Steady-state bifurcation in coupled systems with wreath product symmetry

Author(s)

Luci Any Francisco Roberto

• Document type: Master's Dissertation
• Press: São Carlos.
• Institution: Universidade de São Paulo (USP). Instituto de Ciências Matemáticas e de Computação (ICMC/SB)
• Defense date: 2004-02-26

Examining board members

Miriam Garcia Manoel; Isabel Salgado Labouriau; Hildebrando Munhoz Rodrigues

• Advisor: Miriam Garcia Manoel

Abstract

In this project we study steady-state bifurcation in system of N coupled cells that possess a \"global\" symmetry group G, and in which each cell possess its own \"internai\" symmetry group L, where G is a subgroup of thc permutation group SN of N elements and L is a compact Lie group. The coupling we consider is invariant under the internai symmetries of each cell and the combination of the groups L and G leads to the total symmetry group given by L wreath product G, L ≀ G, i. e., LN ∔ G. We relate the steady-state bifurcations that occur in the coupled system with symmetry group L ≀ G to bifurcations with symmetry L or G. We apply the results to a non-degenerate system of N coupled cells with symmetry O (2) ≀ SN. We see how the invariant theory for O (2) ≀ SN is related to the invariant theories for O(2) and SN. We check that, up to conjugacy, there are exactly N branches, namely, those with axial subgroup. Moreover, we discuss stability and directions of the solution branches. (AU)