Quadratic slow-fast systems on the plane. (August 2021)

• Record Type: Journal Article
• Authors: Meza-Sarmiento, Ingrid S.; Oliveira, Regilene; da Silva, Paulo R.
• Is Part Of: Nonlinear analysis. Volume 60(2021)
• Publication Date: 2021-08
• Subjects: Topological invariant -- Vector field -- Singular perturbation -- Quadratic system; Nonlinear functional analysis -- Periodicals; Analyse fonctionnelle non linéaire -- Périodiques; Nonlinear functional analysis; Periodicals; 515.7248
• DOI: 10.1016/j.nonrwa.2020.103286
• Languages: English
• ISSNs: 1468-1218

Abstract

Abstract: In this paper singularly perturbed quadratic polynomial differential systems ε x ̇ = P ε ( x, y ) = P ( x, y, ε ), y ̇ = Q ε ( x, y ) = Q ( x, y, ε ) with x, y ∈ R, ε ⩾ 0 and ( P ε, Q ε ) = 1 for ε > 0, are considered. We prove that there are 10 classes of equivalence for these systems. We describe the dynamics of these 10 classes on the Poincaré disc when ε = 0 . For ε > 0, we present the possible local behavior of the solutions near of a finite and infinite equilibrium point under suitable conditions. More specifically, if p 0 is a finite equilibrium point then we obtain the local behavior for ε > 0 using Fenichel theory. Assuming that p 0 is an infinite equilibrium point, there exists K ⊂ M 0 normally hyperbolic and p 0 ∈ M 0 ′ ∩ K using the Poincaré compactification and algebraic invariant we describe globally the dynamics for ε > 0 small of some classes of equivalence.