Centers on center manifolds in the Lorenz, Chen and Lü systems

Abstract We provide in a very straightforward manner a proof for the existence of centers on center manifolds, for the generalized Lorenz system, x = a ( y - x ) , y = bx + cy - xz , z = dz + xy . From this result, the presence of this Hopf bifurcation of codimension infinity is trivially deduced for the Lorenz, Chen and Lu systems. Our outcomes are novel for the Lorenz and Chen systems and, in the case of the Lu system, we obtain again, but in an easier way, the result found in the literature. Moreover, we show for this Hopf bifurcation a relationship between the three systems.

[1]  Xijun Deng.  Invariant algebraic surfaces of the generalized Lorenz system , 2013 .

[2]  Luis Fernando Mello,et al.  Degenerate Hopf bifurcations in the Lü system , 2009 .

[3]  M. Escobar,et al.  Geochemistry and chemostratigraphy of the Colón-Mito Juan units (Campanian-Maastrichtian), Venezuela: Implications for provenance, depositional conditions, and stratigraphic subdivision , 2013 .

[4]  Manuel Merino,et al.  Chen's attractor exists if Lorenz repulsor exists: the Chen system is a special case of the Lorenz system. , 2013, Chaos.

[5]  D. Mahler.  Record , 1970 .

[6]  Zhuosheng Lü,et al.  Codimension-2 Bautin bifurcation in the Lü system , 2007 .

[7]  Claudio Pessoa,et al.  Centers on center manifolds in the Lü system , 2011 .

[8]  Guanrong Chen,et al.  Local bifurcations of the Chen System , 2002, Int. J. Bifurc. Chaos.

[9]  Guanrong Chen,et al.  YET ANOTHER CHAOTIC ATTRACTOR , 1999 .

[10]  Alejandro J. Rodríguez-Luis,et al.  The Lü system is a particular case of the Lorenz system , 2013 .

[12]  F. Fernández-Calderón,et al.  [Spanish adaptation of the Substance Dependence Severity Scale: preliminar results]. , 2013, Adicciones.

[13]  Kesheng Wu,et al.  Darboux polynomials and rational first integrals of the generalized Lorenz systems , 2012 .

[14]  Yongguang Yu,et al.  Hopf bifurcation analysis of the Lü system , 2004 .

[15]  Guanrong Chen,et al.  Bifurcation Analysis of Chen's equation , 2000, Int. J. Bifurc. Chaos.

[16]  Alejandro J. Rodríguez-Luis,et al.  On Darboux polynomials and rational first integrals of the generalized Lorenz system , 2014 .

[17]  Colin Sparrow,et al.  The Lorenz equations , 1982 .

[18]  Yongguang Yu,et al.  Hopf bifurcation in the Lü system , 2003 .

[19]  A. Rauh,et al.  Analytical investigation of the hopf bifurcation in the Lorenz model , 1986 .

[20]  Guanrong Chen,et al.  A Note on Hopf bifurcation in Chen's System , 2003, Int. J. Bifurc. Chaos.

[21]  E. Lorenz.  Deterministic nonperiodic flow , 1963 .

[22]  Jinhu Lu,et al.  A New Chaotic Attractor Coined , 2002, Int. J. Bifurc. Chaos.

[23]  Guanrong Chen,et al.  Dynamical Analysis of a New Chaotic Attractor , 2002, Int. J. Bifurc. Chaos.