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Asymptotically isochronous systems

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dc.contributor.authorCalogero, Francesco
dc.contributor.authorGómez-Ullate Otaiza, David
dc.date.accessioned2023-06-20T10:55:06Z
dc.date.available2023-06-20T10:55:06Z
dc.date.issued2008-12
dc.description© Atlantis Press. One of us (FC) would like to thank Francois Leyvraz for several illuminating discussions. The research reported in this paper has profited from visits by each of the two authors in the Department of the other performed in the framework of the exchange program among our two Universities. The research of DGU is supported in part by the Ramón y Cajal program of the Ministerio de Ciencia y Tecnología and by the DGI under grants FIS2005-00752 and MTM2006-00478.
dc.description.abstractMechanisms are elucidated underlying the existence of dynamical systems whose generic solutions approach asymptotically (at large time) isochronous evolutions: all their dependent variables tend asymptotic ally to functions periodic with the sa m e fixed period. We focus on two such mechanisms, emphasizing their generality and illustrating each of them via a representative example. The first example belongs to a recently discovered class of integrable indeed solvable many-body problems. The second example consists of a broad class of (generally nonintegrable) models obtained by deforming appropriately the well-known (integrable and isochronous) many-body problem with inverse-cube two-body forces and a one-body linear ("harmonic oscillator") force.
dc.description.departmentDepto. de Física Teórica
dc.description.facultyFac. de Ciencias Físicas
dc.description.refereedTRUE
dc.description.sponsorshipMinisterio de Ciencia y Tecnologia. Ramón y Cajal program
dc.description.sponsorshipDGI
dc.description.statuspub
dc.eprint.idhttps://eprints.ucm.es/id/eprint/30840
dc.identifier.doi10.2991/jnmp.2008.15.4.5
dc.identifier.issn1402-9251
dc.identifier.officialurlhttp://dx.doi.org/10.2991/jnmp.2008.15.4.5
dc.identifier.relatedurlhttp://www.tandfonline.com
dc.identifier.relatedurlhttp://arxiv.org/abs/0710.1487
dc.identifier.urihttps://hdl.handle.net/20.500.14352/51448
dc.issue.number4
dc.journal.titleJournal of nonlinear mathematical physics
dc.language.isoeng
dc.page.final426
dc.page.initial410
dc.publisherAtlantis Press
dc.relation.projectIDFIS2005-00752
dc.relation.projectIDMTM2006-00478
dc.rights.accessRightsopen access
dc.subject.cdu51-73
dc.subject.keywordMany-body problems
dc.subject.keywordPeriodic-solutions
dc.subject.keywordRiemann surfaces
dc.subject.keywordComplex odes
dc.subject.keywordPotentials
dc.subject.keywordQuantum
dc.subject.keywordMotions
dc.subject.keywordSuperintegrability
dc.subject.keywordHamiltonians
dc.subject.keywordQuantization
dc.subject.ucmFísica-Modelos matemáticos
dc.subject.ucmFísica matemática
dc.titleAsymptotically isochronous systems
dc.typejournal article
dc.volume.number15
dcterms.references[1] F. Calogero, Isochronous systems, in: Encyclopedia of Mathematical Physics edited by J.-P. Françoise, G. Naber and Tsou Sheung Tsun (Oxford: Elsevier) 2006 (ISBN 978-0-1251-2666-3), vol. 3, pp. 166-172. [2] F. Calogero Isochronous systems (Oxford: Oxford University Press) (in press, to appear in February 2008). [3] F. Calogero, D. Gómez-Ullate, A new class of solvable many-body problems with constraints associated with an exceptional polynomial subspace of codimension two, J. Phys. A: Math. Theor. 40 (2007) F573–F580. [4] F. Calogero, Classical Many-Body Problems Amenable to Exact Treatments (Lecture Notes in Physics Monograph Vol. 66) 2001 (Berlin: Springer) [5] F. Calogero, Motion of Poles and Zeros of Special Solutions of Nonlinear and Linear Partial Differential Equations and Related “Solvable” Many Body Problems, Nuovo Cimento 43B (1978) 177–241. [6] D. Gómez-Ullate, M. Sommacal, Periods of the goldfish many-body problem J. Nonlinear Math. Phys. 12 (Suppl. 1) (2005) 351–62. [7] E. Hairer, S. P. Norsett, G. Wanner, Solving Ordinary Differential Equations I: Nonstiff Problems 1987 (Berlin: Springer) pp. 193–195 [8] F. Calogero Periodic solutions of a system of complex ODEs, Phys. Lett. A293 (2002) 146– 150. [9] F. Calogero, M. Sommacal, Periodic solutions of a system of complex ODEs. II. Higher periods J. Nonlinear Math Phys. 9 (2002) 483–516. [10] F. Calogero, D. Gómez-Ullate, P. M. Santini, M. Sommacal, The transition from regular to irregular motions, explained as travel on Riemann surfaces J. Phys. A: Math. Gen. 38 (2005) 8873–8896. [11] F. Calogero, D. Gómez-Ullate, P. M. Santini, M. Sommacal, Towards a theory of chaos explained as travel on Riemann surfaces (in preparation). [12] Yu. Fedorov, D. G´omez-Ullate, Dynamical systems on infinitely sheeted Riemann surfaces, Physica D 227 (2007) 120–134. [13] P. Grinevich, P. M.Santini, Newtonian dynamics in the plane corresponding to straight and cyclic motions on the hyperelliptic curve µ_ 2 = v _n − 1, n ∈ Z: ergodicity, isochrony, periodicity and fractals, Physica D 232, (2007) 22–32
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