Summary
This study delves into the stability analysis of periodic solutions of dynamical systems, a topic of paramount importance across various engineering disciplines. At the core of such investigations lies the differential equation, serving as the foundational model from which periodic vibrations and their stability characteristics are derived. Although numerous methods and algorithms exist for finding the orbits and analysing their stability, they often come with limitations, including the need for equations to be problem-specific, constrained capabilities, limited convergence rates, and substantial computational demands. Our research aims to circumvent these challenges by leveraging well-established solvers/integrators, specifically utilizing the DifferentialEquations.jl package within the Julia programming environment. We propose an approach by developing an algorithm with linear time complexity that not only identifies the fixed point and its spectral properties but also achieves a convergence rate comparable to that of the integrator (e.g., fourth-order Runge-Kutta). This is accomplished through the affine mapping of the initial state (history function) for one period via direct integration. The rounding errors from the small perturbation of the fixed point are effectively eliminated by applying dual numbers. Our comprehensive testing across various scenarios, including nonlinear systems, demonstrates the algorithm's robust performance. This method is applicable to almost any time-periodic system that can be numerically simulated.
