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Conference rusure::math

Title:	Mathematics at DEC

Moderator:	RUSURE::EDP

Created:	Mon Feb 03 1986
Last Modified:	Fri Jun 06 1997
Last Successful Update:	Fri Jun 06 1997
Number of topics:	2083
Total number of notes:	14613

135.0. "Nonlinear periodic recursions" by HARE::STAN () Tue Aug 21 1984 21:04

Shmuel Avital notes that the nonlinear recurrence

		a  + 1
		 n
	a    = --------
	 n+1     a
		  n-1

generates a periodic sequence (of period 5) for any initial a  and a .
							     0      1

He then asks if it is possible to construct nonlinear recursive relations
that generate periodic sequences of any given period length k.

			Reference

Shmuel Avital, Problem 912. Crux Mathematicorum. 2(1984)53.

T.R	Title	User	Date	Lines
135.1		HARE::STAN	`Tue Aug 21 1984 21:17`	24
	I wrote some programs to hunt for periodic sequences generated by nonlinear recurrences of the form p a + q n a = ---------- n+1 r a + s n-1 and the only ones I found were: a(n+1) = r / a(n) generates a sequence of period 2. a(n+1) = -r^2 / ( a(n) + r ) generates a sequence of period 3. a(n+1) = -2r^2 / ( a(n) + 2r ) generates a sequence of period 4. a(n+1) = (r a(n) + r^2) / a(n-1) generates a sequence of period 5. a(n+1) = -3r^2 / ( a(n) + 3r ) generates a sequence of period 6. Have I missed any? Are there any other general or specific forms that generate sequences of period k with k>6?
135.2		HARE::STAN	`Wed Aug 22 1984 12:34`	3
	Oh, and I forgot to mention a(n+1) = a(n) / a(n-1) also generates a sequence of period 6.
135.3		TURTLE::GILBERT	`Wed Aug 22 1984 20:00`	13
	Given the form of a recurrence of a in terms of a and a , one approach n+1 n n-1 for finding cycles of length k is to expand a and a (in terms of a n+k-1 n+k n-1 and a ), and set the expressions equal to a and a , respectively. n n-1 n This may be a very ambitious undertaking for k > 4 -- for humans, but VAXima may be able to offer some solutions to the resulting equations. Note that if c divides k, and solutions having a cycle length of c is known (even if c=1), it should be possible to factor these from the expansions for cycle length k. - Gilbert
135.4		HARE::STAN	`Fri Aug 31 1984 20:24`	9
	A recursion (for real numbers) generating a sequence of period 9 is a = \| a \| - a . n+1 n n-1 Reference --------- Morton Brown, Problem 6439, American Mathematical Monthly. 90(1983)569.
135.5		HARE::STAN	`Wed Sep 26 1984 04:31`	12
	A recursion of period 7 is a = a = a = d 1 2 3 a = \|a - a \| + \|a - a \| . n+3 n+2 n+1 n+1 n Reference --------- David R. Richman, Sums of Absolute Differences. Journal of Recreational Mathematics. 17(no. 1)(1984)38-41.
135.6		TOOLS::STAN	`Tue Jun 04 1985 17:16`	4
	The definitive reference showing how to construct non-linear periodic sequences for any given period length is R. C. Lyness, "Cycles", Mathematical Gazette. 45(1961)207-209.