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

Third-order explicit autonomous differential equations in one scalar variab le or, mechanically interpreted, jerky dynamics constitute an interesting s ubclass of dynamical systems that can exhibit many major features of regula r and irregular or chaotic dynamical behavior. In this paper, we investigat e the circumstances under which three dimensional autonomous dynamical syst ems possess at least one equivalent jerky dynamics. In particular, we deter mine a wide class of three-dimensional vector fields with polynomial and no n-polynomial nonlinearities that possess this property. Taking advantage of this general result, we focus on the jerky dynamics of Sprott's minimal ch aotic dynamical systems and Rossler's toroidal chaos model. Based on the in terrelation between the jerky dynamics of these models, we classify them ac cording to their increasing polynomial complexity. Finally, we also provide a simple criterion that excludes chaotic dynamics for some classes of jerk y dynamics and, therefore, also for some classes of three-dimensional dynam ical systems. [S1063-651X(98)09710-4].