“Logistic Equation”的概念、定义、习题解题方法及翻译-数学辞典

单词

Logistic Equation

释义

Logistic Equation

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

(1)

where

(sometimes also denoted

) is a Positive constant (the ``biotic potential''). We will start

in theinterval

. In order to keep points in the interval, we must find appropriate conditions on

. The maximumvalue

can take is found from

(2)

so the largest value of

occurs for

. Plugging this in,

. Therefore, tokeep the Map in the desired region, we must have

. The Jacobian is

(3)

and the Map is stable at a point

. 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 ifa value of

greater than 3 is chosen. The map becomes unstable and we get a Pitchfork Bifurcation with twostable orbits of period two corresponding to the two stable Fixed Points of

. The fixedpoints 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 aregiven 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 themconvert to Real roots at this value (although this is difficult to show except by plugging in). The4-Cycle therefore starts at

. The Bifurcations comefaster 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 3or 7 as a result of Mode Locking. The period 3 Bifurcation occurs at , as isderived below. Following the 3-Cycle, the Period Doublings then beginagain with Cycles of 6, 12, ...and 7, 14, 28, ..., and then once again break off toChaos.

A set of equations which can be solved to give the onset of an arbitrary -cycle (Saha and Strogatz1995) 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 desiredBifurcation 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 generalizeseasily 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), CapacityDimension 0.538 (Grassberger 1981), and Information Dimension 0.5170976 (Grassberger and Procaccia 1983).

See also Bifurcation, Feigenbaum Constant, Logistic Distribution, LogisticEquation 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.

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