Simplest 3D continuous-Time quadratic Systems as Candidates for Generating multiscroll Chaotic attractors

In many situations, the existence of several equilibrium points in a dynamical system makes its dynamics more complex and allows some special structures. Examples include the well-known multiscroll attractors ([Wang, 2009; Qi et al., 2008; Liu & Chen, 2004; Li, 2008; Wang et al., 2009; Lü et al., 2008; Yu et al., 2006; Yu et al., 2008, 2010a; Wang & Chen, 2012] and references therein) such as chaotic attractors with multiple-merged basins of attraction, scroll grid attractors, and 2n-wing and n×m-wing Lorenz-like attractors. It is remarkable that the vector fields of all these systems are very complex with high dimensions. In some other situations, the number of equilibrium points has no influence on the scroll number of the resulting chaotic attractor. For example, in [Wang & Chen, 2012] the considered system has one equilibrium point, but generates many-petal attractors. So the existence of a system with only zero, one, or fewer than n equilibrium points that generates n-scroll chaotic attractors leaves an important open problem whose solution is not evident. We notice that this type of attractor has several real world applications. For example, the circuit verification of proposed multiscroll chaotic systems is very important for several potential engineering applications such as secure communication and efficient liquid mixing. See [Yu et al., 2010b] for more details. Furthermore, multiscroll chaotic attractors have wide applications in complex networks [Lü & Chen, 2005] and multiagent systems [Zhu et al., 2013].