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Lyness cycles

Published online by Cambridge University Press:  15 June 2017

Stan Dolan
Stan Dolan*
Affiliation:
126A Harpenden Road, St Albans AL3 6BZ e-mail: stan@standolan.co.uk
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Extract

In 1942, R. C. Lyness noted that some recurrence relations generate cycles, irrespective of the initial values. For example, the order 2 recurrence relation

generates a cycle of period 5 for almost all values of u1 and u2 [1].

The globally periodic nature of sequences generated by this recurrence relation can be seen by setting u1 = x and u2 = y. The sequence is then

Lyness gave other examples of such recurrence relations but had been unable to find one with period 7 and challenged readers of the Gazette to find such a recurrence relation or prove it to be impossible.

No answer to this challenge was forthcoming. However, since Lyness's time, interest in these cycles has been maintained due to links with cross-ratios and elliptic curves. In recent years, Jonny Griffiths has done much to popularise these cycles [2].

Type
Articles
Information
The Mathematical Gazette , Volume 101 , Issue 551 , July 2017 , pp. 193 - 207
Copyright
Copyright © Mathematical Association 2017 

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References

1. Lyness, R.C., Cycles, Math. Gaz. 26 (February 1942) p. 62.Google Scholar
2. Griffiths, J., Lyness cycles, elliptic curves, and Hikorsky triangles, accessed January 2017 at: http://www.s253053503.websitehome.co.uk/jg-msc-uea/thesis-final-11-2-2012.pdf Google Scholar
3. Cull, P., Flahive, M., Robson, R., Difference equations; from rabbits to chaos, UTM, Springer (2005).Google Scholar
4. Kulenovic, M., Ladas, G., Dynamics of second order rational difference equations, Chapman and Hall/CRC (2002).Google Scholar