Replacing the logistic equation

(1)

with the quadratic recurrence equation

(2)

where  (sometimes also denoted ) is a positive constant sometimes known as the "biotic potential" gives the so-called logistic map. This quadratic map is capable of very complicated behavior. While John von Neumann had suggested using the logistic map  as a random number generator in the late 1940s, it was not until work by W. Ricker in 1954 and detailed analytic studies of logistic maps beginning in the 1950s with Paul Stein and Stanislaw Ulam that the complicated properties of this type of map beyond simple oscillatory behavior were widely noted (Wolfram 2002, pp. 918-919).

The first few iterations of the logistic map (2) give

			(3)
			(4)
			(5)

where  is the initial value, plotted above through five iterations (with increasing iteration number indicated by colors; 1 is red, 2 is yellow, 3 is green, 4 is blue, and 5 is violet) for various values of .

The logistic map computed using a graphical procedure (Tabor 1989, p. 217) is known as a web diagram. A web diagram showing the first hundred or so iterations of this procedure  and initial value  appears on the cover of Packel (1996; left figure) and is animated in the right figure above.

In general, this recurrence equation cannot be solved in closed form. Wolfram (2002, p. 1098) has postulated that any exact solution must be of the form

(6)

where  is some function and  is its inverse function. M. Trott (pers. comm.) has shown that smooth solutions cannot exist for generic values of , with the possible exception of  even and nonzero. The only exact solutions known are for r=-2, r=2 and r=4, summarized in the table below (Wolfram 2002, p. 1098), and R. Germundsson (pers. comm., Apr. 25, 2002) has proved that no other solutions of this form are possible.

		solution
2
4

The illustration above shows a bifurcation diagram of the logistic map obtained by plotting as a function of  a series of values for  obtained by starting with a random value , iterating many times, and discarding the first points corresponding to values before the iterates converge to the attractor. In other words, the set of fixed points of  corresponding to a given value of  are plotted for values of  increasing to the right.

An enlargement of the previous diagram around  is illustrated above, with value of  at which a -cycle first appears indicated by blue lines.

In order to study the fixed points of the logistic map, let an initial point  lie in the interval . Now find appropriate conditions on  which keep points in the interval. The maximum value  can take is found from

(7)

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

(8)

and the map is stable at a point  if .

Now find the fixed points of the map, which occur when . For convenience, drop the  subscript on 

	(9)
	(10)

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

			(11)
			(12)
			(13)
			(14)

For convenience, drop the  subscripts and rewrite

	(15)
	(16)
	(17)

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

			(18)
			(19)
			(20)
			(21)

These solutions are only real for , so this is where the 2-cycle begins. Note that the 2-cycle can also be found by computing the discriminant of

(22)

which is

(23)

When this equals 0, two roots coincide, so  is the onset of period doubling. For , the solutions  are given by (0, 0, ) and (, , 3), so the first bifurcation occurs at .

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

(24)

The first  of these give , , ..., , and the last uses the fact that the onset of period  occurs by a fold bifurcation, so the th derivative is 1. For small , these can be solved exactly, but the complexity rapidly increases with .

Now look for the onset of the 3-cycle. To eliminate the 1-cycles, consider

(25)

This gives

(26)

The roots of this equation are all imaginary for  less than some cutoff , at which point two of them convert to real roots. The value of  can be found by computing the discriminant of (26),

(27)

When the discriminant is zero, two roots coincide. This happens at

(28)

(OEIS A086178) as first shown by Myrberg in 1958, so the 3-cycle starts at . Saha and Strogatz (1995) give a simplified algebraic treatment for the 3-cycle which involves solving

(29)

together with three other simultaneous equations, where

			(30)
			(31)
			(32)

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

			(33)
			(34)
			(35)

giving

(36)

This has the positive solution found previously, .

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 related to the unique positive root  of the cubic equation

(37)

by

(38)

which is the unique positive root of the sextic polynomial

			(39)
			(40)

(OEIS A086179). For ,

			(41)
			(42)
			(43)

Solving the resulting cubic equation using computer algebra gives

(44)

and , ,  given by

(45)

giving numerical roots

			(46)
			(47)
			(48)
			(49)
			(50)
			(51)
			(52)
			(53)
			(54)
			(55)

where  is the silver constant.

