Deautonomising the Lyness mapping

TL;DR

This paper investigates the deautonomisation of the Lyness mapping, revealing new behaviors in the derivative form and providing insights into the growth of late-confinement conditions in non-integrable cases.

Contribution

It demonstrates that deautonomisation is possible for all N in the derivative form and uncovers a novel exponential dependence in the N=2 case.

Findings

Deautonomisation is limited to N=2 in standard form.

Arbitrary N deautonomisation is possible in the derivative form.

Late singularity confinement growth relates to non-linear conditions.

Abstract

We examine the Lyness mapping (an integrable $N$th-order discrete system which can be generated from a one-dimensional reduction of the Hirota-Miwa equation) from the point of view of deautonomisation. We show that only the $N=2$ case can be deautonomised when one works with the standard form of the mapping. However it turns out that deautonomisation is possible for arbitrary $N$ when one considers the derivative form of the Lyness mapping. The deautonomisation of the derivative of the $N=2$ case leads to a result we have never met before: the secular dependence in the coefficients of the mapping enters through two different exponential terms instead of just a single one. As a consequence, it turns out that a limit of this multiplicative dependence towards an additive one is possible without modifying the dependent variable. Finally, the analysis of the `late' singularity confinement of the $N=2$ case leads to a novel realisation of the full-deautonomisation principle: the dynamical degree is not given (as is customary) simply by the solution of some linear or multiplicative equation, but is present in the growth of the non-linear (and non-integrable) late-confinement conditions.