On the classification of the homoclinic cycles in \(\mathbb{R}^4\) (Q2738884)

The author studies the number of equilibrium points for structurally stable homoclinic cycles of vector fields equivariant under the action of a symmetry group in \(\mathbb{R}^n\), \(n\geq 3\). For the case of \(n=4\), on the basis of the action of the symmetry group such homoclinic cycles have been classified as of type \(A\), \(B\), and \(C\) by \textit{P. Chossat} and \textit{R. Lauterbach} [Methods in equivariant bifurcations and dynamical systems. Advanced Series in Nonlinear Dynamics. 15. Singapore: World Scientific (2000; Zbl 0968.37001)]. NEWLINENEWLINENEWLINEHere, the number of equilibrium points for type \(B\) and \(C\) cycles is investigated using direct calculations in a given basis. The number of equilibrium points for type \(C\) cycles is shown only to be 4 or 8. For type \(B\) cycles this number is shown to be either 2, 3, or 6. A classification of type \(A\) cycles remains an open problem.