Small Amplitude Periodic Orbits in Three-Dimensional Quadratic Vector Fields with a Zero-Hopf Singularity

We consider some families of three-dimensional quadratic vector fields having a fixed zero-Hopf equilibrium. We are interested in the bifurcation of periodic ν -orbits from the singularity, that is, those small amplitude orbits that make a fixed arbitrary number ν of revolutions about a rotation axi...

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Bibliographic Details
Published in:Journal of dynamics and differential equations 2024-06, Vol.36 (2), p.1325-1346
Main Author: García, Isaac A.
Format: Article
Language:English
Subjects:
Applications of Mathematics
Mathematics
Mathematics and Statistics
Ordinary Differential Equations
Partial Differential Equations
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Summary:We consider some families of three-dimensional quadratic vector fields having a fixed zero-Hopf equilibrium. We are interested in the bifurcation of periodic ν -orbits from the singularity, that is, those small amplitude orbits that make a fixed arbitrary number ν of revolutions about a rotation axis and then returns to the initial point closing the orbit. When the parameters of the family are restricted to certain explicitly computable open semi-algebraic sets Λ , we characterize those parameters that give rise to the appearance of local two-dimensional periodic invariant manifolds through the singularity. Also we use a Bautin-type analysis to study the maximum number of small-amplitude ν -limit cycles that can be made to bifurcate from the equilibrium when the parameters of the family are restricted to Λ . We obtain global upper bounds on the number of bifurcated ν -limit cycles.
ISSN:1040-7294
1572-9222
DOI:10.1007/s10884-022-10208-4