TOPICS
Search

Logistic Map


LogisticEquationIterations

Replacing the logistic equation

 (dx)/(dt)=rx(1-x)
(1)

with the quadratic recurrence equation

 x_(n+1)=rx_n(1-x_n),
(2)

where r (sometimes also denoted mu) 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 x_(n+1)=4x_n(1-x_n) 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

x_1=rx_0(1-x_0)
(3)
x_2=r^2(1-x_0)x_0(1-rx_0+rx_0^2)
(4)
x_3=r^3(1-x_0)x_0(1-rx_0+rx_0^2)×(1-r^2x_0+r^2x_0^2+r^3x_0^2-2r^3x_0^3+r^3x_0^4),
(5)

where x_0 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 r.

Web diagram of the logistic map

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 r=3.741 and initial value x_0 approx 0.00079 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

 x_n=1/2{1-f[r^nf^(-1)(1-2x_0)]},
(6)

where f is some function and f^(-1) is its inverse function. M. Trott (pers. comm.) has shown that smooth solutions cannot exist for generic values of r, with the possible exception of r 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.

rf(x)solution
-22cos(1/3(pi-sqrt(3)x))1/2-cos{1/3[pi-(-2)^n(pi-3cos^(-1)(1/2-x_0))]}
2e^x1/2{1-exp[2^nln(1-2x_0)]}
4cosx1/2{1-cos[2^ncos^(-1)(1-2x_0)]}
LogisticEquationBifurcation

The illustration above shows a bifurcation diagram of the logistic map obtained by plotting as a function of r a series of values for x_n obtained by starting with a random value x_0, 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 x_n corresponding to a given value of r are plotted for values of r increasing to the right.

LogisticMapBifurcations

An enlargement of the previous diagram around r=3.5 is illustrated above, with value of r at which a 2^n-cycle first appears indicated by blue lines.

In order to study the fixed points of the logistic map, let an initial point x_0 lie in the interval [0,1]. Now find appropriate conditions on r which keep points in the interval. The maximum value x_(n+1) can take is found from

 (dx_(n+1))/(dx_n)=r(1-2x_n)=0,
(7)

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=|(dx_(n+1))/(dx_n)|=|r(1-2x_n)|,
(8)

and the map is stable at a point x_0 if J(x_0)<1.

Now find the fixed points of the map, which occur when x_(n+1)=x_n. For convenience, drop the n subscript on x_n

f(x)=rx(1-x)=x
(9)
x[1-r(1-x)]=x(1-r+rx)=rx[x-(1-r^(-1))]=0,
(10)

so the fixed points are x_1^((1))=0 and x_2^((1))=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)=rx_(n+1)(1-x_(n+1))
(11)
=r[rx_n(1-x_n)][1-rx_n(1-x_n)]
(12)
=r^2x_n(1-x_n)(1-rx_n+rx_n^2)
(13)
=x_n.
(14)

For convenience, drop the n subscripts and rewrite

x{r^2[1-x(1+r)+2rx^2-rx^3]-1}=0
(15)
x[-r^3x^3+2r^3x^2-r^2(1+r)x+(r^2-1)]=0
(16)
-r^3x[x-(1-r^(-1))][x^2-(1+r^(-1))x+r^(-1)(1+r^(-1))]=0.
(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

x_+/-^((2))=1/2[(1+r^(-1))+/-sqrt((1+r^(-1))^2-4r^(-1)(1+r^(-1)))]
(18)
=1/2[(1+r^(-1))+/-sqrt(1+2r^(-1)+r^(-2)-4r^(-1)-4r^(-2))]
(19)
=1/2[(1+r^(-1))+/-sqrt(1-2r^(-1)-3r^(-2))]
(20)
=1/2[(1+r^(-1))+/-r^(-1)sqrt((r-3)(r+1))].
(21)

These solutions are only real for r>=3, so this is where the 2-cycle begins. Note that the 2-cycle can also be found by computing the discriminant of

 (f^2(x)-x)/(f(x)-x)=r^2x^2-r(1+r)x+(1+r)=0,
(22)

which is

 ((1+r)(3-r))/(r^2).
(23)

When this equals 0, two roots coincide, so r_2=3 is the onset of period doubling. For n=2, the solutions (x_1,x_2,r) are given by (0, 0, +/-1) and (2/3, 2/3, 3), so the first bifurcation occurs at r_2=3.

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

 {x_2=rx_1(1-x_1); x_3=rx_2(1-x_2); |; x_n=rx_(n-1)(1-x_(n-1)); x_1=rx_n(1-x_n); r^nproduct_(k=1)^(n)(1-2x_k)=1.
(24)

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 fold bifurcation, so the nth derivative is 1. For small n, these can be solved exactly, but the complexity rapidly increases with n.

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

 (f^3(x)-x)/(f(x)-x)=0.
(25)

This gives

 1+r+r^2-(r^4+2r^3+2r^2+r)x+(2r^5+3r^4+3r^3+r^2)x^2-(r^6+5r^5+3r^4+r^3)x^3+(3r^6+4r^5+r^4)x^4-(3r^6-r^5)x^5+r^6x^6=0.
(26)

The roots of this equation are all imaginary for r less than some cutoff r_3, at which point two of them convert to real roots. The value of r_3 can be found by computing the discriminant of (26),

 D=((r^2-5r+7)^2(r^2-2r-7)^3(1+r+r^2)^2)/(r^(30)).
(27)

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

 r_3=1+2sqrt(2)=3.828427...
(28)

(OEIS A086178) as first shown by Myrberg in 1958, so the 3-cycle starts at r_3. Saha and Strogatz (1995) give a simplified algebraic treatment for the 3-cycle which involves solving

 r^3(1-2alpha+4beta-8gamma)=1,
(29)

together with three other simultaneous equations, where

alpha=x_1+x_2+x_3
(30)
beta=x_1x_2+x_1x_3+x_2x_3
(31)
gamma=x_1x_2x_3.
(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

2alpha=3+r^(-1)
(33)
4beta=3/2+5r^(-1)+3/2r^(-2)
(34)
8gamma=-1/2+7/2r^(-1)+5/2r^(-2)+5/2r^(-3),
(35)

giving

 r^2-2r-7=0.
(36)

This has the positive solution found previously, r_3=1+2sqrt(2).

