Periodic orbits of difference equations

Abstract

The real difference equationan+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 ΛPof 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.