Symmetry, Integrability and Geometry: Methods and Applications
 RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Archive Impact factor Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 SIGMA: Year: Volume: Issue: Page: Find

 SIGMA, 2011, Volume 7, 082, 35 pp. (Mi sigma640)

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

Full text: PDF file (506 kB)
Full text: http://emis.mi.ras.ru/.../082
References: PDF file   HTML file

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
Language:

Citation: Francesco Calogero, “Discrete-Time Goldfishing”, SIGMA, 7 (2011), 082, 35 pp.

Citation in format AMSBIB
\Bibitem{Cal11} \by Francesco Calogero \paper Discrete-Time Goldfishing \jour SIGMA \yr 2011 \vol 7 \papernumber 082 \totalpages 35 \mathnet{http://mi.mathnet.ru/sigma640} \crossref{https://doi.org/10.3842/SIGMA.2011.082} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2861194} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000294118000002} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84855219561} 

• http://mi.mathnet.ru/eng/sigma640
• http://mi.mathnet.ru/eng/sigma/v7/p82

 SHARE:

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
2. F. Calogero, “Another new goldfish model”, Theoret. and Math. Phys., 171:2 (2012), 629–640
3. F. Calogero, “On a technique to identify solvable discrete-time many-body problems”, Theoret. and Math. Phys., 172:2 (2012), 1052–1072
4. Calogero F., Leyvraz F., “New Solvable Discrete-Time Many-Body Problem Featuring Several Arbitrary Parameters”, J. Math. Phys., 53:8 (2012), 082702
5. Calogero F., Leyvraz F., “New Solvable Discrete-Time Many-Body Problem Featuring Several Arbitrary Parameters. II”, J. Math. Phys., 54:10 (2013), 102702
6. Bruschi M., Calogero F., Leyvraz F., “a Large Class of Solvable Discrete-Time Many-Body Problems”, J. Math. Phys., 55:8 (2014), 082703
7. Calogero F., Leyvraz F., “a Nonautonomous Yet Solvable Discrete-Time N-Bodyproblem”, J. Phys. A-Math. Theor., 47:10 (2014), 105203
8. Bihun O., Calogero F., “Generations of Solvable Discrete-Time Dynamical Systems”, J. Math. Phys., 58:5 (2017), 052701
•  Number of views: This page: 168 Full text: 30 References: 22