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Hopf bifurcation for some analytic differential systems in $\R^3$ via averaging theory

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  • We study the Hopf bifurcation from the singular point with eigenvalues $a$ε$ \ \pm\ bi$ and $c $ε located at the origen of an analytic differential system of the form $ \dot x= f( x)$, where $x \in \R^3$. Under convenient assumptions we prove that the Hopf bifurcation can produce $1$, $2$ or $3$ limit cycles. We also characterize the stability of these limit cycles. The main tool for proving these results is the averaging theory of first and second order.
    Mathematics Subject Classification: Primary: 37G15, 37D45.


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  • [1]

    C. A. Biuca and J. Llibre, Averaging methods for finding periodic orbits via Brownew degree, Bull. Sci. Math, 128 (2004), 7-22.doi: 10.1016/j.bulsci.2003.09.002.


    C. A. Buzzi, J. Llibre and P. R. Da Silva, Generalized $3$-dimensional Hopf bifurcation via averaging theory, Discrete Continuous Dynam. Systems - A, 17 (2007), 529-540.


    J. Llibre, Averaging theory and limit cycles for quadratic systems, Radovi Matematicki, 11 (2002), 215-228.


    J. A. Sanders and F. Verhulst, "Averaging Methods in Nonlinear Dynamical Systems," Applied Mathematical Sci., 59, Springer-Verlag, New York, 1985.


    F. Verhulst, "Nonlinear Differential Equations and Dynamical Systems," Universitext. Springer-Verlag, Berlin, 1996.

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