Simple Chaotic Flows with a Curve 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 [Sprott, 2010]. Recent research has involved categorizing periodic and chaotic attractors as either self-excited or hidden [Bragin et al., 2011; Leonov & Kuznetsov, 2010, 2011, 2013a, 2013b, 2014; Leonov et al., 2011, 2012; Leonov et al., 2014; Leonov et al., 2015a, 2015b; Sharma et al., 2015a, 2015b]. A self-excited attractor has a basin of attraction that is associated with an unstable equilibrium, whereas a hidden attractor has a basin of attraction that does not intersect with small neighborhoods of any equilibrium points. The classical attractors of Lorenz, Rössler, Chua, Chen, Sprott systems (cases B to S) and other widely-known attractors are those excited from unstable equilibria. From a computational point of view this allows one to use a numerical method in which a trajectory that started from a point on the unstable manifold in the neighborhood of an unstable equilibrium, reaches an attractor and identifies it [Leonov et al., 2011]. Hidden attractors cannot be found by this method and are important in engineering applications because they allow unexpected and potentially disastrous responses to perturbations in a structure like a bridge or an airplane wing.