A Simple Chaotic Flow with a Plane of Equilibria

It is widely recognized that mathematically simple systems of nonlinear differential equations can exhibit chaos. With the advent of fast computers, it is now possible to explore the entire parameter space of these systems with the goal of finding parameters that result in some desired characteristics of the system. 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 et al., 2013; Molaie et al., 2013; Jafari & Sprott, 2013, 2015; Jafari et al., 2015; Jafari et al., 2014; Pham et al., 2014a; Pham et al., 2015; Pham et al., 2014b; Kingni et al., 2014; Shahzad et al., 2015; Sprott et al., 2015; Pham et al., 2014c; Tahir et al., 2015; Gotthans & Petržela, 2015; Wang & Chen, 2012, 2013; Wei, 2011; Pham et al., 2014d]. The attractors for such systems have been called hidden attractors [Kuznetsov & Leonov, 2011; Leonov et al., 2014; Leonov & Kuznetsov, 2013; Leonov et al., 2015; Leonov et al., 2011; Leonov et al., 2012; Leonov & Kuznetsov, 2013; Bragin et al., 2011], and that accounts for the difficulty of discovering them since there is no systematic way to choose initial conditions except by extensive numerical search. 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.