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Periodic orbits of difference equations

Published online by Cambridge University Press:  14 November 2011

A. F. Beardon ,
S. R. Bullett  and
P. J. Rippon
A. F. Beardon
Affiliation:
D.P.M.M.S, University of Cambridge, 16 Mill Lane, Cambridge CB2 1SB, U.K.
S. R. Bullett
Affiliation:
School of Mathematical Sciences, Queen Mary & Westfield College, Mile End Road, London El 4NS, U.K.
P. J. Rippon
Affiliation:
Faculty of Mathematics and Computing, Open University, Milton Keynes MK7 6AA, U.K.
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Abstract

The real difference equation an+2 − (λ|an+1| + μan+1) + an = 0 may be interpreted as a dynamical system Φ:(an, an+1) ↦ (an+1, an+2) acting in the plane. The set ΛP of points (λ, μ) for which the mapping Φ is periodic has a rich structure. In this paper, we derive some geometric properties of ΛP (for example, we show that it is unbounded and uncountable), and we derive criteria for Φ to be periodic. We also investigate when Φ is conjugate to a rotation of the plane, and we describe how the rotation numbers of the corresponding circle maps Φ/|Φ| are related to the structure of ΛP.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 125 , Issue 4 , 1995 , pp. 657 - 674
Copyright
Copyright © Royal Society of Edinburgh 1995

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