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

Jaume Llibre; Clàudia Valls

doi:10.3934/dcds.2011.30.779

Article Contents

2011, Volume 30, Issue 3: 779-790. Doi: 10.3934/dcds.2011.30.779

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

Jaume Llibre^1, and
Clàudia Valls^2,

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

Received: November 2009

Revised: January 2011

Published online: March 2011

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Abstract

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.

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Mathematics Subject Classification: Primary: 37G15, 37D45.

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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.