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SIGMA, 2011, Volume 7, 082, 35 pages (Mi sigma640)  

This article is cited in 8 scientific papers (total in 8 papers)

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 $\ell=0,1,2,…$ 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

DOI: https://doi.org/10.3842/SIGMA.2011.082

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Full text: http://emis.mi.ras.ru/.../082
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Bibliographic databases:

ArXiv: 1108.4492
MSC: 37J35; 37C27; 70F10; 70H06
Received: May 4, 2011; in final form July 29, 2011; Published online August 23, 2011
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Citation: Francesco Calogero, “Discrete-Time Goldfishing”, SIGMA, 7 (2011), 082, 35 pp.

Citation in format AMSBIB
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\by Francesco Calogero
\paper Discrete-Time Goldfishing
\jour SIGMA
\yr 2011
\vol 7
\papernumber 082
\totalpages 35
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\crossref{https://doi.org/10.3842/SIGMA.2011.082}
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\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84855219561}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. F. Calogero, “A new goldfish model”, Theoret. and Math. Phys., 167:3 (2011), 714–724  mathnet  crossref  crossref  mathscinet  adsnasa  isi
    2. F. Calogero, “Another new goldfish model”, Theoret. and Math. Phys., 171:2 (2012), 629–640  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    3. F. Calogero, “On a technique to identify solvable discrete-time many-body problems”, Theoret. and Math. Phys., 172:2 (2012), 1052–1072  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib  elib
    4. Calogero F., Leyvraz F., “New Solvable Discrete-Time Many-Body Problem Featuring Several Arbitrary Parameters”, J. Math. Phys., 53:8 (2012), 082702  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    5. Calogero F., Leyvraz F., “New Solvable Discrete-Time Many-Body Problem Featuring Several Arbitrary Parameters. II”, J. Math. Phys., 54:10 (2013), 102702  crossref  mathscinet  zmath  adsnasa  isi  scopus
    6. Bruschi M., Calogero F., Leyvraz F., “a Large Class of Solvable Discrete-Time Many-Body Problems”, J. Math. Phys., 55:8 (2014), 082703  crossref  mathscinet  zmath  adsnasa  isi  scopus
    7. Calogero F., Leyvraz F., “a Nonautonomous Yet Solvable Discrete-Time N-Bodyproblem”, J. Phys. A-Math. Theor., 47:10 (2014), 105203  crossref  mathscinet  zmath  adsnasa  isi  scopus
    8. Bihun O., Calogero F., “Generations of Solvable Discrete-Time Dynamical Systems”, J. Math. Phys., 58:5 (2017), 052701  crossref  mathscinet  zmath  isi  scopus
  • Symmetry, Integrability and Geometry: Methods and Applications
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