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JAY KAPPRAFF and GARY W. ADAMSON
Name: Jay Kappraff Address: Department of Mathematics, New Jersey Institute of Technology, U.S.A. E-mail: Kappraff@aol.com
Name: Gary W. Adamson Address: P.O. Box 124571, San Diego, CA 92112-4571,
U.S.A.
Abstract: An infinite number of periodic trajectories
are derived for the logistic equation of dynamic systems theory at a value
of the parameter corresponding to the extreme point on the real axis of
the Mandelbrot set. Beginning with the edge of a family of star n-gons
as the seed, the trajectory of the logistic map cycles through a sequence
of edges of other star n-gons. Each n-gon for n odd is shown
to have its own characteristic cycle length. The logistic map is shown
to be one of an infinite families of maps, all exhibiting periodic trajectories,
derived from a family pf polynomials related to the Lucas sequence.
1 The Relationship between Polygons
Consider a sequence of polynomials (see Table 1) whose coefficients, disregarding signs, sum to the Lucas sequence: 1, 3, 4, 7, 11, …, a Fibonacci-type sequence.
These polynomials are generated, starting with 2 and
x,
by the recursive formula:  (1)Consider the second Lucas polynomial L2,
x2 – 2 and its iterative map,
 (2)also known as the logistic map. It represents the extreme left-hand point on the real axis of the Mandelbrot set at the onset of full-blown chaos in which its Julia set is disconnected, comprising a Cantor set (Schroeder, 1990; Peitgen, 1992). Starting with a seed value x0 and placing
it into Equation 2, the sequence x0,
x1, x2, … is generated and referred
to as the trajectory of the map. If xn = x0,
the trajectory repeats and is said to be an n-cycle. As a result
of our analysis, there exist cycles of all lengths which can be characterized
as edges of star 2n-gons for n odd in which each value of
n
has its own characteristic cycle length.
2 Star Polygons and the Cyclotomic n-gon Consider zn – 1 = 0 for z a complex number and n an odd integer. The complex roots of this equation form a regular n-gon
of unit radius known as an n-cyclotomic polygon (Kappraff,
2001). It can be shown that for arbitrary k-values, there is a j-value
such that,
is the length of the j-th diagonal (including the edge, j=1)
of a regular 2n-gon of radius 1 (Kappraff, 2002).
Also when j is relatively prime to 2n the diagonal can be
thought of as the edge of the star n-gon {2n/j} for
j>0
and {2n/2n+j} for j<0. Therefore according
to Equations 1 and 3, an edge of
a star 2n-gon maps to the edge of another star 2n-gon.
3 Polygons and Chaos for the Cyclotomic 7-gon Consider the cyclotomic 7-gon. Beginning with a seed value
of, if x0 =
Since 8 º 1 (mod 7) the sequence repeats with the 3-cycle, x0 ®
x1 ® x2
® x0, or ® Since k (mod n) correspond to identical edge lengths. Therefore 4 (mod 7) º -3 is equivalent to k = 3 in Sequence 4, and so the 3-cycle is represented by the sequence of k-values: 1® 2® 3® 1… The corresponding sequence of j-values , according to Equation 3, is: 3® -1® -5® 3. These are listed in Table 2 along with the corresponding sequence of star 14-gons: {14/3}® {14/13}® {14/9}® {14/3} also denoted by the sequence: 3® 13® 9® 3… If the vertices of the polygon are numbered from 0 to 13 this sequence can be associated with the edges of star 14-gons drawn from vertex number 0 of the 14-gon to a vertex of this sequence. The order of the edges in the cycle are also indicated in Table 2 beginning with the seed i = 0. Results for the cyclotomic 9-, 11-, 13- and 17-gons and a general algorithm for determining the cycle of any cyclotomic n-gon can be found in (Kappraff, 2001). Table 2. Cycles for the Logistic
Equation
4 Polygons and Chaos for
We have found that the entire sequence of Lucas polynomials Lm(x) in Table 1 represent generalized "logistic" maps exhibiting cycles due to their property shown in Equation 1 which states that an edge of a star 2n-gon maps to the edge of another star 2n-gon. We have also proven that the cycle length corresponding to any cyclotomic n-gon for n odd is equal to the smallest exponent p such that,
Where m is the index of the m-th Lucas polynomial. Values of p are listed in Table 3 for values of n = 7,9,11,13,17 and m = 2,3,4,5. 
From this table we see that the cyclotomic 7-gon has a
3-cycle for m = 2,3,4, and 5 as described above for m = 2.
Notice that the 9-gon has only a 3-cycle corresponding to 5 Conclusion We have shown that at a critical point of the Mandelbrot
set where orbits of the logistic equation begin to escape, each of these
periods can be characterized by a sequence of edge lengths of a family
of star 2n-gons, for odd n, each n having a characteristic
cycle length.
References Kappraff, J. (2001) Polygons and chaos, In: Jablan, S., ed., Bridges, Winfield: Great Plains Press. Kappraff, J. (2002) Beyond Measure: A Guided Tour through Nature, Myth, and Number, Singapore: World Scientific Publ., In press. Peitgen, H.-O., Jurgens, H., and Saupe, D. (1992) Chaos and Fractals, New York: Springer-Verlag. Schroeder, M. (1990) Fractals, Chaos,
Power Laws, New York: W.H. Freman Press, 1990.
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and exhibit the crucial property,
for k = 1,2,3,…, 
and n odd, ignoring the signs. These expressions are the real parts
of n-th roots of unity given by the equation,
= 
where 4k + j = n. (3)
,
the iterates are the sequence of edge lengths of different species of star
14-gon corresponding to
for, k
º1,2,4,8,…(mod
7). (4)
=
-0.44509…
=-1.80189…® 
=1.2469…
=
positive and negative values of the index
(5)
for k = 1,2,4 = 
with k = 3 missing since 3 and 9 are not relatively prime. We have
also found that the period length is a divisor (or factor) of the number
of integers 1,2,3,…,