To find the onset of the 4-cycle, eliminate the 2- and 1-cycles by considering

(56)

This gives a 12th-order polynomial in . The value of  can be found by computing the discriminant of this polynomial,

(57)

whose only real positive roots are

			(58)
			(59)
			(60)
			(61)

The 4-cycle therefore starts at

(62)

(OEIS A086180).

The onset of 5-cycles can be found analytically, and gives a 22nd-order polynomial in  whose real positive roots are  (OEIS A118452), 3.90557..., and 3.99026....

The onset of 6-cycles can be found analytically, and gives a 40th-order polynomial in  whose real positive roots are  (OEIS A118453), 3.93751..., 3.97776..., and 3.99758....

The onset of 7-cycles can be found analytically, and gives a 114th-order polynomial in  whose real positive roots are  (OEIS A118746), 3.77413..., 3.88602..., 3.92218..., 3.95102..., 3.96897..., 3.98474..., 3.99453..., and 3.99939....

The onset of 8-cycles can be found analytically as the polynomial root of the 8th-order polynomial

			(63)
			(64)

(OEIS A086181; Bailey 1993; Bailey and Broadhurst 2000; Borwein and Bailey 2003, pp. 51-52).

The onset of 16-cycles at  (OEIS A091517) was originally found using an integer relation calculation which determined that  is the root of a 120th-degree integer polynomial with coefficients that decrease monotonically from  to 1 (Bailey and Broadhurst 2000; Borwein and Bailey 2003, pp. 52-53). This result was subsequently verified exactly using computer algebra (Borwein and Bailey 2003, p. 53; Kotsireas and Karamanos 2004), and is an algebraic number of degree 240.

The following table summarizes the values  at which the -cycle first appears. For , 2, ..., these have algebraic degrees 1, 1, 2, 2, 22, 40, 114, 12, ... (OEIS A118454).

		OEIS	value
1	1		1
2	1		3
3	2	A086178	3.82842712...
4	2	A086180	3.44948974...
5	22	A118452	3.73817237...
6	40	A118453	3.62655316...
7	114	A118746	3.70164076...
8	12	A086181	3.54409035...
9
10
16	240	A091517	3.56440726...

The algebraic orders of the values of  (i.e., the onset of the -cycle) for , 2, ... are therefore given by 1, 2, 12, 240, ... (OEIS A087046). A table of the cycle type and value of  at which the cycle  appears is given below.

	cycle ()		OEIS
1	2	3
2	4	3.449490	A086180
3	8	3.544090	A086181
4	16	3.564407	A091517
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
	accumulation point	3.569945672	A098587

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). The period doubling 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, as illustrated below. 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 , and period doublings then begin again with cycles of 6, 12, ... and 7, 14, 28, ..., and then once again break off to chaos. However, note that considerable structure can be found within this chaos (Mayoral and Robledo 2005ab).

It is relatively easy to show that the logistic map is chaotic on an invariant Cantor set for  (Devaney 1989, pp. 31-50; Gulik 1992, pp. 112-126; Holmgren 1996, pp. 69-85), but in fact, it is also chaotic for all  (Robinson 1995, pp. 33-37; Kraft 1999).

The logistic map has correlation exponent  (Grassberger and Procaccia 1983), capacity dimension 0.538 (Grassberger 1981), and information dimension 0.5170976 (Grassberger and Procaccia 1983).

The logistic map can be used to generate random numbers (Umeno 1998; Andrecut 1998; Gonzáles and Pino 1999, 2000; Gonzáles et al. 2001ab; Wong et al. 2001, Trott 2004, p. 105).

REFERENCES:

Bailey, D. H. "Multiprecision Translation and Execution of Fortran Programs." ACM Trans. Math. Software 19, 288-319, 1993.

Bailey, D. H.; Borwein, J. M.; Calkin, N. J.; Girgensohn, R.; Luke, D. R.; and Moll, V. H. "Bifurcation Points in Chaos Theory." §2.3.2 in Experimental Mathematics in Action. Wellesley, MA: A K Peters, pp. 33-36, 2007.

Bailey, D. H.; Borwein, J. M.; Kapoor, V.; and Weisstein, E. W. "Ten Problems in Experimental Mathematics." Amer. Math. Monthly 113, 481-509, 2006.

Bailey, D. H. and Broadhurst, D. J. "Parallel Integer Relation Detection: Techniques and Applications." Math. Comput. 70, 1719-1736, 2000.

