Local bifurcations of the Chen System

Chaos has been extensively studied by scientists, physicists and mathematicians for more than three decades. Recently, this study has evolved from the traditional trend of understanding and analyzing chaos to the new intention of controlling and utilizing it, especially within the engineering community [Chen & Dong, 1998; Wang & Chen, 2000; Lü et al., 2002c]. Lorenz found the first canonical chaotic attractor in 1963 [Lorenz, 1963; Stewart, 2000], and Chen found another similar but topologically nonequivalent chaotic attractor in 1999 [Chen & Ueta, 1999; Ueta & Chen, 2000]. It has recently been proven that the Chen system is dual to the Lorenz system, in a sense defined by Vanĕc̆ek and C̆elikovský [1996]: The Lorenz system satisfies the condition a12a21 > 0 while the Chen system satisfies a12a21 < 0, where aij are elements of the constant system matrix of their linear parts, A = [aij ]3×3. Very recently, Lü and Chen found a new chaotic system [Lü & Chen, 2002], which satisfies the condition a12a21 = 0 and represents the transition between the Lorenz and the Chen attractors [Lü et al., 2002a]. Over the last two years, there are some detailed investigations and studies of the Chen system [Agiza & Yassen, 2001; C̆elikovský & Chen, 2002; Chang et al., 2000; Lü & Zhang, 2001;