Perturbation Function Iteration Method: A New Framework for Solving Periodic Solutions of Non-linear and Non-smooth Systems

TL;DR

PFIM addresses the challenge of computing periodic responses in nonlinear and non-smooth dynamical systems by recasting the problem via perturbation theory into a time-varying linear framework and solving corrections with a basis-free, PCA-based integration over one period. It derives a linear correction for $\mu=\Delta x$ and $\nu=\Delta\omega$ using $\mu' = Q(\tau)\mu + P(\tau) + F(\tau)\nu$ and enforces periodicity with a phase condition, augmented by a period-wise, piecewise-constant split with $\Phi_i=\exp(Q_i\Delta\tau)$. PFIM achieves quadratic convergence in smooth problems and degrades gracefully with reduced smoothness, while delivering substantial computational savings in high-dimensional non-smooth cases (up to two orders of magnitude faster than HBM) and robust continuation capabilities. Case studies on Van der Pol, Duffing, and a high-dimensional FE cantilever demonstrate PFIM’s accuracy, efficiency, and potential for engineering applications, with clear pathways for parallelization and model-order reduction. Overall, PFIM provides a robust, scalable alternative for nonlinear periodic analysis in complex systems, particularly where non-smooth dynamics and large model sizes render traditional methods costly or impractical.

Abstract

Computing accurate periodic responses in strongly nonlinear or even non-smooth vibration systems remains a fundamental challenge in nonlinear dynamics. Existing numerical methods, such as the Harmonic Balance Method (HBM) and the Shooting Method (SM), have achieved notable success but face intrinsic limitations when applied to complex, high-dimensional, or non-smooth systems. A key bottleneck is the construction of Jacobian matrices for the associated algebraic equations; although numerical approximations can avoid explicit analytical derivation, they become unreliable and computationally expensive for large-scale or non-smooth problems. To overcome these challenges, this study proposes the Perturbation Function Iteration Method (PFIM), a novel framework built upon perturbation theory. PFIM transforms nonlinear equations into time-varying linear systems and solves their periodic responses via a piecewise constant approximation scheme. Unlike HBM, PFIM avoids the trade-off between Fourier truncation errors and the Gibbs phenomenon in non-smooth problems by employing a basis-free iterative formulation, while significantly simplifying the Jacobian computation. Extensive numerical studies, including self-excited systems, parameter continuation, systems with varying smoothness, and high-dimensional finite element models, demonstrate that PFIM achieves quadratic convergence in smooth systems and maintains robust linear convergence in highly non-smooth cases. Moreover, comparative analyses show that, for high-dimensional non-smooth systems, PFIM attains solutions of comparable accuracy with computational costs up to two orders of magnitude lower than HBM. These results indicate that PFIM provides a robust and efficient alternative for periodic response analysis in complex nonlinear dynamical systems, with strong potential for practical engineering applications.

Figures (16)

• Figure 1: Schematic diagram of harmonic balance method.
• Figure 2: Schematic diagram of shooting balance method.
• Figure 3: Convergence of HBM and PFIM solutions for Eq. \ref{['s_1']}. (a) Displacement error: $x^{(0)}$ (HBM); $x^{(1)}$, $x^{(2)}$ (PFIM iterations). Dashed lines $C(x^{(0)})^{2}$, $C(x^{(0)})^{4}$ are scaled error powers. (b) Frequency error with analogous notation.
• Figure 4: Error convergence for Eq. \ref{['s_1']} using linear initial guess. (a) Displacement error (b) Frequency error.
• Figure 5: Response of Eq. \ref{['s_2']}. (a) Amplitude-frequency curve (stable: solid, unstable: dashed) (b) Floquet multiplier evolution (filled: stable, hollow: unstable).