Potentials Unbounded Below

TL;DR

This paper introduces continuous interpolations for discrete dynamical systems using functional conjugation, revealing unbounded potentials with switchbacks in models like Beverton-Holt, Skellam, and logistic map.

Contribution

It develops a novel method to interpolate discrete systems with unbounded potentials, highlighting switchback phenomena in classical population models.

Findings

Interpolations correspond to particle motion in unbounded potentials.

Switchbacks occur where the potential changes form at turning points.

Applications to population dynamics models demonstrate these features.

Abstract

Continuous interpolates are described for classical dynamical systems defined by discrete time-steps. Functional conjugation methods play a central role in obtaining the interpolations. The interpolates correspond to particle motion in an underlying potential, $V$. Typically, $V$ has no lower bound and can exhibit switchbacks wherein $V$ changes form when turning points are encountered by the particle. The Beverton-Holt and Skellam models of population dynamics, and particular cases of the logistic map are used to illustrate these features.