Explicit error bounds and guaranteed convergence of the Koopman-Hill projection stability method for linear time-periodic dynamics

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.