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Regul. Chaotic Dyn., 2018, Volume 23, Issue 1, Pages 102–119 (Mi rcd311)  

Multiple Reductions, Foliations and the Dynamics of Cluster Maps

Inês Cruza, Helena Mena-Matosa, M. Esmeralda Sousa-Diasb

a Centro de Matemática da Universidade do Porto (CMUP), Departamento de Matemática, Faculdade de Ciências da Universidade do Porto, R. Campo Alegre, 687, 4169-007 Porto, Portugal
b Center for Mathematical Analysis, Geometry and Dynamical Systems (CAMGSD), Departamento de Matemática, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisbon, Portugal

Abstract: Reduction of cluster maps via presymplectic and Poisson structures is described in terms of the canonical foliations defined by these structures. In the case where multiple reductions coexist, we establish conditions on the underlying presymplectic and Poisson structures that allow for an interplay between the respective foliations. It is also shown how this interplay may be explored to simplify the analysis and obtain an effective geometric description of the dynamics of the original (not reduced) map. Consequences of the approach we developed to the description of several features of some cluster maps dynamics are illustrated by two examples which include the Somos-5 map and an instance of a Somos-7 map.

Keywords: presymplectic manifolds, Poisson manifolds, foliations, cluster maps

Funding Agency Grant Number
Fundação para a Ciência e a Tecnologia PEst-C/MAT/UI0144/2013
UID/MAT/04459/2013
EXCL/MAT-GEO/0222/2012
The work of I. Cruz and H. Mena-Matos was partially funded by FCT under the project PEst-C/MAT/UI0144/2013. The work of M.E. Sousa-Dias was partially funded by FCT/Portugal through the projects UID/MAT/04459/2013 and EXCL/MAT-GEO/0222/2012.


DOI: https://doi.org/10.1134/S1560354718010082

References: PDF file   HTML file

Bibliographic databases:

MSC: 53D17, 53D05, 53C12, 39A20, 13F60
Received: 19.07.2017
Accepted:22.10.2017
Language:

Citation: Inês Cruz, Helena Mena-Matos, M. Esmeralda Sousa-Dias, “Multiple Reductions, Foliations and the Dynamics of Cluster Maps”, Regul. Chaotic Dyn., 23:1 (2018), 102–119

Citation in format AMSBIB
\Bibitem{CruMenSou18}
\by In\^es Cruz, Helena Mena-Matos, M. Esmeralda Sousa-Dias
\paper Multiple Reductions, Foliations and the Dynamics of Cluster Maps
\jour Regul. Chaotic Dyn.
\yr 2018
\vol 23
\issue 1
\pages 102--119
\mathnet{http://mi.mathnet.ru/rcd311}
\crossref{https://doi.org/10.1134/S1560354718010082}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3759973}
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\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85041381821}


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