Realization problems for limit cycles of planar polynomial vector fields

We show that for any finite configuration of closed curves Gamma subset of R-2, one can construct an explicit planar polynomial vector field that realizes Gamma, up to homeomorphism, as the set of its limit cycles with prescribed periods, multiplicities and stabilities. The only obstruction given on this data is the obvious compatibility relation between the stabilities and the parity of the multiplicities. The constructed vector fields are Darboux integrable and admit a polynomial inverse integrating factor. (C) 2015 Elsevier Inc. All rights reserved.

limit cycle; polynomial vector field; integrating factor; realization problem; configurations; existence; systems

Realization problems for limit cycles of planar polynomial vector fields Articles