The bifurcation of limit cycles in \(Z_n\)-equivariant vector fields
The authors study bifurcation of limit cycles from several diverse (weak) foci for polynomial dynamical systems in the plane. In the case of \(G\)-equivariant dynamical systems (\(G\) a Lie group), singular points appear in \(G\)-orbits. The authors are specially interested in the case \(G=\mathbb Z_n\), and, in particular, focus their attention on a certain \(\mathbb Z_8\)-equivariant system \[ dx/dt = f(x,y),\quad dy/dy=g(x,y) \] with \(f\) and \(g\) polynomials of degree 7 depending on nine parameters (with four relations between these, so five free parameters), and a varying parameter \(\delta\). They show that, for \(\delta = 0\), this specific system has eight foci (sitting on the unit circle) with the same topological structure. As the parameters are varied, and, in particular, if \(\delta\) becomes positive (of order ten in the perturbation parameter \(\varepsilon\)), then the system undergoes bifurcations; in particular, the author prove that, by a suitable choice of the perturbation, the (weak foci of the) system can bifurcate 40 limit cycles, and there are 24 stable limit cycles among them. These, as well as the perturbed and unperturbed parameters, are identified by completely explicit formulas. A short reference list is given including previous work by the authors dealing with the same problem of bifurcation of limit cycles from \(\mathbb Z_n\)-equivariant dynamical systems of degree \(k\), with \(n\) and \(k\) smaller than in the present note.
scientific article; zbMATH DE number 5817291