I am interested in the Mathieu equation, $\frac{d^2x}{dt^2}+(\delta+\epsilon\cos t)x=0$. I often see diagrams illustrating the stability of solutions, like this one (S=stable, U=unstable):
Does anyone know of a good source which discusses how to generate the stability curves? I've looked at Advanced Methods of Scientists and Engineers by Bender and Orszag, but I feel like there's something that I'm missing, though I'm not sure what it is. I understand where the $\delta$-intercepts of $0.25, 1, 2.25$ come from, but I do not see how to generate the shaded part of the diagram. I am hoping that another reference or two will fill in the missing gaps.
Best Answer
A highly thorough and accessible reference on the Mathieu equation and the parameters that characterize its stability/instability curves can be found in Chapter 6 of Rand's Lecture Notes on Nonlinear Vibrations.
Hope this helps!