Stability boundaries and the exact calculation of mixed-stability limit cycles in higher-dimensional state spaces

A multiple-shooting algorithm for calculating limit cycles of nonlinear systems is developed, and the relationship of mixed-stability limit cycles to the stability boundaries of equilibria is demonstrated. Exact determination of mixed-stability limit cycles in higher-dimensional state spaces can be an important tool for use in exact calculation of an equilibrium point´s domain of attraction. The algorithm is presented along with three test limit cycle calculations: one for a system with saturation and integrator wind-up; one for a balancing peg-leg robot system; and one for a system with negative Van der Pol damping. Also presented is a way to determine the local tangent to a system´s stability boundary by considering limit cycle properties. This method is demonstrated by prediction of stable and unstable initial conditions and verification of the predictions via numerical simulation