Small limit cycles bifurcating from fine focus points in cubic order \(Z_{2}\)-equivariant vector fields. (English) Zbl 1067.34033
Summary: The existence of 12 small limit cycles is proved for cubic-order \(Z_2\)-equivariant vector fields, which bifurcate from fine focus points. This is a new result in the study of the second part of the 16th Hilbert problem. The system under consideration has a saddle point, or a node, or a focus point (including center) at the origin, and two weak focus points which are symmetric about the origin. It is shown that the system can exhibit 10 and 12 small limit cycles for some special cases. Further studies are given in this paper to consider all possible cases, and prove that such a \(Z_2\)-equivariant vector field can have maximal 12 small limit cycles. Fourteen or sixteen small limit cycles, as expected before, are not possible.
MSC:
| 34C07 | Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations |
| 34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |
| 34C23 | Bifurcation theory for ordinary differential equations |
| 37G15 | Bifurcations of limit cycles and periodic orbits in dynamical systems |
| 37G40 | Dynamical aspects of symmetries, equivariant bifurcation theory |
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