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Bifurcation Analysis in Planar Quadratic Differential Systems with Boundary

Jocelyn A. Castro

Departamento de Matemáticas, Universidad de Sonora, Hermosillo, Sonora 83000, México

Instituto Tecnológico y de Estudios Superiores de Monterrey, Sonora Norte 83000, México

E-mail Address: jocelyn.castro@unison.mx

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and

Fernando Verduzco

http://orcid.org/0000-0002-3237-2090

Departamento de Matemáticas, Universidad de Sonora, Hermosillo, Sonora 83000, México

E-mail Address: verduzco@mat.uson.mx

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https://doi.org/10.1142/S0218127420300177Cited by:0 (Source: Crossref)

Abstract

Given a planar quadratic differential system delimited by a straight line, we are interested in studying the bifurcation phenomena that can arise when the position on the boundary of two tangency points are considered as parameters of bifurcation. First, under generic conditions, we find a two-parametric family of quadratic differential systems with at least one tangency point. After that, we find a normal form for this parameterized family. Next, we study two subfamilies, one of them characterized by the existence of two fold points of different nature, and the other one, characterized by the existence of one fold point and one boundary equilibrium point. For the first family, we find sufficient conditions for the existence of stationary bifurcations: saddle-node, transcritical and pitchfork, while for the second family, the existence of the called transcritical Bogdanov–Takens bifurcation is proved. Finally, the results are illustrated with two examples.

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