Multi-parameter homotopy for finding periodic solutions of power electronic circuits

Commonly used methods for calculating periodic steady state, such as forward integration and shooting, may fail for highly nonlinear circuits with multiple solutions and/or multiple time scales. Homotopy continuation methods, because of their potentially large or global regions of convergence, and suitability for finding multiple solutions, have been applied to the calculation of periodic steady state for such systems. This paper applies real and complex multi-parameter homotopy to finding periodic solutions of power electronic circuits. We show that multi-parameter homotopy methods can avoid period-doubling and cyclic fold bifurcations along solution paths, and find all stable and unstable periodic solutions along folding or period-doubling paths. We distinguish between circuit-direct and formulation-indirect homotopy, and show that the latter (with two real parameters) can avoid period-doubling bifurcations, while the former cannot