
SIGMA 7 (2011), 082, 35 pages arXiv:1108.4492
https://doi.org/10.3842/SIGMA.2011.082
Contribution to the Proceedings of the Conference “Symmetries and Integrability of Difference Equations (SIDE9)”
DiscreteTime Goldfishing
Francesco Calogero
Physics Department, University of Rome ''La Sapienza'', Istituto Nazionale di Fisica Nucleare, Sezione di Roma, Italy
Received May 04, 2011, in final form July 29, 2011; Published online August 23, 2011
Abstract
The original continuoustime ''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 initialvalue problem.
Several other, more general, solvable dynamical systems ''of goldfish type'' have been identified over
time, featuring, in the righthand (''forces'') side of their Newtonian
equations of motion, in addition to other contributions, a
velocitydependent term such as that appearing in the righthand 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 discretetime 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 discretetime independent variable l=0,1,2,... becomes the standard continuoustime independent
variable t, 0≤t<∞.
Key words:
nonlinear discretetime dynamical systems; integrable and solvable maps; isochronous discretetime dynamical systems; discretetime dynamical systems of goldfish type.
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