Geometry and integrability of quadratic systems with invariant hyperbolas

Oliveira, Regilene; Schlomiuk, Dana; Travaglini, Ana Maria

doi:10.14232/ejqtde.2021.1.6

Cited by 3 publications

(1 citation statement)

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“…On QSL 3 and hence on QSL ≥3 the geometrical classification is done in [Schlomiuk & Vulpe, 2010], and the topological classification (phase portraits) was done in [Schlomiuk & Vulpe, 2012;Schlomiuk & Zhang, 2018]. On QSH the geometrical classification was done in [Oliveira et al, 2017], the topological classification and the integrability of families in this class is done in [Oliveira et al, 2021a,b]. Additional references on quadratic polynomial differential where important global tools were introduced and used in studied of two families of quadratic differential systems are [Llibre & Schlomiuk, 2004;.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Global Phase Portraits of the Quadratic Systems Having a Singular and Irreducible Invariant Curve of Degree 3

Llibre

Pantazi

2023

Int. J. Bifurcation Chaos

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Any singular irreducible cubic curve (or simply, cubic) after an affine transformation can be written as either [Formula: see text], or [Formula: see text], or [Formula: see text]. We classify the phase portraits of all quadratic polynomial differential systems having the invariant cubic [Formula: see text]. We prove that there are [Formula: see text] different topological phase portraits for such quadratic polynomial differential systems. We control all the bifurcations among these distinct topological phase portraits. These systems have no limit cycles. Only three phase portraits have a center, 19 of these phase portraits have one polycycle, three of these phase portraits have two polycycles. The maximum number of separartices that have these phase portraits is 26 and the minimum number is nine, the maximum number of canonical regions of these phase portraits is seven and the minimum is three.

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Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%