Dynamics of Nonpolar Solutions to the Discrete Painlevé I Equation

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