I am using the Mathieu equation
$$x''(t) + a x(t) + 2 q \cos(2 t) x(t) = 0,$$
as a model of a physical system. The conditions on the parameters $a$ and $q$ for stable solutions to exist are well known and are even implemented as built-in functions in systems such as Mathematica.
Now, I would like to extend my model to the form $$x''(t) + c x^2(t)+ a x(t) + 2 q \cos(2 t) x(t) = 0,$$ and establish the stability criteria for this modified model.
Is this a known model, or can it be transformed to one?
Does it even make sense to talk about stability conditions for this case -- as far as I can see Floquet's theorem does not apply?
Although the most general form would be most interesting, it would also be interesting to know the stability criteria for the less general case of $a=0$.