IMPROVED NUMERICAL FLOQUET MULTIPLIERS

Abstract

This paper studies numerical methods for linear stability analysis of periodic solutions in codes for bifurcation analysis of small systems of ordinary differential equations (ODEs). Popular techniques in use today (including the AUTO97 method) produce very inaccurate Floquet multipliers if the system has very large or small multipliers. These codes compute the monodromy matrix explicitly or as a matrix pencil of two matrices. The monodromy matrix arises naturally as a product of many matrices in many numerical methods, but this is not exploited. In this case, all Floquet multipliers can be computed with very high precision by using the periodic Schur decomposition and corresponding algorithm [Bojanczyk et al., 1992]. The time discretisation of the periodic orbit becomes the limiting factor for the accuracy. We present just enough of the numerical methods to show how the Floquet multipliers are currently computed and how the periodic Schur decomposition can be fitted into existing codes but omit all details. However, we show extensive test results for a few artificial matrices and for two four-dimensional systems with some very large and very small Floquet multipliers to illustrate the problems experienced by current techniques and the better results obtained using the periodic Schur decomposition. We use a modified version of AUTO97 [Doedel et al., 1997] in our experiments.