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

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2022Suggested citation
García, I. A. (Isaac A.);
.
(2022)
.
Small Amplitude Periodic Orbits in Three-Dimensional Quadratic Vector Fields with a Zero-Hopf Singularity.
Journal of Dynamics and Differential Equations, 2022.
https://doi.org/10.1007/s10884-022-10208-4.
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We consider some families of three-dimensional quadratic vector fields having a fixed zeroHopf 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 ,
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 . We obtain global
upper bounds on the number of bifurcated ν-limit cycles.
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Journal of Dynamics and Differential Equations, 2022European research projects
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