Secondary bifurcations in systems with all-to-all coupling. Part II.

• Tipo: Artigo em Revista Científica Internacional
• Data: 2006

Autores

dias, aps (Autor) FCUP; rodrigues, a (Autor) Outra

Revista

Título: Dynamical Systems

Vol. 21 Nº 3

Páginas: 439-463

ISSN: 1468-9367

Editora: Taylor & Francis

Indexação

ISI Web of Knowledge - 6 Citações

Scopus - 7 Citações

Classificação Científica

FOS: Ciências exactas e naturais > Matemática

Outras Informações

ID Authenticus: P-004-FN7

DOI: 10.1080/14689360600759689

Abstract (EN): In a recent paper Dias and Stewart studied the existence, branching geometry, and stability of secondary branches of equilibria in all-to-all coupled systems of differential equations, that is, equations that are equivariant under the permutation action of the symmetric group S-N. They consider the most general cubic order system of this type. Primary branches in such systems correspond to partitions of N into two parts p, q with p + q = N. Secondary branches correspond to partitions of N into three parts a, b, c with a + b + C = N. They prove that except in the case a = b = c secondary branches exist and are (generically) globally unstable in the cubic order system. In this work they realized that the cubic order system is too degenerate to provide secondary branches if a = b = c. In this paper we consider a general system of ordinary differential equations commuting with the permutation action of the symmetric group S-3n on R-3n. Using singularity theory results, we find sufficient conditions on the coefficients of the fifth order truncation of the general smooth S-3n-equivariant vector field for the existence of a secondary branch of equilibria near the origin with S-n x S-n x S-n symmetry of such system. Moreover, we prove that under such conditions the solutions are (generically) globally unstable except in the cases where two tertiary bifurcations occur along the secondary branch. In these cases, the instability result holds only for the equilibria near the secondary bifurcation points. We show an example where stability between tertiary bifurcation points on the secondary branch occurs.

Idioma: Inglês

Tipo (Avaliação Docente): Científica

Nº de páginas: 25