Floquet Theory as a Computational Tool

We describe how classical Floquet theory may be utilized, in a continuation framework, to construct an efficient Fourier spectral algorithm for approximating periodic orbits. At each continuation step, only a single square matrix, whose size equals the dimension of the phase-space, needs to be factorized; the rest of the required numerical linear algebra just consists of back-substitutions with this matrix. The eigenvalues of this key matrix are the Floquet exponents, whose crossing of the imaginary axis indicates bifurcation and change-in-stability. Hence we also describe how the new periodic orbits created at a period-doubling bifurcation point may be efficiently computed using our approach.