. pt.	3.569945672

For additional values, see Rasband (1990, p. 23). Note that the table in Tabor (1989, p. 222) is incorrect, as is the $n=2$ entry in Lauwerier (1991). In the middle of the complexity, a window suddenly appears with a regular period like 3 or 7 as a result of Mode Locking. The period 3 Bifurcation occurs at $r=1+2\sqrt{2}=3.828427\ldots$, as is derived below. Following the 3-Cycle, the Period Doublings then begin again with Cycles of 6, 12, ...and 7, 14, 28, ..., and then once again break off to Chaos.

A set of $n+1$ equations which can be solved to give the onset of an arbitrary $n$-cycle (Saha and Strogatz 1995) is

$$\cases{ x_2=rx_1(1-x_1)\cr x_3=rx_2(1-x_2)\cr \vdots\cr ... ...n-1})\cr x_1=rx_n(1-x_n)\cr r^n\prod_{k=1}^n (1-2x_k)=1.\cr}$$ (13)

The first $n$ of these give $f(x)$, $f^2(x)$, ..., $f^n(x)$, and the last uses the fact that the onset of period $n$ occurs by a Tangent Bifurcation, so the $n$th Derivative is 1.

For $n=2$, the solutions ($x_1$, ..., $x_n$, $r$) are (0, 0, $\pm 1$) and (${2/3}$, ${2/3}$, 3), so the desired Bifurcation occurs at $r_2=3$. Taking $n=3$ gives

$${d[f^3(x)]\over dx} = {d[f^3(x)]\over d[f^2(x)]} {d[f^2(x)]\over d[f(x)]} {d[f(x)]\over dx}$$

$$= {d[f(z)]\over dz} {d[f(y)]\over dy} {d[f(x)]\over dx}$$

$$= r^3(1-2z)(1-2y)(1-2x).$$ (14)

Solving the resulting Cubic Equation using computer algebra gives

$$x_1 = -\left({{2^{5^6}\over 63\cdot 7^{1/3}}+{1\over 63\cdot 28^{1/3}}}\right)c^2$$

$$\phantom{=} - {1\over 9\cdot 98^{1/3}} c +{10+\sqrt{2}\over 21} +\left({{4\cdot 2^{5/6}\over 9\cdot 7^{1/3}}-{2^{1/3}\over 7^{1/3}}}\right)c^{-1}$$

$$\phantom{=} +{25\cdot 28^{1/3}-44\cdot 2^{1/6}7^{1/3}\over 9} c^{-2}$$ (15)

$$x_2 = \left({{1\over 63\cdot 28^{1/3}}+{2^{5/6}\over 63\cdot 7^{1/3}}}\right)c^2 -{2^{2/3}\over 9\cdot 7^{2/3}} c+{10+\sqrt{2}\over 21}$$

$$\phantom{=} +\left({{8\cdot 2^{5/6}\over 9\cdot 7^{1/3}}-{2\cdot 2^{1/3}\over 7^{1/3}}}\right)c^{-1}$$

$$\phantom{=} +{44\cdot 2^{1/6}7^{1/3}-25\cdot 28^{1/3}\over 9} c^{-2}$$ (16)

$$x_3 = {1\over 3\cdot 98^{1/3}}c+{10+\sqrt{2}\over 21}+{2^{1/3}(9-4\sqrt{2}\,)\over 3\cdot 7^{1/3}} c^{-1}$$ (17)

$$r = 1+2\sqrt{2},$$ (18)

where

$$c\equiv(-25+22 \sqrt{2}+3 \sqrt{3}\sqrt{1100\sqrt{2}-1593}\,)^{1/3}.$$ (19)

Numerically,

$$x_1 = 0.514355\ldots$$ (20)

$$x_2 = 0.956318\ldots$$ (21)

$$x_3 = 0.159929\ldots$$ (22)

$$r = 3.828427\ldots.$$ (23)

Saha and Strogatz (1995) give a simplified algebraic treatment which involves solving

$$r^3(1-2\alpha+4\beta-8\gamma)=1,$$ (24)

together with three other simultaneous equations, where

$$\alpha \equiv x_1+x_2+x_3$$ (25)

$$\beta \equiv x_1x_2+x_1x_3+x_2x_3$$ (26)

$$\gamma \equiv x_1x_2x_3.$$ (27)

