The Koopman-Hill projection method is used to approximate the fundamental solution matrix of linear time-periodic ordinary differential equations, possibly stemming from linearization around a periodic solution of a nonlinear dynamical system. By expressing both the true fundamental solution and its approximation as series, we derive an upper bound for the approximation error that decays exponentially with the size of the Hill matrix. Exponential decay of the Fourier coefficients of the system dynamics is key to guarantee convergence. The paper also analyzes a subharmonic formulation that improves the convergence rate. Two numerical examples, including a Duffing oscillator, illustrate the theoretical findings.

翻译：Koopman-Hill投影方法用于近似线性时周期常微分方程的基本解矩阵，此类方程可能源于非线性动力系统周期解附近的线性化。通过将真实基本解及其近似均表示为级数形式，我们推导出近似误差的上界，该误差随Hill矩阵维度的增大呈指数衰减。系统动力学傅里叶系数的指数衰减是保证收敛性的关键。本文还分析了一种能提升收敛速度的次谐波表述形式。两个数值算例（包括Duffing振子）验证了理论结果。