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

Beck, C.; and Schlögl, F. Thermodynamics of Chaotic Systems. Cambridge, England: Cambridge University Press, 1993.

Bogomolny, A. "Chaos Creation (There is Order in Chaos)." http://www.cut-the-knot.org/blue/chaos.shtml.

Borwein, J. and Bailey, D. "Bifurcation Points in the Logistic Iteration." §2.3 in Mathematics by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A K Peters, pp. 50-53, 2003.

Costa, U. M. S.; Lyra, M. L.; Plastino, A. R.; and Tsallis, C. "Power-Law Sensitivity to Initial Conditions Within a Logistic-Like Family of Maps: Fractality and Nonextensivity." Phys. Rev. E 56, 245-250, 1997.

Devaney, R. An Introduction to Chaotic Dynamical Systems, 2nd ed. Redwood City, CA: Addison-Wesley, 1989.

Dickau, R. M. "Bifurcation Diagram." http://mathforum.org/advanced/robertd/bifurcation.html.

Elaydi, S. N. Discrete Chaos. Boca Raton, FL: Chapman & Hall, 2000.

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.

Gulick, D. Encounters with Chaos. New York: McGraw-Hill, 1992.

Holmgren, R. A First Course in Discrete Dynamical Systems, 2nd ed. New York: Springer-Verlag, 1996.

 Jaffe, S. "The Logistic Equation: Computable Chaos." http://library.wolfram.com/infocenter/MathSource/579/.

Kotsireas, I. S. and Karamanos, K. "Exact Computation of the Bifurcation Point  of the Logistic Map and the Bailey-Broadhurst Conjectures." Internat. J. Bifurcation and Chaos 14, 2417-2423, 2004. http://www.lacim.uqam.ca/~plouffe/OEIS/archive_in_pdf/costas-cecm.pdf

Kraft, R. L. "Chaos, Cantor Sets, and Hyperbolicity for the Logistic Maps." Amer. Math. Monthly 106, 400-408, 1999.

Latora, V.; Rapisarda, A.; Tsallis, C.; and Baranger, M. "The Rate of Entropy Increase at the Edge of Chaos." Phys. Lett. A 273, 97, 2000.

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

MathPages. "Closed Forms for the Logistic Map." http://www.mathpages.com/home/kmath188.htm.

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

Mayoral, E. and Robledo, A. "A Recent Appreciation of the Singular Dynamics at the Edge of Chaos." Jan. 17, 2005a. http://arxiv.org/abs/cond-mat/0501398.

Mayoral, E. and Robledo, A. "Tsallis'  Index and Mori's  Phase Transitions at Edge of Chaos." Phys. Rev. E 72, 026209, 2005b.

Packel, E. Mathematica for Mathematics Teachers. Inst. of Computation, 1996.

Pearl, R. Ch. 18 in The Biology of Population Growth. New York: Knopf, 1978.

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

Quetelet, A. and Verhulst, P. F. Annuaire de l'Académie royale des sciences de Belgique 16, 97-124, 1850.

Ramasubramanian, K. and Sriram, M. S. "A Comparative Study of Computation of Lyapunov Spectra with Different Algorithms." 17 Sep 1999. http://arxiv.org/abs/chao-dyn/9909029.

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

Robinson, C. Stability, Symbolic Dynamics, and Chaos. Boca Raton, FL: CRC Press, 1995.

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.

Sloane, N. J. A. Sequences A086178, A086179, A086180, A086181, A087046, A091517, A098587, A118452, A118453, A118454, and A118746 in "The On-Line Encyclopedia of Integer Sequences."

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.

Tsallis, C.; Plastino, A. R.; and Zheng, W.-M. "Power-Law Sensitivity to Initial Conditions--New Entropic Representation." Chaos, Solitons & Fractals 8, 885-891, 1997.

Trott, M. The Mathematica GuideBook for Programming. New York: Springer-Verlag, pp. 24-25, 2004. http://www.mathematicaguidebooks.org/.

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

Wolfram Research, Inc. "Logistic Map." http://documents.wolfram.com/mathematica/Demos/SoundGallery/LogisticMap.html.

Wolfram, S. A New Kind of Science. Champaign, IL: Wolfram Media, pp. 918-921 and 1098, 2002.