Simple polynomial classes of chaotic jerky dynamics

Third-order explicit autonomous differential equations, commonly called jer ky dynamics, constitute a powerful approach to understand the properties of functionally very simple but nonlinear three-dimensional dynamical systems that can exhibit chaotic longtime behavior. In this paper, we investigate the dynamics that can be generated by the two simplest polynomial jerky dyn amics that, up to these days, are known to show chaotic behavior in some pa rameter range. After deriving several analytical properties of these system s, we systematically determine the dependence of the long-time dynamical be havior on the system parameters by numerical evaluation of Lyapunov spectra . Some features of the systems that are related to the dependence on initia l conditions are also addressed. The observed dynamical complexity of the t wo systems is discussed in connection with the existence of homoclinic orbi ts. (C) 2001 Elsevier Science Ltd. All rights reserved.