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Name: Jay Kappraff 

Address: Department of Mathematics, New Jersey Institute of Technology, U.S.A.


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 
and Chaos for the Cyclotomic 7-gon

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. 

Table 1. The Lucas Polynomials

L1 x 1
L2 x2 2 3
L3 x3 3x 4
L4 x4 4x2 + 2 7
L5 x5 5x3 + 5x 11
L6 x6 6x4 9x2 2 18 
L7 x7 7x5 + 14x3 7x 29
... ... ...

These polynomials are generated, starting with 2 and x, by the recursive formula:  and exhibit the crucial property,


Consider the second Lucas polynomial L2, x2 2 and its iterative map,


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  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, 

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,

where 4k + j = n. (3)

Furthermore it can be shown that, corresponding to an n-cyclotomic polygon for n odd,  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, 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)

Since 8 º 1 (mod 7) the sequence repeats with the 3-cycle, 

x0 ® x1 ® x2 ® x0, 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 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 
Corresponding to the Cyclotomic 7-gon.






















4 Polygons and Chaos for 
Generalized Logistic Equations

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. 

Table 3 Exponents p such that 
m2 N = 7  n = 9  n = 11  n = 13 n =15 N =17




3 3















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  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 15-gon the only integers from 1,2,,7 that are relatively prime to 15 are 1,2,4,7 so that periods for the cyclotomic 15-gon must be factors of 4. We see from Table 3 that periods of 2 and 4 occur for the L2 and L4 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 2n-gons, for odd n, each n having a characteristic cycle length. 


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.