Floquet Multipliers.

AUTO extracts an approximation to the linearized Poincaré map from the Jacobian of the linearized collocation system that arises in Newton's method. This procedure is very efficient; the map is computed at negligible extra cost. The linear equations solver of AUTO is described in DoKeKe:91b DoKeKe:91b. The actual Floquet multiplier solver was written by Fa:94 Fa:94. For a detailed description of the algorithm see FaJe:91 FaJe:91.

For periodic solutions, the exact linearized Poincaré map always has a multiplier $z=1$. A good accuracy check is to inspect this multiplier in the diagnostics output-file fort.9. If this multiplier becomes inaccurate then the automatic detection of potential secondary periodic bifurcations (if ISP=2) is discontinued and a warning is printed in fort.9. It is strongly recommended that the contents of this file be habitually inspected, in particular to verify whether solutions labeled as BP or TR (cf. Table 6.1) have indeed been correctly classified.