3-dimensional Hopf bifurcation via averaging theory

Pages: 529 - 540, Volume 17, Issue 3, March 2007

doi:10.3934/dcds.2007.17.529       Abstract        Full Text (167.9K)       Related Articles

Jaume Llibre - Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain (email)
Claudio A. Buzzi - Departamento de Matemática, Universidade Estadual Paulista–UNESP, 15054-000 São José do Rio Preto, São Paulo, Brazil (email)
Paulo R. da Silva - Departamento de Matemática, Universidade Estadual Paulista–UNESP, 15054-000 São José do Rio Preto, São Paulo, Brazil (email)

Abstract: 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.

Keywords:  Hopf bifurcation, Lorenz system, averaging theory.
Mathematics Subject Classification:  37G15, 37D45.

Received: January 2006;      Revised: October 2006;      Published: December 2006.