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Discrete-Time Goldfishing
Francesco Calogero Physics Department, University of Rome "La Sapienza", Istituto Nazionale di Fisica Nucleare, Sezione di Roma, Italy
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
$\ell=0,1,2,\dots$ becomes the standard
continuous-time independent variable
$t$,
$0\leq t<\infty$.
Keywords:
nonlinear discrete-time dynamical systems; integrable
and solvable maps; isochronous discrete-time dynamical systems;
discrete-time dynamical systems of goldfish type.
MSC: 37J35;
37C27;
70F10;
70H06 Received: May 4, 2011; in final form
July 29, 2011; Published online
August 23, 2011
Language: English
DOI:
10.3842/SIGMA.2011.082
Fulltext:
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