A simple approach for the computation of Lyapunov–Floquet transformations for general time-periodic systems

Ordinary differential equations with time-periodic coefficients (so-called time-periodic systems) can be analyzed using Lyapunov–Floquet (L–F) transformations. These transformations reduce the linear part of a time-periodic equation to the time-invariant form and facilitate the application of well-established techniques tailored for time-invariant systems. In the previous work, the construction of L–F transformations relied on Chebyshev polynomials and their properties, which may often prove challenging to grasp and apply effectively. This paper endeavors to present a more intuitive and straightforward approach for computing L–F transformations. The solution of a linear time-periodic system can be expressed as a product of an exponential function and a vector-valued polynomial in time with time-periodic coefficients. Substitution of the solution reduces a time-periodic equation to an eigenvalue problem, which can be solved to obtain the general solution. Rearranging the solution yields the state transition matrix, which can be used in the Lyapunov–Floquet theorem to compute the L–F transformation. The inverse of these transformations is important for the nonlinear analysis and control and can be determined by defining the adjoint system to the time-periodic system. As examples, L–F transformations and their inverses are generated for the Mathieu equation and a double inverted pendulum subjected to a time-periodic force. In the end, the usefulness of L–F transformations is showcased by performing the bifurcation study of a nonlinear Mathieu equation using the center manifold theorem.