The extended periodic motion concept for fast limit cycle detection of self-excited systems

Abstract

Limit cycle solutions of self-excited dynamic systems can be determined by continuation of solutions along a system parameter variation or by brute-force testing. While the brute-force search for basins of attraction is computationally intractable, continuation methods compute only those branches that are connected to others, thus neglecting a-priori unknown solutions and detached branches, such as isolas. In this work, a method is proposed for finding limit cycles of self-excited dynamic systems. The method is based on the continuation of nonlinear modes for non-conservative systems, for which the Extended Periodic Motion Concept (E-PMC) is applied. The E-PMC allows for finding stable and unstable periodic solutions along the nonlinear mode and is especially helpful for determining solutions that are detached from other solution branches. Hence, the a-priori selection of proper initial conditions for the limit cycle computation is no longer necessary. A self-excited frictional oscillator with cubic stiffness terms is studied. The proposed technique proves to be robust and finds all isolated periodic solutions that were published previously by other authors. In an extended model configuration, the E-PMC finds co-existing stable limit cycles and unstable periodic orbits, one of which gives rise to hyper-chaotic motion with multiple positive Lyapunov exponents.