Conditions for the occurrence of limit cycles in autonomous potential damped systems

Abstract

The local dynamic stability of discrete, weakly damped, systems under step loading of infinite duration either constant directional (potential load) or partial follower (non-potential load) is thoroughly re-examined. In particular, for such autonomous damped systems we seek conditions for the existence of: (a) a limit cycle response in case of a potential loading (symmetric) system and (b) a non-singular transformation which transforms a non-potential (asymmetric) system into an equivalent symmetric system. Using a 2-DOF as a model new findings are established that contradict existing widely accepted results. Thus, symmetric systems under certain conditions can exhibit a limit cycle response due to either a double-zero Jacobian eigenvalue or to a Hopf bifurcation. Also asymmetric (non-self-adjoint) systems, regardless of the degree of asymmetry, can always be transformed into equivalent symmetric systems. A variety of numerical examples confirm the validity of the theoretical findings presented herein.