Discrete-Time Goldfishing

TL;DR

This paper introduces and analyzes discrete-time solvable dynamical systems that generalize the continuous-time goldfish model, exhibiting behaviors like isochrony and asymptotic isochrony, and reduce to the classical model in the limit.

Contribution

The authors develop a class of discrete-time goldfish-type dynamical systems that are solvable and exhibit interesting behaviors, extending the continuous-time models.

Findings

Discrete-time models are solvable and exhibit isochronous behavior.

Models reduce to classical goldfish systems in the continuous limit.

Discrete models display remarkable dynamical properties similar to continuous counterparts.

Abstract

The original continuous-time "goldfish" dynamical system is characterized by two neat formulas, the first of which provides the $N$ Newtonian equations of motion of this dynamical system, while the second provides the solution of the corresponding initial-value problem. Several other, more general, solvable dynamical systems "of goldfish type" have been identified over time, featuring, in the right-hand ("forces") side of their Newtonian equations of motion, in addition to other contributions, a velocity-dependent term such as that appearing in the right-hand side of the first formula mentioned above. The solvable character of these models allows detailed analyses of their behavior, which in some cases is quite remarkable (for instance isochronous or asymptotically isochronous). In this paper we introduce and discuss various discrete-time dynamical systems, which are as well solvable, which also display interesting behaviors (including isochrony and asymptotic isochrony) and which reduce to dynamical systems of goldfish type in the limit when the discrete-time independent variable $\ell=0,1,2,...$ becomes the standard continuous-time independent variable $t$, $0\leq t<\infty $.