Categorizing Chaotic Flows from the Viewpoint of Fixed Points and Perpetual Points

Recently, many new chaotic flows have been discovered that are not associated with a saddle point, including ones without any equilibrium points, with only stable equilibria, or with a line containing infinitely many equilibrium points [Jafari & Sprott, 2013, 2015; Jafari et al., 2013; Jafari et al., 2015b; Kingni et al., 2014; Lao et al., 2014; Molaie et al., 2013; Pham et al., 2014a; Pham et al., 2014b; Pham et al., 2014c; Pham et al., 2014d; Pham et al., 2015; Shahzad et al., 2015; Tahir et al., 2015; Pham et al., 2016; Goudarzi et al., 2016]. The attractors of these categories have been called hidden attractors [Leonov & Kuznetsov, 2014; Leonov et al., 2014; Leonov & Kuznetsov, 2011; Leonov et al., 2011, 2012; Leonov et al., 2015; Leonov & Kuznetsov, 2013a, 2013b, 2013c; Bragin et al., 2011; Kuznetsov et al., 2010; Kuznetsov et al., 2011; Leonov, 2010; Kiseleva et al., 2017; Andrievsky et al., 2016; Kiseleva et al., 2016; Bianchi et al., 2016; Kuznetsov et al., 2016b]. Hidden attractors are important in engineering applications because they allow unexpected and potentially disastrous responses to perturbations in a structure like a bridge or aircraft wing. The classical attractors of Lorenz [Lorenz, 1963], Rössler [Rössler, 1976], Chen [Chen & Ueta, 1999], Sprott (cases B to S) [Sprott, 1994], and other well-known attractors are excited from unstable equilibria. Thus, one can find these attractors by starting a trajectory from a point on the unstable manifold in the neighborhood of an unstable equilibrium [Leonov et al., 2011]. One of the interesting topics in nonlinear dynamics that was recently proposed is perpetual points [Prasad, 2015a; Dudkowski et al., 2015; Prasad, 2015b; Jafari et al., 2015a]. It has been shown that these points can be used to locate hidden attractors and to find coexisting attractors in multistable systems [Prasad, 2015a].