Bifurcations of limit cycles in a \(Z_6\)-equivariant planar vector field of degree 5. (English) Zbl 1107.34317
Summary: A concrete numerical example of \(Z_6\)-equivariant planar perturbed Hamiltonian polynomial vector fields of degree 5 having at least 24 limit cycles and the configurations of compound eyes are given by using the bifurcation theory of planar dynamical systems and the method of detection functions. There is reason to conjecture that the Hilbert number \(H(2k+1)\geq(2k+1)^2-1\) for the perturbed Hamiltonian systems.
MSC:
| 34C23 | Bifurcation theory for ordinary differential equations |
| 34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |
| 34C14 | Symmetries, invariants of ordinary differential equations |
| 37G40 | Dynamical aspects of symmetries, equivariant bifurcation theory |
| 37J45 | Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010) |
