A Hopf bifurcation with spherical symmetry (Q1201305)

The authors consider Hopf bifurcation on a ten dimensional center manifold in a \(O(3)\)-symmetric system. The main hypothesis is that the representation on the critical modes is given by the representation on \(V_ 5\oplus V_ 5\), the sum of two copies of the absolutely irreducible representations of \(O(3)\) on the spherical harmonics of order two. This problem has been considered in a paper by \textit{G. Iooss} and \textit{M. Rossi} [SIAM J. Math. Anal. 20, 511-539 (1989; Zbl 0681.58030)], who found the same qualitative behaviour as in the present paper. However, the methods are entirely different. In this paper, the authors use the representation of \(O(3)\) on the \(3\times 3\) symmetric, traceless matrices which is equivalent to the given representation to reduce the computations to matrix manipulations, an advantage which has not been explored in the Iooss and Rossi paper. This makes the results much more transparent.