Singular dynamics of a $q$-difference Painlev\'e equation in its initial-value space

TL;DR

This paper explicitly constructs the initial-value space of a $q$-discrete Painlevé equation and analyzes the complex dynamical behaviors of its solutions, especially near singularities and exceptional lines.

Contribution

It provides a detailed geometric construction of the initial-value space and describes the solution behaviors near singularities for a $q$-discrete Painlevé equation.

Findings

Solutions are repelled from singular lines away from the origin.

Near the origin, solutions exhibit saddle-point behavior due to merging base points.

The dynamics are complex near the anti-canonical divisor components.

Abstract

We construct the initial-value space of a $q$-discrete first Painlev\'e equation explicitly and describe the behaviours of its solutions $w(n)$ in this space as $n\to\infty$, with particular attention paid to neighbourhoods of exceptional lines and irreducible components of the anti-canonical divisor. These results show that trajectories starting in domains bounded away from the origin in initial value space are repelled away from such singular lines. However, the dynamical behaviours in neighbourhoods containing the origin are complicated by the merger of two simple base points at the origin in the limit. We show that these lead to a saddle-point-type behaviour in a punctured neighbourhood of the origin.