基于Tikhonov正则化改进的IHB法求解Mathieu-Duffing系统多重解

Multiple solutions of the Mathieu-Duffing system obtained by the improved IHB method based on Tikhonov regularization

增量谐波平衡法(IHB法)是研究强非线性振动系统的一种半数值半解析方法,然而已有研究表明,在求解含多重解的系统时该方法的收敛性强烈地依赖于初值的选择。Tikhonov正则化常被用于优化问题中来解决可能出现的病态问题。文章通过在原始的IHB法中引入Tikhonov正则化,提出一种改进的IHB法(TIHB法)来求解具有多重解的Mathieu-Duffing系统。结果表明,改进的TIHB法可以快速、高效地获得系统的多个稳定或不稳定解,且算法的收敛性能要远远优于原始的IHB法。

The incremental harmonic balance method(IHB method) is a semi-numerical and semi-analytical method for strongly nonlinear dynamic systems.However,previous studies have shown that the convergence performance of the original IHB method in solving systems with multiple solutions strongly depends on the selection of initial values.The Tikhonov regularization is often used in optimization problems to solve potential ill-posed problems.In this paper,by incorporating the Tikhonov regularization into the original IHB method,an improved IHB method(TIHB method) is proposed to obtain the multiple solutions of the Mathieu-Duffing system.The results show that the improved TIHB method can obtain the stable and unstable solutions of the Mathieu-Duffing system quickly and efficiently,and the convergence performance of the TIHB method is much better than the original IHB method.