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 phasespace, needs to be factorized; the rest of the required numerical linear algebra just consists of backsubstitutions 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.