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.

提出了非线性系统拟周期解的连续性和稳定性分析方法。所提出的连续方法通过将缩减空间顺序二次规划方法与多维谐波平衡方法耦合而改进了预测器-校正器连续框架，并得出了连续问题所需的梯度。为了确定准周期解的稳定性，提出了一种新的方法，该方法基于Floquet理论，将谐波项方程应用到已知的拟周期解中，并且基于谐波平衡方程的解析公式。还对准周期解的稳定性因子进行了敏感性分析。最后，通过两个数值实例验证了所提出方法的有效性和适用性。已经证明所提出的方法能够追踪非线性系统的非周期解并分析其稳定性。