An iterative procedure for determining limit cycles using Lagrangian mechanics

In many nonlinear autonomous mechanical systems, determination of the amplitudes and periods of limit cycles by direct numerical integration of the system differential equations can be very time-consuming computationally. In this investigation, an efficient iterative algorithm which converges to limit cycles of single-degree-of-freedom systems is presented. It is based on the work done by nonconservative forces in the system. Numerical results for several illustrative examples are given, including the van der Pol equation, a feedback control system with hysteresis, a system having an infinite number of stable and unstable limit cycles, and an oscillator with nonlinear dry friction.