Stability region control for a parametrically forced Mathieu equation

We exhibit common features of how the size of parametric regions of stability for the Mathieu equation can be enlarged. The paper shows that the mechanisms for these changes via parametric forcing follow the pattern established earlier for the Arnold circle map which provides a discrete model for external forcing. The various types of behaviour of the standard Mathieu equation for a given set of parameters can be classified as having either (i) all solutions bounded, (ii) at least one unbounded solution, or (iii) periodic solutions of period -2 pi or -4 pi . The marginal case (iii) forms the boundary of the regions of stability and instability. We consider a parametric method for changing the shapes of the stability regions and show how maximally stable regions can be produced.