TY - JOUR

T1 - Dynamics of Nonpolar Solutions to the Discrete Painlevé I Equation

AU - Ercolani, Nicholas

AU - Lega, Joceline

AU - Tippings, Brandon

N1 - Funding Information: \ast Received by the editors September 8, 2021; accepted for publication (in revised form) by T. Kaper January 3, 2022; published electronically June 1, 2022. https://doi.org/10.1137/21M1445156 Funding: The work of the first author was partially supported by NSF grant DMS-1615921. \dagger Department of Mathematics, University of Arizona, Tucson, AZ 85721 USA (ercolani@math.arizona.edu; lega@math.arizona.edu, http://www.math.arizona.edu/\sim lega/; tippings@email.arizona.edu). Publisher Copyright: © 2022 Society for Industrial and Applied Mathematics.

PY - 2022

Y1 - 2022

N2 - This manuscript develops a novel understanding of nonpolar solutions of the discrete Painlevé I equation (dP1). As the nonautonomous counterpart of an analytically completely integrable difference equation, this system is endowed with a rich dynamical structure. In addition, its nonpolar 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 nonpolar 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 three-dimensional 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.

AB - This manuscript develops a novel understanding of nonpolar solutions of the discrete Painlevé I equation (dP1). As the nonautonomous counterpart of an analytically completely integrable difference equation, this system is endowed with a rich dynamical structure. In addition, its nonpolar 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 nonpolar 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 three-dimensional 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.

KW - Painlevé property

KW - asymptotic expansions

KW - center manifold theory

KW - nonautonomous discrete dynamical system

KW - orthogonal polynomials

KW - recurrence coefficients

KW - singularity confinement

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U2 - https://doi.org/10.1137/21M1445156

DO - https://doi.org/10.1137/21M1445156

M3 - Article

SN - 1536-0040

VL - 21

SP - 1322

EP - 1351

JO - SIAM Journal on Applied Dynamical Systems

JF - SIAM Journal on Applied Dynamical Systems

IS - 2

ER -