2023/06/27 by David Marín, Marín, David, Lucas Queiroz +3 · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Dynamical Systems (math.DS) #FOS: Mathematics #Nonlinear Dynamics and Pattern Formation #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.2306.15473
openalex publication_date 2023/06/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.