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$3$ - dimensional Hopf bifurcation via averaging theory of second order

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  • We study the Hopf bifurcation occurring in polynomial quadratic vector fields in $\R^3$. By applying the averaging theory of second order to these systems we show that at most $3$ limit cycles can bifurcate from a singular point having eigenvalues of the form $\pm bi$ and $0$. We provide an example of a quadratic polynomial differential system for which exactly $3$ limit cycles bifurcate from a such singular point.
    Mathematics Subject Classification: Primary: 37G15; Secondary: 37D45.


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