2014/07/10 by Isabel S. Labouriau, Labouriau, Isabel S., Adrian C. Murza +1
Computer Science · Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Dynamical Systems (math.DS) #FOS: Mathematics #Nonlinear Dynamics and Pattern Formation #Quantum chaos and dynamical systems #math.DS
paper · pdf · doi:10.48550/arxiv.1407.2866
Changes: introduction rewritten, minor corrections everywhere
openalex publication_date 2014/07/10 · arxiv created 2015/07/30 · arxiv updated 2015/07/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In the study of the periodic solutions of a Γ-equivariant dynamical system, the H~mod~K theorem gives all possible periodic solutions, based on group-theoretical aspects. By contrast, the equivariant Hopf theorem guarantees the existence of families of small-amplitude periodic solutions bifurcating from the origin for each C-axial subgroup of Γ×\mathbbS1. In this article we compare the bifurcation of periodic solutions for generic differential equations equivariant under the full group of symmetries of the tetrahedron and the group of rotational symmetries of the cube. The two groups are the image of inequivalent representations of the symmetric group S4. The possible spatial symmetries of bifurcating solutions are different, even though the two groups yield the same group of matrices Γ×\mathbbS1. The same group of matrices occurs again as the extension Γ×\mathbbS1 when Γ is the full group of symmetries of the cube. For these three groups, while characterizing the Hopf bifurcation, we identify which periodic solution types, whose existence is guaranteed by the H~mod~K theorem, are obtainable by Hopf bifurcation from the origin.