Instability Maps of a System of Mathieu-Equations Using Different Perturbation methods

It is well known, that the Mathieu-equation possesses several instability regions. Dealing with systems of Mathieu- equations results in an increase of the number and size of the instability regions. The so called combination frequencies, an addition of the natural frequencies of the time invariant system, appear. Applying the Floquet theory and numerical integration yields a grid of stable and unstable areas. Even if this procedure delivers good results it is not suitable for a fast calculation, because it is very CPU- time consuming. Therefore several Perturbation methods are compared in this paper. The first one is the Lindstedt-Poincaré method, delivering some of the instability areas. The second one is the Multiple- Scales- method used with an approximation up to second order. Assuming the periodicity of the equations solution, with parameters on the boundary curve, facilitates the mathematical description of the borderline. An example is carried out; the analytical stability boundaries are validated by the Floquet theory.