The period of the limit cycle bifurcating from a persistent polycycle

Identifier:  imarina:9391714

Handle: https://hdl.handle.net/20.500.11797/imarina9391714

Authors:  Marín, David; Queiroz, Lucas; Villadelprat, Jordi

Abstract:
We consider smooth families of planar polynomial vector fields {Xμ}μ∈Λ, where Λ is an open subset of RN, for which there is a hyperbolic polycycle Γ that is persistent (i.e., such that none of the separatrix connections is broken along the family). It is well known that in this case the cyclicity of Γ at μ0 is zero unless its graphic number r(μ0) is equal to one. It is also well known that if r(μ0)=1 (and some generic conditions on the return map are verified) then the cyclicity of Γ at μ0 is one, i.e., exactly one limit cycle bifurcates from Γ. In this paper we prove that this limit cycle approaches Γ exponentially fast and that its period goes to infinity as 1/|r(μ)−1| when μ→μ0. Moreover, we prove that if those generic conditions are not satisfied, although the cyclicity may be exactly 1, the behavior of the period of the limit cycle is not determined.