August  2007, 17(3): 529-540. doi: 10.3934/dcds.2007.17.529

3-dimensional Hopf bifurcation via averaging theory

1. 

Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain

2. 

Departamento de Matemática, Universidade Estadual Paulista–UNESP, 15054-000 São José do Rio Preto, São Paulo, Brazil, Brazil

Received  January 2006 Revised  October 2006 Published  December 2006

We consider the Lorenz system $\dot x = \s (y-x)$, $\dot y =rx -y-xz$ and $\dot z = -bz + xy$; and the Rössler system $\dot x = -(y+z)$, $\dot y = x +ay$ and $\dot z = b-cz + xz$. Here, we study the Hopf bifurcation which takes place at $q_{\pm}=(\pm\sqrt{br-b},\pm\sqrt{br-b},r-1),$ in the Lorenz case, and at $s_{\pm}=(\frac{c+\sqrt{c^2-4ab}}{2},-\frac{c+\sqrt{c^2-4ab}}{2a}, \frac{c\pm\sqrt{c^2-4ab}}{2a})$ in the Rössler case. As usual this Hopf bifurcation is in the sense that an one -parameter family in ε of limit cycles bifurcates from the singular point when ε=0. Moreover, we can determine the kind of stability of these limit cycles. In fact, for both systems we can prove that all the bifurcated limit cycles in a neighborhood of the singular point are either a local attractor, or a local repeller, or they have two invariant manifolds, one stable and the other unstable, which locally are formed by two $2$-dimensional cylinders. These results are proved using averaging theory. The method of studying the Hopf bifurcation using the averaging theory is relatively general and can be applied to other $3$- or $n$-dimensional differential systems.
Citation: Jaume Llibre, Claudio A. Buzzi, Paulo R. da Silva. 3-dimensional Hopf bifurcation via averaging theory. Discrete & Continuous Dynamical Systems - A, 2007, 17 (3) : 529-540. doi: 10.3934/dcds.2007.17.529
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