Structurally unstable quadratic vector fields of codimension two: families possessing a finite saddle-node and an infinite saddle-node

We consider a bifurcation that occurs in some two-dimensional vector fields, namely a codimension-one bifurcation in which there is simultaneously the creation of a pair of equilibria via a steady state bifurcation and the destruction of a large amplitude periodic orbit. We show that this phenomenon may occur in an unfolding of the saddle-node/pitchfork normal form equations, although not near the saddle-node/pitchfork instability. By construction and analysis of a return map, we show that the codimension-one bifurcation emerges from a codimension-two point at which there is a heteroclinic bifurcation between two saddle equilibria, one hyperbolic and the other nonhyperbolic. We find the same phenomenon occurs in the normal form equations for the hysteresis/pitchfork bifurcation, in this case arbitrarily close to the instability, and show there are restrictions regarding the way in which such dynamics can occur near pitchfork/pitchfork bifurcations. These conclusions carry over to analogous phenomena in normal forms for steady state/Hopf bifurcations.

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Structurally Unstable Quadratic Vector Fields of Codimension Two: Families Possessing Either a Cusp Point or Two Finite Saddle-Nodes

Journal of Dynamics and Differential Equations ◽

10.1007/s10884-020-09871-2 ◽

2020 ◽

Author(s):

Joan C. Artés ◽

Regilene D. S. Oliveira ◽

Alex C. Rezende

Keyword(s):

Vector Fields ◽

Cusp Point ◽

Codimension Two ◽

Quadratic Vector Fields

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On the singularity structure of Kahan discretizations of a class of quadratic vector fields

European Journal of Mathematics ◽

10.1007/s40879-021-00479-4 ◽

2021 ◽

Author(s):

René Zander

Keyword(s):

Elliptic Curves ◽

Vector Fields ◽

Singularity Structure ◽

Quadratic Vector Fields ◽

Parameter Values

AbstractWe discuss the singularity structure of Kahan discretizations of a class of quadratic vector fields and provide a classification of the parameter values such that the corresponding Kahan map is integrable, in particular, admits an invariant pencil of elliptic curves.

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Planar quadratic vector fields with finite saddle connection on a straight line (convex case)

Qualitative Theory of Dynamical Systems ◽

10.1007/bf02972672 ◽

2005 ◽

Vol 6 (2) ◽

pp. 187-204 ◽

Cited By ~ 1

Author(s):

Paulo César Carrião ◽

Maria Elasir Seabra Gomes ◽

Antonio Augusto Gaspar Ruas

Keyword(s):

Vector Fields ◽

Saddle Connection ◽

Quadratic Vector Fields ◽

Straight Line ◽

Convex Case

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Twelve Limit Cycles in 3D Quadratic Vector Fields with Z3 Symmetry

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127418501390 ◽

2018 ◽

Vol 28 (11) ◽

pp. 1850139 ◽

Cited By ~ 3

Author(s):

Laigang Guo ◽

Pei Yu ◽

Yufu Chen

Keyword(s):

Limit Cycles ◽

Vector Fields ◽

Center Manifold ◽

Three Dimensional ◽

Center Manifold Theory ◽

Normal Form Theory ◽

Quadratic Vector Fields ◽

Fine Focus ◽

Form Theory ◽

Manifold Theory

This paper is concerned with the number of limit cycles bifurcating in three-dimensional quadratic vector fields with [Formula: see text] symmetry. The system under consideration has three fine focus points which are symmetric about the [Formula: see text]-axis. Center manifold theory and normal form theory are applied to prove the existence of 12 limit cycles with [Formula: see text]–[Formula: see text]–[Formula: see text] distribution in the neighborhood of three singular points. This is a new lower bound on the number of limit cycles in three-dimensional quadratic systems.

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