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) 
(2) 
(3) 
(4) 
(5) 
(6) 
(7) 
(8) 
(9) 
(10) 
(11) 

(12) 
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 3Cycle, 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) 
For , the solutions (, ..., , ) are (0, 0, ) and (, , 3), so the desired
Bifurcation occurs at . Taking gives
(14) 
(15)  
(16)  
(17)  
(18) 
(19) 
(20)  
(21)  
(22)  
(23) 
Saha and Strogatz (1995) give a simplified algebraic treatment which involves solving
(24) 
(25)  
(26)  
(27) 
(28)  
(29)  
(30) 
(31) 
(32) 
(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, 115118, 1996.
Bogomolny, A. ``Chaos Creation (There is Order in Chaos).'' http://www.cuttheknot.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. 6980, 1988.
Gordon, W. B. ``Period Three Trajectories of the Logistic Map.'' Math. Mag. 69, 118120, 1996.
Grassberger, P. ``On the Hausdorff Dimension of Fractal Attractors.'' J. Stat. Phys. 26, 173179, 1981.
Grassberger, P. and Procaccia, I. ``Measuring the Strangeness of Strange Attractors.'' Physica D 9, 189208, 1983.
Lauwerier, H. Fractals: Endlessly Repeated Geometrical Figures. Princeton, NJ: Princeton University Press, pp. 119122, 1991.
May, R. M. ``Simple Mathematical Models with Very Complicated Dynamics.'' Nature 261, 459467, 1976.
Peitgen, H.O.; Jürgens, H.; and Saupe, D. Chaos and Fractals: New Frontiers of Science. New York: SpringerVerlag, pp. 585653, 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, 11751178, 1980.
Saha, P. and Strogatz, S. H. ``The Birth of Period Three.'' Math. Mag. 68, 4247, 1995.
Strogatz, S. H. Nonlinear Dynamics and Chaos. Reading, MA: AddisonWesley, 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. 117140, 1991.
© 19969 Eric W. Weisstein