On the 3-dimensional Hopf bifurcation via averaging theory of third order

Authors: ELOUAHMA BENDIB, SABRINA BADI, AMMAR MAKHLOUF

Abstract: We apply the averaging theory of third order to polynomial quadratic vector fields in $\mathbb{R}^3$ to study the Hopf bifurcation occurring in that polynomial. Our main result shows that at most $10$ 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 $10$ limit cycles bifurcate from a such singular point.

Keywords: Hopf bifurcation, limit cycle, averaging theory of third order

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