# American Institute of Mathematical Sciences

August  2011, 30(3): 779-790. doi: 10.3934/dcds.2011.30.779

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

 1 Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona 2 Departamento de Matemática, Instituto Superior Técnico, Av. Rovisco Pais 1049-001, Lisboa, Portugal

Received  November 2009 Revised  January 2011 Published  March 2011

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.
Citation: Jaume Llibre, Clàudia Valls. Hopf bifurcation for some analytic differential systems in $\R^3$ via averaging theory. Discrete and Continuous Dynamical Systems, 2011, 30 (3) : 779-790. doi: 10.3934/dcds.2011.30.779
##### References:
 [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. [2] 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. [3] J. Llibre, Averaging theory and limit cycles for quadratic systems, Radovi Matematicki, 11 (2002), 215-228. [4] J. A. Sanders and F. Verhulst, "Averaging Methods in Nonlinear Dynamical Systems," Applied Mathematical Sci., 59, Springer-Verlag, New York, 1985. [5] F. Verhulst, "Nonlinear Differential Equations and Dynamical Systems," Universitext. Springer-Verlag, Berlin, 1996.

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##### References:
 [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. [2] 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. [3] J. Llibre, Averaging theory and limit cycles for quadratic systems, Radovi Matematicki, 11 (2002), 215-228. [4] J. A. Sanders and F. Verhulst, "Averaging Methods in Nonlinear Dynamical Systems," Applied Mathematical Sci., 59, Springer-Verlag, New York, 1985. [5] F. Verhulst, "Nonlinear Differential Equations and Dynamical Systems," Universitext. Springer-Verlag, Berlin, 1996.
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