On some methods for investigating stability (Q2759316)

The article presents several algorithms for investigating the stability of certain classes of nonautonomous systems of linear and quasi-linear ordinary differential equations. It is assumed that the system depends on a small perturbation parameter \(\varepsilon\) and that, for \(\varepsilon=0\), the Jacobian is a constant matrix with simple or resonant eigenvalues. Making intensive use of normal forms, the nonautonomous systems are transformed to systems with almost constant diagonal matrices. These procedures not only permit the calculation of the approximate eigenvalues, but also admit the determination of the stability for time-periodic systems. The methods are applied to a number of interesting test cases, like Mathieu's equation and the oscillations of a gyroscope with variable angular momentum.