Gordon (1996) derives not only the value for the onset of the 3-cycle, but also an upper bound r^' for the r-values supporting stable period-3 orbits. This value is related to the unique positive root s_1 of the cubic equation

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

by

 r^'=1+sqrt(s_1),
(38)

which is the unique positive root of the sextic polynomial

r^'=(x^6-6x^5+4x^4+24x^3-14x^2-36x-81)_2
(39)
=3.841499007543...
(40)

(OEIS A086179). For n=3,

(d[f^3(x)])/(dx)=(d[f^3(x)])/(d[f^2(x)])(d[f^2(x)])/(d[f(x)])(d[f(x)])/(dx)
(41)
=(d[f(z)])/(dz)(d[f(y)])/(dy)(d[f(x)])/(dx)
(42)
=r^3(1-2z)(1-2y)(1-2x).
(43)

Solving the resulting cubic equation using computer algebra gives

 r=1+2sqrt(2)
(44)

and x_1, x_2, x_3 given by

 343x^6-980x^5+868x^4-134x^3-161x^2+70x-7=0,
(45)

giving numerical roots

x_1=(343x^6-980x^5+868x^4-134x^3-161x^2+70x-7)_2
(46)
=sqrt(2)-2/7(1-2sqrt(2))[1+cos(2/7pi)-5]
(47)
=0.514355...
(48)
x_2=(343x^6-980x^5+868x^4-134x^3-161x^2+70x-7)_4
(49)
=sqrt(2)-2/7(1-2sqrt(2))[1+cos(4/7pi)-5]
(50)
=0.956318...
(51)
x_3=(343x^6-980x^5+868x^4-134x^3-161x^2+70x-7)_5
(52)
=sqrt(2)-2/7(1-2sqrt(2))[1+cos(6/7pi)-5]
(53)
=0.159929...
(54)
r=3.828427...,
(55)

where 2+2cos(2/7pi) is the silver constant.

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

 (f^4(x)-x)/(f^2(x)-x)=0.
(56)

This gives a 12th-order polynomial in x. The value of r_4 can be found by computing the discriminant of this polynomial,

 D=((r^2+1)^3(r^2-4r+5)^3(r^2-2r-5)^2)/(r^(132))(r^6-6r^5+3r^4+28r^3-9r^2-54r-135),
(57)

whose only real positive roots are

r_4^'=1+sqrt(6)
(58)
=3.449489...
(59)
r_4^('')=1+sqrt(4+3(4^(1/3)))
(60)
=3.906010...
(61)

The 4-cycle therefore starts at

 r_4=1+sqrt(6)=3.449489...
(62)

(OEIS A086180).

The onset of 5-cycles can be found analytically, and gives a 22nd-order polynomial in r whose real positive roots are r_5=3.73817... (OEIS A118452), 3.90557..., and 3.99026....

The onset of 6-cycles can be found analytically, and gives a 40th-order polynomial in r whose real positive roots are r_6=3.62655... (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 r whose real positive roots are r_7=3.70164... (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

r_8=(4913+2108t^2-604t^3-977t^4+8t^5+44t^6+392t^7-193t^8-40t^9+48t^(10)-12t^(11)+t^(12))_3
(63)
=3.54409...
(64)

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

The onset of 16-cycles at r_(16)=3.564407266095... (OEIS A091517) was originally found using an integer relation calculation which determined that alpha=r_(16)(r_(16)-2) is the root of a 120th-degree integer polynomial with coefficients that decrease monotonically from 257^(30) 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 r_n at which the n-cycle first appears. For n=1, 2, ..., these have algebraic degrees 1, 1, 2, 2, 22, 40, 114, 12, ... (OEIS A118454).

ndeg(r_n)OEISvalue
111
213
32A0861783.82842712...
42A0861803.44948974...
522A1184523.73817237...
640A1184533.62655316...
7114A1187463.70164076...
812A0861813.54409035...
9
10
16240A0915173.56440726...

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

ncycle (2^n)r_(2^n)OEIS
123
243.449490A086180
383.544090A086181
4163.564407A091517
5323.568750
6643.56969
71283.56989
82563.569934
95123.569943
1010243.5699451
1120483.569945557
inftyaccumulation point3.569945672A098587

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). 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 r=1+2sqrt(2)=3.828427..., 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 r>2+sqrt(5) approx 4.236 (Devaney 1989, pp. 31-50; Gulik 1992, pp. 112-126; Holmgren 1996, pp. 69-85), but in fact, it is also chaotic for all r>4 (Robinson 1995, pp. 33-37; Kraft 1999).

The logistic map has correlation exponent 0.500+/-0.005 (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).


See also

2x mod 1 Map, Bifurcation, Feigenbaum Constant, Logistic Map--r=2, Logistic Map--r=2, Logistic Map--r=4, Period Three Theorem, Quadratic Map, Silver Constant, Tent Map

Explore with Wolfram|Alpha

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 b_4 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.pdfKraft, 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' q Index and Mori's q 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.

Referenced on Wolfram|Alpha

Logistic Map

Cite this as:

Weisstein, Eric W. "Logistic Map." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LogisticMap.html

Subject classifications