标题： 离散时间金鱼运动 显示英文标题

标题： Discrete-Time Goldfishing

摘要： 原始连续时间"金鱼"动力系统由两个简洁的公式表征，第一个公式提供了该动力系统的$N$牛顿运动方程，而第二个公式提供了相应的初值问题的解。 随着时间的推移，已经发现了其他一些更一般的、可解的"金鱼型"动力系统，它们的牛顿运动方程的右边（"力"的方面）除了其他贡献外，还包含一个与上述第一个公式右边出现的类似的速度相关项。 这些模型的可解性允许对其行为进行详细分析，其中一些行为相当显著（例如等时性或渐近等时性）。 在本文中，我们引入并讨论各种离散时间动力系统，它们同样可解，也表现出有趣的行为（包括等时性和渐近等时性），并且当离散时间自变量$\ell=0,1,2,...$变为标准连续时间自变量$t$、$0\leq t<\infty $时，它们会退化为金鱼型动力系统。 显示英文摘要

摘要： 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 $.