Bifurcation Analysis in Planar Quadratic Differential Systems with Boundary

Jocelyn A. Castro, Fernando Verduzco

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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.

Original languageEnglish
Article number2030017
JournalInternational Journal of Bifurcation and Chaos
Issue number7
StatePublished - 15 Jun 2020

Bibliographical note

Publisher Copyright:
© 2020 World Scientific Publishing Company.


  • Stationary bifurcation
  • differential systems with boundary
  • quadratic system
  • transcritical Bogdanov-Takens bifurcation


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