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Il Nuovo Cimento B (1971-1996)

, Volume 102, Issue 5, pp 451–470 | Cite as

Bifurcations, symmetries and maximal isotropy subgroups

  • G. Cicogna
  • G. Gaeta
Article

Summary

We consider the occurrence of bifurcating solutions— stationary, or periodic (of Hopf or «quaternionic» type)—in problems exhibiting symmetry properties. We focus our attention on the case in which the isotropy subgroup of the solution is maximal: this permits not only a reduction of the dimensionality of the problem, but also a simple classification of the various types of bifurcations. We deal separately with various possible situations (in particular, the cases of fixed point subspace with minimal or higher dimension); we discuss also some group-theoretical results and several examples.

Keywords

PACS 03.20 Classical mechanics of discrete systems general mathematical aspects PACS 02.20 Group theory 

Бифуркации, симметрии и подгруппы максимальной изотронии

Резюме

Мы рассаатриваем появление бифуркационных решений —стационарных или периодических—в проблемах, обладающих свойствами симметрии. Особое внимание уделяется случаю, в котором подгруппа изотропии решения является максимальной. Это позволяет не только уменьшить размерность проблемы, но также допускает простую классификацию различных типов бифуркаций. Мы анализируем различные возможные ситуации, а также обсуждаем теоретико-групповые результаты и некоторые примеры.

Riassunto

In questo lavoro viene studiata la presenza di soluzioni di biforcazione—stazionarie o periodiche (del tipo di Hopf o «quaternioniche»)—di problemi dotati di proprietà di simmetria. L’attenzione viene concentrata sul caso in cui il sottogruppo di isotropia della soluzione è massimale: ciò permette non solo una riduzione della dimensione del problema, ma anche una semplice classificazione dei vari tipi di biforcazione. Vengono trattate separatamente varie situazioni possibili (in particolare i casi di sottospazi fissi di dimensione minima o di dimensione maggiore); sono anche presi in esame diversi risultati gruppali e numerosi esempi.

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Copyright information

© Società Italiana di Fisica 1988

Authors and Affiliations

  • G. Cicogna
    • 1
  • G. Gaeta
    • 2
  1. 1.Dipartimento di Fisica dell’UniversitàPisaItalia
  2. 2.CRMUniversité de MontrealMontrealCanada

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