JAY KAPPRAFF and GARY W. ADAMSON
Name: Jay Kappraff Address: Department of Mathematics, New Jersey Institute of Technology, U.S.A. Email: Kappraff@aol.com
Name: Gary W. Adamson Address: P.O. Box 124571, San Diego, CA 921124571,
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 ngons
as the seed, the trajectory of the logistic map cycles through a sequence
of edges of other star ngons. Each ngon 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 Fibonaccitype sequence.
These polynomials are generated, starting with 2 and
x,
by the recursive formula:
and exhibit the crucial property,
Consider the second Lucas polynomial L_{2},
x^{2} – 2 and its iterative map,
also known as the logistic map. It represents the extreme lefthand point on the real axis of the Mandelbrot set at the onset of fullblown chaos in which its Julia set is disconnected, comprising a Cantor set (Schroeder, 1990; Peitgen, 1992). Starting with a seed value x_{0 }and placing
it into Equation 2, the sequence x_{0},
x_{1}, x_{2}, … is generated and referred
to as the trajectory of the map. If x_{n} = x_{0},
the trajectory repeats and is said to be an ncycle. As a result
of our analysis, there exist cycles of all lengths which can be characterized
as edges of star 2ngons for n odd in which each value of
n
has its own characteristic cycle length.
2 Star Polygons and the Cyclotomic ngon Consider for k = 1,2,3,…, and n odd, ignoring the signs. These expressions are the real parts of nth roots of unity given by the equation, z^{n} – 1 = 0 for z a complex number and n an odd integer. The complex roots of this equation form a regular ngon
of unit radius known as an ncyclotomic polygon (Kappraff,
2001). It can be shown that for arbitrary kvalues, there is a jvalue
such that,
= where 4k + j = n. (3) 3 Polygons and Chaos for the Cyclotomic 7gon Consider the cyclotomic 7gon. Beginning with a seed value
of, if x_{0} = ,
the iterates are the sequence of edge lengths of different species of star
14gon corresponding to
for, k º1,2,4,8,…(mod 7). (4) Since 8 º 1 (mod 7) the sequence repeats with the 3cycle, x_{0} ® x_{1} ® x_{2} ® x_{0,} or = 1.2469…®= 0.44509… ® =1.80189…® =1.2469… Since = positive and negative values of the index 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 3cycle is represented by the sequence of kvalues: 1® 2® 3® 1… The corresponding sequence of jvalues , according to Equation 3, is: 3® 1® 5® 3. These are listed in Table 2 along with the corresponding sequence of star 14gons: {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 14gons drawn from vertex number 0 of the 14gon 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 17gons and a general algorithm for determining the cycle of any cyclotomic ngon 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 L_{m}(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 2ngon maps to the edge of another star 2ngon. We have also proven that the cycle length corresponding to any cyclotomic ngon for n odd is equal to the smallest exponent p such that, (5) Where m is the index of the mth 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 7gon has a
3cycle for m = 2,3,4, and 5 as described above for m = 2.
Notice that the 9gon has only a 3cycle corresponding to
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,…, relatively
prime to n. For example, for a cyclotomic 15gon the only integers
from 1,2,…,7 that are relatively prime to 15 are 1,2,4,7 so that periods
for the cyclotomic 15gon must be factors of 4. We see from Table
3 that periods of 2 and 4 occur for the L_{2} and L_{4}
Logistic Maps.
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 2ngons, 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: SpringerVerlag. Schroeder, M. (1990) Fractals, Chaos,
Power Laws, New York: W.H. Freman Press, 1990.
