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

 (1)

where (sometimes also denoted ) is a Positive constant (the biotic potential''). We will start in the interval . In order to keep points in the interval, we must find appropriate conditions on . The maximum value can take is found from
 (2)

so the largest value of occurs for . Plugging this in, . Therefore, to keep the Map in the desired region, we must have . The Jacobian is
 (3)

and the Map is stable at a point if . Now we wish to find the Fixed Points of the Map, which occur when . Drop the subscript on
 (4)

 (5)

so the Fixed Points are and . An interesting thing happens if a value of 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 . The fixed points of order two must satisfy , so
 (6)

Now, drop the subscripts and rewrite
 (7)

 (8)

 (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
 (10)

These solutions are only Real for , 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
 (11)

This gives
 (12)
The Roots of this equation are all Imaginary for , 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 . 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 at which the cycle appears is given below.

 cycle () 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 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 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 , 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 equations which can be solved to give the onset of an arbitrary -cycle (Saha and Strogatz 1995) is

 (13)

The first of these give , , ..., , and the last uses the fact that the onset of period occurs by a Tangent Bifurcation, so the th Derivative is 1.

For , the solutions (, ..., , ) are (0, 0, ) and (, , 3), so the desired Bifurcation occurs at . Taking gives

 (14)

Solving the resulting Cubic Equation using computer algebra gives
 (15) (16) (17) (18)

where
 (19)

Numerically,
 (20) (21) (22) (23)

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

 (24)

together with three other simultaneous equations, where
 (25) (26) (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
 (28) (29) (30)

giving
 (31)

Gordon (1996) derives not only the value for the onset of the 3-Cycle, but also an upper bound for the -values supporting stable period 3 orbits. This value is obtained by solving the Cubic Equation
 (32)

for , then
 (33) (34)

The logistic equation has Correlation Exponent (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.

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