Dynamics of Non-polar Solutions to the Discrete Painlev\'e I Equation

TL;DR

This paper investigates the complex dynamics of non-polar solutions to the discrete Painlevé I equation, revealing their connection to heteroclinic orbits, invariant manifolds, and asymptotic behaviors, with implications for understanding discrete integrable systems.

Contribution

It provides a novel analysis of non-polar solutions of dP1, including their asymptotic behavior, connection to the Freud orbit, and a new method for asymptotic expansion of iterates.

Findings

Non-polar solutions form heteroclinic connections between fixed points at infinity.

The Freud orbit is a singular limit of other non-polar solutions.

Solutions asymptotically converge to the Freud orbit, following invariant curves.

Abstract

This manuscript develops a novel understanding of non-polar solutions of the discrete Painlev\'e I equation (dP1). As the non-autonomous counterpart of an analytically completely integrable difference equation, this system is endowed with a rich dynamical structure. In addition, its non-polar solutions, which grow without bounds as the iteration index $n$ increases, are of particular relevance to other areas of mathematics. We combine theory and asymptotics with high-precision numerical simulations to arrive at the following picture: when extended to include backward iterates, known non-polar solutions of dP1 form a family of heteroclinic connections between two fixed points at infinity. One of these solutions, the Freud orbit of orthogonal polynomial theory, is a singular limit of the other solutions in the family. Near their asymptotic limits, all solutions converge to the Freud orbit, which follows invariant curves of dP1, when written as a 3-D autonomous system, and reaches the point at positive infinity along a center manifold. This description leads to two important results. First, the Freud orbit tracks sequences of period-1 and 2 points of the autonomous counterpart of dP1 for large positive and negative values of $n$, respectively. Second, we identify an elegant method to obtain an asymptotic expansion of the iterates on the Freud orbit for large positive values of $n$. The structure of invariant manifolds emerging from this picture contributes to a deeper understanding of the global analysis of an interesting class of discrete dynamical systems.