In the determination of the asymptotic stability and bifurcation behaviour of periodic orbits it is necessary to accurately compute the Floquet multipliers having modulus nearly 1 and to determine the number of multipliers that lie well inside and well outside the unit circle. Current numerical methods for computing Floquet multipliers suffer a loss of accuracy in two cases of importance, namely when the multiplier at 1 is (nearly) defective and when there are large multipliers. An alternative approach is proposed which can be derived from a reexamination of the underlying algebraic eigenvalue problem and which is suitable for systems of ordinary differential equations. Numerical results indicate that avoiding the explicit construction of a matrix commonly called the circuit matrix and using known properties of a fixed Floquet multiplier at 1 will circumvent the difficulties mentioned above. Convergence results are given.
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