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.

Dynamics of a family of piecewise-linear area-preserving plane maps I. Rational rotation numbers

“…Numerical evidence indicates that it is nonincreasing for µ > 0, hence nondecreasing for µ < 0 (recall that r(S µν ) = r(S −µ,ν ) since T µν is topologically conjugate to T −µ,ν ). This was observed by Bedford, Bullett and Rippon [2], who proved this holds for ν = 0. One can show that the rotation number is nonincreasing for µ > |ν|, using an extension of the approach used in Theorem 3.2(i).…”

“…These are of interest in part because we have no formulae for generic a and b and in part because they also provide several examples of parameter values for which T ab is periodic. These results all appear in Beardon, Bullett and Rippon [2]. , which is rotation by θ.…”

“…As remarked in the introduction, our methods and results overlap with those used in Beardon, Bullett and Rippon [2]. Their paper contains additional results about periodic maps beyond those given here.…”

“…These include one-parameter families, and cases where the map T ab is purely periodic. These examples were all given in Bedford, Bullett and Rippon [2].…”

“…However he also conjectured the existence of maps in the Froeschlé group having dense orbits in the plane ( [11, p. 221] [2] studied periodic orbits of the maps (1.5). They also studied the set of parameter values Ω B for which all orbits are bounded, determined restrictions on the parameters in this set, and observed that the equality Ω R = Ω B follows from results of Herman [11,VIII.2.4].…”