Symmetric Hybrid Hessian Adaptive Trust-Region Incremental Harmonic Balance Method for Strongly Nonlinear Systems

To address the insufficient convergence robustness, strong dependence on the initial guess, and computational-efficiency bottlenecks of the incremental harmonic balance (IHB) method for strongly nonlinear systems, this paper develops an adaptive framework that integrates a trust-region strategy with a hybrid Hessian matrix. The framework reformulates the harmonic-balance iteration as a constrained nonlinear least-squares optimization problem, constructs a symmetric hybrid Hessian matrix by blending the Gauss–Newton and Newton directions, and uses a trust-region algorithm to adaptively regulate the step size, thereby jointly optimizing the iterative path and convergence behavior. Numerical results show that the proposed method significantly enhances convergence robustness, reduces sensitivity to initial guesses, and improves computational efficiency while maintaining high accuracy. It also captures both stable and unstable periodic solutions in strongly nonlinear systems.