The continuation and stability analysis methods for quasi-periodic solutions of nonlinear systems

The continuation and stability analysis methods for quasi-periodic solutions of nonlinear systems are proposed. The proposed continuation method advances the predictor-corrector continuation framework by coupling the reduced space sequential quadratic programming method with the multi-dimensional harmonic balance method and the gradients required for the continuation problem are derived. In order to determine the stability of quasi-periodic solution, a novel approach based on the analytical formulation of the harmonic balance equations is presented by using the Floquet theory with the perturbation term applied to the known quasi-periodic solution. Sensitivity analysis about the stability factor of quasi-periodic solution is also carried out. Finally, the effectiveness and applicability of the proposed methodology is verified and illustrated by two numerical examples. The proposed approaches have been demonstrated to be able to trace the aperiodic solutions of nonlinear systems and analyze their stabilities.