Further simplifications still are provided in Bechhoeffer (1996) and Gordon (1996), but neither of these techniques generalizes easily to higher Cycles. Bechhoeffer (1996) expresses the three additional equations as

$$2\alpha = 3+r^{-1}$$ (28)

$$4\beta = {\textstyle{3\over 2}}+5r^{-1}+{\textstyle{3\over 2}}r^{-2}$$ (29)

$$8\gamma = -{\textstyle{1\over 2}}+{\textstyle{7\over 2}}r^{-1}+{\textstyle{5\over 2}}r^{-2}+{\textstyle{5\over 2}}r^{-3},$$ (30)

giving

$$r^2-2r-7=0.$$ (31)

Gordon (1996) derives not only the value for the onset of the 3-Cycle, but also an upper bound for the $r$-values supporting stable period 3 orbits. This value is obtained by solving the Cubic Equation

$$s^3-11s^2+37s-108=0$$ (32)

for $s$, then

$r'=1+\sqrt{s}$	(33)
$=1+\sqrt{{\textstyle{11\over 3}}+({\textstyle{1915\over 54}}+{\textstyle{5\over... ...,)^{1/3}+({\textstyle{1915\over 54}}-{\textstyle{5\over 2}}\sqrt{201}\,)^{1/3}}$
$=3.841499007543\ldots.$	(34)

The logistic equation has Correlation Exponent $0.500\pm 0.005$ (Grassberger and Procaccia 1983), Capacity Dimension 0.538 (Grassberger 1981), and Information Dimension 0.5170976 (Grassberger and Procaccia 1983).

See also Bifurcation, Feigenbaum Constant, Logistic Distribution, Logistic Equation r=4, Logistic Growth Curve, Period Three Theorem, Quadratic Map

References

Bechhoeffer, J. ``The Birth of Period 3, Revisited.'' Math. Mag. 69, 115-118, 1996.

Bogomolny, A. ``Chaos Creation (There is Order in Chaos).'' http://www.cut-the-knot.com/blue/chaos.html.

Dickau, R. M. ``Bifurcation Diagram.'' http://forum.swarthmore.edu/advanced/robertd/bifurcation.html.

Gleick, J. Chaos: Making a New Science. New York: Penguin Books, pp. 69-80, 1988.

Gordon, W. B. ``Period Three Trajectories of the Logistic Map.'' Math. Mag. 69, 118-120, 1996.

Grassberger, P. ``On the Hausdorff Dimension of Fractal Attractors.'' J. Stat. Phys. 26, 173-179, 1981.

Grassberger, P. and Procaccia, I. ``Measuring the Strangeness of Strange Attractors.'' Physica D 9, 189-208, 1983.

Lauwerier, H. Fractals: Endlessly Repeated Geometrical Figures. Princeton, NJ: Princeton University Press, pp. 119-122, 1991.

May, R. M. ``Simple Mathematical Models with Very Complicated Dynamics.'' Nature 261, 459-467, 1976.

Peitgen, H.-O.; Jürgens, H.; and Saupe, D. Chaos and Fractals: New Frontiers of Science. New York: Springer-Verlag, pp. 585-653, 1992.

Rasband, S. N. Chaotic Dynamics of Nonlinear Systems. New York: Wiley, p. 23, 1990.

Russell, D. A.; Hanson, J. D.; and Ott, E. ``Dimension of Strange Attractors.'' Phys. Rev. Let. 45, 1175-1178, 1980.

Saha, P. and Strogatz, S. H. ``The Birth of Period Three.'' Math. Mag. 68, 42-47, 1995.

Strogatz, S. H. Nonlinear Dynamics and Chaos. Reading, MA: Addison-Wesley, 1994.

Tabor, M. Chaos and Integrability in Nonlinear Dynamics: An Introduction. New York: Wiley, 1989.

Wagon, S. ``The Dynamics of the Quadratic Map.'' §4.4 in Mathematica in Action. New York: W. H. Freeman, pp. 117-140, 1991.