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 & Continuous Dynamical Systems - A, 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.  doi: 10.1016/j.bulsci.2003.09.002.  Google Scholar

[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.   Google Scholar

[3]

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

[4]

J. A. Sanders and F. Verhulst, "Averaging Methods in Nonlinear Dynamical Systems,", Applied Mathematical Sci., 59 (1985).   Google Scholar

[5]

F. Verhulst, "Nonlinear Differential Equations and Dynamical Systems,", Universitext. Springer-Verlag, (1996).   Google Scholar

show all references

References:
[1]

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

[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.   Google Scholar

[3]

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

[4]

J. A. Sanders and F. Verhulst, "Averaging Methods in Nonlinear Dynamical Systems,", Applied Mathematical Sci., 59 (1985).   Google Scholar

[5]

F. Verhulst, "Nonlinear Differential Equations and Dynamical Systems,", Universitext. Springer-Verlag, (1996).   Google Scholar

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