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Isochronicity into a family of time-reversible cubic vector fields. (English) Zbl 1025.37011
The authors consider planar cubic vector fields with a nondegenerate center at the origin of the $(x,y)$-plane. Without restriction, the linear part is $(-y,x)$. The aim is to characterize, within a certain subfamily, those vector fields for which the center is isochronous. The subfamily chosen consists of all time-reversible vector fields possessing an integrating factor of the form $(1+x)^{-k}$. Recall that a vector field is time-reversible, if it is invariant under some fixed rotation combined with time reversion. It is proved that in this subfamily, there exist exactly six genuinely cubic vector fields with an isochronous center at the origin. They are written down explicitly. The proof consists of two parts: 1) The first five period constants, obtained by computer algebra methods, must vanish and thus yield necessary conditions. 2) Then sufficient conditions, such as the existence of a transversal commuting vector field, are used to select the final list of six vector fields.

37C10Vector fields, flows, ordinary differential equations
34C14Symmetries, invariants (ODE)
34C25Periodic solutions of ODE
37C80Symmetries, equivariant dynamical systems
Full Text: DOI
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