Transformations of nonlinear dynamical systems to jerky motion and its application to minimal chaotic flows

Abstract

Third-order explicit autonomous differential equations in one scalar variable or, mechanically interpreted, jerky dynamics constitute an interesting subclass of dynamical systems that can exhibit many major features of regular and irregular or chaotic dynamical behavior. In this paper, we investigate the circumstances under which three dimensional autonomous dynamical systems possess at least one equivalent jerky dynamics. In particular, we determine a wide class of three-dimensional vector fields with polynomial and non-polynomial nonlinearities that possess this property. Taking advantage of this general result, we focus on the jerky dynamics of Sprott's minimal chaotic dynamical systems and R\"ossler's toroidal chaos model. Based on the interrelation between the jerky dynamics of these models, we classify them according to their increasing polynomial complexity. Finally, we also provide a simple criterion that excludes chaotic dynamics for some classes of jerky dynamics and, therefore, also for some classes of three-dimensional dynamical systems.