The cyclicity of hyperbolic hemicycles

2025/01/28 by David Marín, Marín, David, Jordi Villadelprat +1 · 1 citation
Chemistry · Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Dynamical Systems (math.DS) #FOS: Mathematics #Molecular spectroscopy and chirality #Quantum chaos and dynamical systems

paper · pdf · doi:10.48550/arxiv.2501.16924

openalex publication_date 2025/01/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

We consider families of planar polynomial vector fields of degree n and study the cyclicity of a type of unbounded polycycle~Γ called hemicycle. Compactified to the Poincaré disc,~Γ consists of an affine straight line together with half of the line at infinity and has two singular points, which are hyperbolic saddles located at infinity. We prove four main results. Theorem A deals with the cyclicity of~Γ when perturbed without breaking the saddle connections. For the other results we consider the case n=2. More concretely they are addressed to the quadratic integrable systems belonging to the class Q₃^R and having two hemicycles, Γ_u and Γ_ℓ, surrounding each one a center. Theorem B gives the cyclicity of Γ_u and Γ_ℓ when perturbed inside the whole family of quadratic systems. In Theorem C we study the number of limit cycles bifurcating simultaneously from Γ_u and Γ_ℓ when perturbed as well inside the whole family of quadratic systems. Finally, in Theorem D we show that for three specific cases there exists a simultaneous alien limit cycle bifurcation from Γ_u and Γ_ℓ.

Cited by

• Generalized Pseudo-Hopf Bifurcation: Limit Cycle Position and Period

Related

• The period of the limit cycle bifurcating from a persistent polycycle
• Local Uniform Finite Cyclicity of the H₁₄³ Semihyperbolic Hemicycle
• On the number of limit cycles which appear by perturbation of two-saddle cycles of planar vector fields
• On the cyclicity of persistent hyperbolic polycycles
• Transversal conics and the existence of limit cycles